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<title data-rh="true">The MDSECheck method: choosing secure square MDS matrices for P-SP-networks | Vac Research</title><meta data-rh="true" name="viewport" content="width=device-width,initial-scale=1"><meta data-rh="true" name="twitter:card" content="summary_large_image"><meta data-rh="true" property="og:url" content="https://vac.dev/rlog/mdsecheck-method"><meta data-rh="true" property="og:locale" content="en"><meta data-rh="true" name="docusaurus_locale" content="en"><meta data-rh="true" name="docusaurus_tag" content="default"><meta data-rh="true" name="docsearch:language" content="en"><meta data-rh="true" name="docsearch:docusaurus_tag" content="default"><meta data-rh="true" name="image" content="https://vac.dev/_og/3ef9132dff3526ac9748fdf93b437ed276abf3e7.png"><meta data-rh="true" property="og:title" content="The MDSECheck method: choosing secure square MDS matrices for P-SP-networks | Vac Research"><meta data-rh="true" name="description" content="This article introduces MDSECheck method — a novel approach"><meta data-rh="true" property="og:description" content="This article introduces MDSECheck method — a novel approach"><meta data-rh="true" property="og:type" content="article"><meta data-rh="true" property="article:published_time" content="2025-02-28T23:00:00.000Z"><link data-rh="true" rel="icon" href="/theme/image/favicon.ico"><link data-rh="true" rel="canonical" href="https://vac.dev/rlog/mdsecheck-method"><link data-rh="true" rel="alternate" href="https://vac.dev/rlog/mdsecheck-method" hreflang="en"><link data-rh="true" rel="alternate" href="https://vac.dev/rlog/mdsecheck-method" hreflang="x-default"><script data-rh="true" type="application/ld+json">{"@context":"https://schema.org","@type":"BlogPosting","@id":"https://vac.dev/rlog/mdsecheck-method","mainEntityOfPage":"https://vac.dev/rlog/mdsecheck-method","url":"https://vac.dev/rlog/mdsecheck-method","headline":"The MDSECheck method: choosing secure square MDS matrices for P-SP-networks","name":"The MDSECheck method: choosing secure square MDS matrices for P-SP-networks","description":"This article introduces MDSECheck method — a novel approach","datePublished":"2025-02-28T23:00:00.000Z","author":{"@type":"Person","name":"Aleksei Vambol"},"keywords":[],"isPartOf":{"@type":"Blog","@id":"https://vac.dev/rlog","name":"Research Blog"}}</script><link rel="alternate icon" type="image/png" href="/theme/image/favicon.png">
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class="col col--9 col--offset-1" itemscope="" itemtype="http://schema.org/Blog"><article class=""><header><nav class="theme-doc-breadcrumbs breadcrumbsContainer_RLvU" aria-label="Breadcrumbs"><ul class="breadcrumbs" itemscope="" itemtype="https://schema.org/BreadcrumbList"><a class="breadcrumbs__item" href="/blog"><span class="lsd-typography lsd-typography--body3">All posts</span></a><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="lsd-typography lsd-typography--body3 breadcrumbs__link" itemprop="name">The MDSECheck method: choosing secure square MDS matrices for P-SP-networks</span><meta itemprop="position" content="1"></li></ul></nav><h1 class="title_f1Hy">The MDSECheck method: choosing secure square MDS matrices for P-SP-networks</h1><div class="container_PkUo margin-vert--md"><time datetime="2025-02-28T23:00:00.000Z" itemprop="datePublished">February 28, 2025</time><div class="authors_dZ4g"><span class="lsd-typography lsd-typography--body2">by </span><div class="col col--6 authorCol_y4tx"><div class="avatar margin-bottom--sm"><div class="avatar__intro" itemprop="author" itemscope="" itemtype="https://schema.org/Person"><div class="avatar__name"><span class="lsd-typography lsd-typography--body2" itemprop="name">Aleksei Vambol</span></div></div></div></div></div><div>18 min read</div></div><hr class="blog-divider"></header><div id="__blog-post-container" class="markdown"><p>This article introduces MDSECheck method — a novel approach
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to checking square MDS matrices for unconditional security
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as the components of affine permutation layers of P-SP-networks.</p>
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<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="introduction">Introduction<a href="#introduction" class="hash-link" aria-label="Direct link to Introduction" title="Direct link to Introduction"></a></h2>
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<p>Maximum distance separable (MDS) matrices play a significant role
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in algebraic coding theory and symmetric cryptography.
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In particular, square MDS matrices are commonly used in
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affine permutation layers of
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partial substitution-permutation networks (P-SPNs).
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These are widespread designs of
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the modern symmetric ciphers and hash functions.
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A classic example of the latter is Poseidon <a href="#references">[1]</a>,
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a well-known hash function used in zk-SNARK proving systems.</p>
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<p>Square MDS matrices differ in terms of security
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that they are able to provide for P-SPNs.
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The use of some such matrices in certain P-SPNs may result in existence
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of infinitely long subspace trails of small period for the latter,
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which make them vulnerable to differential cryptanalysis <a href="#references">[2]</a>.</p>
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<p>Two methods for security checking of square MDS matrices for P-SPNs
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have been proposed in <a href="#references">[2]</a>.
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The first one, which is referred to as the three tests method
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in the rest of the article, is aimed at security checking for
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a specified structure of the substitution layer of a P-SPN.
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The second method, which is referred here as the sufficient test method,
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has been designed to determine whether a square MDS matrix satisfies
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a sufficient condition of being secure regardless of the structure of
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a P-SPN substitution layer, i.e. to check whether the matrix belongs to
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the class of square MDS matrices, which are referred to
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as unconditionally secure in the current article.</p>
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<p>This article aims to introduce MDSECheck method —
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a novel approach to checking square MDS matrices for unconditional security,
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which has already been implemented in the Rust programming language as
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the library crate <a href="#references">[3]</a>.
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The next sections explain the notions mentioned above,
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describe the MDSECheck method as well as its mathematical foundations,
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provide a brief overview of the MDSECheck library crate
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and outline possible future research directions.</p>
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<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="mds-matrix-how-to-define-and-construct">MDS matrix: how to define and construct<a href="#mds-matrix-how-to-define-and-construct" class="hash-link" aria-label="Direct link to MDS matrix: how to define and construct" title="Direct link to MDS matrix: how to define and construct"></a></h2>
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<p>An <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">m</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> over a finite field is called MDS,
|
||
if and only if for distinct <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-dimensional column vectors <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">v_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>
|
||
the column vectors <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>1</mn></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>M</mi><msub><mi>v</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_1 \: | \: M v_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mn>2</mn></msub><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>M</mi><msub><mi>v</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">v_2 \: | \: M v_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>,
|
||
where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">∣</span></span></span></span> stands for vertical concatenation,
|
||
do not coincide in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> or more components.
|
||
The set of all possible column vectors <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mtext> </mtext><mi mathvariant="normal">∣</mi><mtext> </mtext><mi>M</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">v \: | \: M v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span> for
|
||
some fixed matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is a systematic MDS code, i.e.
|
||
a linear code, which contains input symbols on their original positions
|
||
and achieves the Singleton bound.
|
||
The latter property results in good error-correction capability.</p>
|
||
<p>There are several equivalent definitions of MDS matrices,
|
||
but the next one is especially useful for constructing them
|
||
directly by means of algebraic methods.
|
||
A matrix over a finite field is called MDS,
|
||
if and only if all its square submatrices are nonsingular.
|
||
The matrix entries and the matrix itself are also considered submatrices.</p>
|
||
<p>One of the most efficient and straightforward methods to directly construct
|
||
an MDS matrix is generating a Cauchy matrix <a href="#references">[4]</a>.
|
||
Such an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">m</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix is defined using
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">m</span></span></span></span>-dimensional vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-dimensional vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span></span></span>,
|
||
for which all entries in the concatenation of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">x</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span></span></span> are distinct.
|
||
The entries of the Cauchy matrix are described by the formula
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_{i, j} = 1 \: / \: (x_i - y_j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.
|
||
It is obvious that any submatrix of a Cauchy matrix is also a Cauchy matrix.
|
||
The Cauchy determinant formula <a href="#references">[5]</a> implies that
|
||
every square Cauchy matrix is nonsingular.
|
||
Thus, Cauchy matrices satisfy the second definition of MDS matrices.</p>
|
||
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="partial-substitution-permutation-networks">Partial substitution-permutation networks<a href="#partial-substitution-permutation-networks" class="hash-link" aria-label="Direct link to Partial substitution-permutation networks" title="Direct link to Partial substitution-permutation networks"></a></h2>
|
||
<p>Describing SPNs in algebraic terms is convenient,
|
||
so this approach has been chosen for this article.
|
||
SPNs are designs of the symmetric cryptoprimitives,
|
||
which operate on an internal state, which is represented
|
||
as an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-dimensional vector over some finite field,
|
||
and update this state iteratively by means of
|
||
the round transformations described below.</p>
|
||
<p>Each round begins with an optional update of the internal state by
|
||
adding to its components some input data or extraction of
|
||
some of these components as the output data.
|
||
This optional step depends on the specific cryptoprimitive
|
||
and the current round number.
|
||
The next step is called the nonlinear substitution layer
|
||
and lies in replacing the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em"></span><span class="mord mathnormal">i</span></span></span></span>-th component of the internal state
|
||
with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S_i(c)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mclose">)</span></span></span></span> for each <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>1..</mn><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [1..n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord">1..</span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">c</span></span></span></span> is the component value
|
||
and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> is a nonlinear invertible function over the finite field.
|
||
The function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> is specific to the cryptoprimitive and called an S-Box.
|
||
The final step, which is known as the affine permutation layer,
|
||
replaces the internal state with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>X</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">M X + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em"></span><span class="mord mathnormal" style="margin-right:0.07847em">MX</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">c</span></span></span></span>,
|
||
where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.07847em">X</span></span></span></span> is the current internal state,
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is a nonsingular square matrix and
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">c</span></span></span></span> is the vector of the round constants.
|
||
The value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">c</span></span></span></span> is specific to the cryptoprimitive
|
||
and the current round number,
|
||
while <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> typically depends only on the cryptoprimitive.
|
||
The data flow diagram for an SPN is given below.</p>
|
||
<div class="codeBlockContainer_EB2s codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#F8F8F2;--prism-background-color:rgba(var(--lsd-surface-secondary), 0.08)"><div class="codeBlockContent_ugSV"><pre tabindex="0" class="prism-code language-text codeBlock_TWhw thin-scrollbar"><code class="codeBlockLines_LDrR"><span class="token-line" style="color:#F8F8F2"><span class="token plain">.................................. </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> │ │ │ │ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">┌────────────────────────────────┐ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">│ Optional addition / extraction │ <─────> Input / output</span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">└──┬────────┬────────┬────────┬──┘ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">│S₁(x)│ │S₂(x)│ │ ... │ │Sₙ(x)│ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">└──┬──┘ └──┬──┘ └──┬──┘ └──┬──┘ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">┌────────────────────────────────┐ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">│ Affine permutation │ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">└──┬────────┬────────┬────────┬──┘ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">┌────────────────────────────────┐ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">│ Optional addition / extraction │ <─────> Input / output</span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">└──┬────────┬────────┬────────┬──┘ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">│S₁(x)│ │S₂(x)│ │ ... │ │Sₙ(x)│ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">└──┬──┘ └──┬──┘ └──┬──┘ └──┬──┘ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">┌────────────────────────────────┐ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">│ Affine permutation │ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">└──┬────────┬────────┬────────┬──┘ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain"> ▼ ▼ ▼ ▼ </span><br></span><span class="token-line" style="color:#F8F8F2"><span class="token plain">.................................. </span><br></span></code></pre><div class="buttonGroup_Qu4e"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_an20" aria-hidden="true"><div class="icon_S7Kx m_thRi copyButtonIcon_ZL7v"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" fill="none" viewBox="0 0 14 14"><path fill="#fff" d="M2.917 12.833q-.482 0-.825-.343a1.12 1.12 0 0 1-.342-.823V3.5h1.167v8.167h6.416v1.166zM5.25 10.5q-.481 0-.824-.343a1.12 1.12 0 0 1-.343-.824v-7q0-.48.343-.824t.824-.342h5.25q.481 0 .824.343t.343.823v7q0 .482-.343.825a1.12 1.12 0 0 1-.824.342zm0-1.167h5.25v-7H5.25z"></path></svg></div></span></button></div></div></div>
|
||
<p>Partial SPNs are modifications of SPNs,
|
||
where for certain rounds some S-Boxes are replaced with
|
||
the identity functions to reduce computational efforts <a href="#references">[2]</a>.
|
||
For example, the nonlinear substitution layers of the partial rounds of
|
||
Poseidon update only the first internal state component <a href="#references">[1]</a>.
|
||
In the case of P-SPNs, security considerations commonly demand to choose <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>
|
||
as a square MDS matrix, because these matrices provide
|
||
perfect diffusion property for the affine permutation layer <a href="#references">[6]</a>.
|
||
Possessing this property means ensuring that
|
||
any two <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-dimensional internal states,
|
||
which differ in exactly <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em"></span><span class="mord mathnormal">t</span></span></span></span> components,
|
||
are mapped by the affine permutation layer to
|
||
two new internal states that differ in at least <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n - t + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6984em;vertical-align:-0.0833em"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span> components.</p>
|
||
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="square-mds-matrix-security-check-in-the-context-of-p-spns">Square MDS matrix security check in the context of P-SPNs<a href="#square-mds-matrix-security-check-in-the-context-of-p-spns" class="hash-link" aria-label="Direct link to Square MDS matrix security check in the context of P-SPNs" title="Direct link to Square MDS matrix security check in the context of P-SPNs"></a></h2>
|
||
<p>Certain square MDS matrices should not be used in certain P-SPNs
|
||
to avoid making them vulnerable to differential cryptanalysis,
|
||
since it may exploit the existence of infinitely long subspace trails
|
||
of small period for vulnerable P-SPNs. <a href="#references">[2]</a>.
|
||
Such matrices are called insecure with respect to particular P-SPNs.</p>
|
||
<p>An infinitely long subspace trail of period <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span> exists for a P-SPN,
|
||
if and only if there is a proper subspace
|
||
of differences of internal state vectors,
|
||
such that if for a pair of initial internal states
|
||
the difference belongs to this subspace,
|
||
then the difference for the new internal states,
|
||
which are obtained from the initial ones
|
||
by means of the same <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span>-round transformation,
|
||
also belongs to this subspace <a href="#references">[2]</a>.</p>
|
||
<p>Two methods for checking square MDS matrices for suitability for P-SPNs
|
||
in terms of existence of infinitely long subspace trails
|
||
have been proposed in <a href="#references">[2]</a>.
|
||
The three tests method is aimed at checking
|
||
whether using a specified matrix for a P-SPN
|
||
with a specified structure of the substitution layer
|
||
leads to existence of infinitely long subspace trails of period <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">p</span></span></span></span>
|
||
for this P-SPN for all <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">p</span></span></span></span> no larger than a given <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span>.
|
||
The sufficient test method has been designed to determine
|
||
whether a square MDS matrix satisfies a sufficient condition
|
||
of non-existence of infinitely long subspace trails of period <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">p</span></span></span></span>
|
||
for P-SPNs using this matrix for all <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">p</span></span></span></span> no larger than a specified <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span>.</p>
|
||
<p>The sufficient test method is a direct consequence of
|
||
Theorem 8 in <a href="#references">[2]</a> and consists in checking that
|
||
the minimal polynomial of the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal">p</span></span></span></span>-th power of the tested matrix
|
||
has maximum degree and is irreducible for all <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>1..</mn><mi>l</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">p \in [1..l]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord">1..</span><span class="mord mathnormal" style="margin-right:0.01968em">l</span><span class="mclose">]</span></span></span></span>.
|
||
The aforesaid sufficient non-existence condition is satisfied by the matrix,
|
||
if and only if all the checks yield positive results.</p>
|
||
<p>It is convenient to define
|
||
the unconditional P-SPN security level of the square MDS matrix as follows:
|
||
this level is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span> for the matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>,
|
||
if and only if the minimal polynomials of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">M^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">M^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span>
|
||
have maximum degree and are irreducible,
|
||
but for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mrow><mi>l</mi><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">M^{l \: + \: 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">+</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> the minimal polynomial does not have this property.
|
||
Using this definition, the purpose of the sufficient test method
|
||
can be described as checking whether
|
||
the unconditional P-SPN security level of the specified matrix
|
||
is no less than a given bound.</p>
|
||
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="mdsecheck-method-getting-rid-of-the-matrix-powers">MDSECheck method: getting rid of the matrix powers<a href="#mdsecheck-method-getting-rid-of-the-matrix-powers" class="hash-link" aria-label="Direct link to MDSECheck method: getting rid of the matrix powers" title="Direct link to MDSECheck method: getting rid of the matrix powers"></a></h2>
|
||
<p>The MDSECheck method, whose name is derived from
|
||
the words "MDS", "security", "elaborated" and "check",
|
||
has the same purpose as the sufficient test method,
|
||
but achieves it differently.
|
||
The differences of the first method from the latter
|
||
and approaches to implementing them can be described as follows:</p>
|
||
<ol>
|
||
<li>
|
||
<p>Computation and verification of minimal polynomials
|
||
of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">M^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">M^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">M^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is
|
||
the tested <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mclose">)</span></span></span></span>
|
||
and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span> is the security level bound,
|
||
has been replaced with checks for the corresponding powers
|
||
of a root of the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>
|
||
for non-presence in nontrivial subfields of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
|
||
<ol>
|
||
<li>
|
||
<p>The non-presence check is performed without
|
||
straightforward consideration of all nontrivial subfields of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.
|
||
The root is checked only for non-presence in the subfields
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^{n \: / \: p_1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^{n \: / \: p_2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>,
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>d</mi></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^{n \: / \: p_d})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">p_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>
|
||
are all prime divisors of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>.</p>
|
||
</li>
|
||
<li>
|
||
<p>The non-presence check reuses some data computed during
|
||
the checking for irreducibility the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>,
|
||
which in this case coincides with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>
|
||
designating the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>.
|
||
The values of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^{q^{n \: / \: p_j}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2624em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> are saved
|
||
for each <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>1..</mn><mi>d</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">j \in [1..d]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.05724em">j</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord">1..</span><span class="mord mathnormal">d</span><span class="mclose">]</span></span></span></span> during the irreducibility check
|
||
to replace exponentiations with sequential computations
|
||
of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mi>i</mi></msup><msup><mo stretchy="false">)</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y^i)^{q^{n \: / \: p_j}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.318em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> from
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msup><mo stretchy="false">)</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y^{(i \: - \: 1)})^{q^{n \: / \: p_j}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.318em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">i</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>
|
||
as its product with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^{q^{n \: / \: p_j}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2624em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>.</p>
|
||
</li>
|
||
</ol>
|
||
</li>
|
||
<li>
|
||
<p>The check of the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>
|
||
for irreducibility and maximum degree
|
||
is performed without unconditional computation of this polynomial.
|
||
This computation has been replaced with the Krylov method fragment,
|
||
which consists in building and solving
|
||
only one system of linear equations over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mclose">)</span></span></span></span>.
|
||
If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> has an irreducible minimal polynomial of maximum degree,
|
||
then its coefficients are trivially determined from the system solution.
|
||
If the system is degenerate,
|
||
then the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> does not have such properties.</p>
|
||
</li>
|
||
</ol>
|
||
<p>The correctness of the first distinctive feature can be proven as follows.
|
||
Verifying that the minimal polynomial of a matrix
|
||
is of maximum degree and irreducible
|
||
is equivalent to verifying that
|
||
the characteristic polynomial of this matrix is irreducible,
|
||
because the minimal polynomial divides the characteristic one.
|
||
Also, it is trivially proven that for a matrix with such a minimal polynomial
|
||
it is equal to the characteristic polynomial.
|
||
Thus, the required checks for the matrices <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">M^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">M^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">M^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span>
|
||
can be done by checking their characteristic polynomials for irreducibility.</p>
|
||
<p>Let <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> be <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mclose">)</span></span></span></span>,
|
||
whose minimal polynomial is of maximum degree and irreducible.
|
||
The statements in the previous paragraph imply that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>,
|
||
which is the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-degree characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>, is irreducible.
|
||
Consider <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> over the extension field <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,
|
||
which is the splitting field of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>.
|
||
Let <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>∈</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">α \in GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> be a root of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>.
|
||
According to standard results from the Galois field theory,
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.0037em">α</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>q</mi></msup></mrow><annotation encoding="application/x-tex">α^q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><msup><mi>q</mi><mn>2</mn></msup></msup></mrow><annotation encoding="application/x-tex">α^{q^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msup></msup></mrow><annotation encoding="application/x-tex">α^{q^{n \: - \: 1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>
|
||
are distinct roots of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> <a href="#references">[7]</a>.
|
||
Thus, these powers of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.0037em">α</span></span></span></span> are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> distinct eigenvalues of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>.
|
||
Hence, due to matrix similarity properties, there is some matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.05764em">S</span></span></span></span>
|
||
such that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>M</mi><msup><mi>S</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">S M S^{-1} = D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord mathnormal" style="margin-right:0.10903em">SM</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.02778em">D</span></span></span></span> is the diagonal matrix,
|
||
whose nonzero elements are
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.0037em">α</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>q</mi></msup></mrow><annotation encoding="application/x-tex">α^q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><msup><mi>q</mi><mn>2</mn></msup></msup></mrow><annotation encoding="application/x-tex">α^{q^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msup></msup></mrow><annotation encoding="application/x-tex">α^{q^{n \: - \: 1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>.
|
||
Therefore, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><msup><mi>M</mi><mi>i</mi></msup><msup><mi>S</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>D</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">S M^i S^{-1} = D^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord mathnormal" style="margin-right:0.05764em">S</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em">D</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>,
|
||
so the roots of the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">M^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>
|
||
are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>α</mi><mi>q</mi></msup><msup><mo stretchy="false">)</mo><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">(α^q)^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>α</mi><msup><mi>q</mi><mn>2</mn></msup></msup><msup><mo stretchy="false">)</mo><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">(α^{q^2})^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2369em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>α</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msup></msup><msup><mo stretchy="false">)</mo><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">(α^{q^{n \: - \: 1}})^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2369em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>.
|
||
If the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> has degree less than <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>,
|
||
then the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">M^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> is divisible
|
||
by this minimal polynomial,
|
||
while <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> lies in some nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.
|
||
One of the fields isomorphic to this subfield is a residue class ring
|
||
of polynomials modulo the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> <a href="#references">[7]</a>.
|
||
If the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> is of degree <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>,
|
||
then the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">M^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>
|
||
equals this minimal polynomial and therefore is irreducible,
|
||
while <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> does not lie in any nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.
|
||
In this case, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>α</mi><mi>i</mi></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(α^i)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>α</mi><mi>i</mi></msup><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">(α^i)^{n \: - \: 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>
|
||
are linearly independent as distinct roots of
|
||
an irreducible polynomial over a finite field <a href="#references">[7]</a>,
|
||
so any field containing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> has at least <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>q</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">q^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8588em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> elements
|
||
and therefore cannot be a trivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.
|
||
Thus, checking the characteristic polynomials of
|
||
the matrices <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">M^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">M^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">M^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span> for irreducibility
|
||
is equivalent to verifying that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">α^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">α^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">α^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span>
|
||
do not lie in any nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
|
||
<p>The last sentences of the two previous paragraphs imply the following:
|
||
verifying that the minimal polynomials of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">M^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">M^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">M^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span>
|
||
are of maximum degree and irreducible can be performed
|
||
by verifying that the corresponding powers of a root of
|
||
the characteristic polynomial of the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mclose">)</span></span></span></span>
|
||
do not belong to any nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>. <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">■</mi></mrow><annotation encoding="application/x-tex">\blacksquare</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.675em"></span><span class="mord amsrm">■</span></span></span></span></p>
|
||
<p>The approaches to implementing the first distinctive feature
|
||
can be explained and proven to be correct as follows.
|
||
Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>w</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^w)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em">w</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is a nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>u</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>
|
||
if and only if <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.02691em">w</span></span></span></span> divides <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">u</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo><</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">w < u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathnormal" style="margin-right:0.02691em">w</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">u</span></span></span></span> <a href="#references">[7]</a>,
|
||
the presence of some <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">ε</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">ε</span></span></span></span> in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>h</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^h)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">h</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,
|
||
which is a nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,
|
||
implies that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi><mo>∈</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><mi>ν</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ε \in GF(q^{n \: / \: ν})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em"></span><span class="mord mathnormal">ε</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mathnormal mtight" style="margin-right:0.06366em">ν</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> for some prime <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">ν</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.06366em">ν</span></span></span></span> dividing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>,
|
||
because <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">h</span></span></span></span> divides the quotient of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> and some of its prime factors.
|
||
Thus, checking that some value does not belong to subfields
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^{n \: / \: p_1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^{n \: / \: p_2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>,
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>d</mi></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^{n \: / \: p_d})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,
|
||
where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">p_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span> are all prime divisors of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>,
|
||
is equivalent to checking this value for
|
||
non-presence in nontrivial subfields of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.</p>
|
||
<p>Checking for irreducibility the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>
|
||
is performed by means of Algorithm 2.2.9 in <a href="#references">[8]</a>
|
||
and consists in sequential computation of
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mi>p</mi></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^p \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8588em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><msup><mi>p</mi><mn>2</mn></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^{p^2} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1814em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>,
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><msup><mi>p</mi><mrow><mo stretchy="false">⌊</mo><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><mn>2</mn><mo stretchy="false">⌋</mo></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^{p^{\lfloor n \: / \: 2 \rfloor}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2341em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0397em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9667em"><span style="top:-2.9667em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">2</span><span class="mclose mtight">⌋</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>
|
||
and checking that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>C</mi><mi>D</mi><mo stretchy="false">(</mo><msup><mi>y</mi><msup><mi>p</mi><mi>i</mi></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>−</mo><mtext> </mtext><mi>y</mi><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">GCD(y^{p^i} \mod f(y) \: - \: y, f(y)) = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2445em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.07153em">GC</span><span class="mord mathnormal" style="margin-right:0.02778em">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9945em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9021em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>
|
||
for each <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>1..</mn><mo stretchy="false">⌊</mo><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><mn>2</mn><mo stretchy="false">⌋</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [1..\lfloor n \: / \: 2 \rfloor]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord">1..</span><span class="mopen">⌊</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">/</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mord">2</span><span class="mclose">⌋]</span></span></span></span>,
|
||
where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> is the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>
|
||
and coincides with the minimal polynomial in this case.
|
||
The optimized root non-presence check is performed
|
||
by checking that for each <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>2..</mn><mi>l</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [2..l]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord">2..</span><span class="mord mathnormal" style="margin-right:0.01968em">l</span><span class="mclose">]</span></span></span></span> for each <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>1..</mn><mi>d</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">j \in [1..d]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.05724em">j</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord">1..</span><span class="mord mathnormal">d</span><span class="mclose">]</span></span></span></span>
|
||
the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>y</mi><mi>i</mi></msup><msup><mo stretchy="false">)</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mo>−</mo><msup><mi>y</mi><mi>i</mi></msup><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((y^i)^{q^{n \: / \: p_j}} - y^i) \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.318em;vertical-align:-0.25em"></span><span class="mopen">((</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> is nonzero.
|
||
This approach is based on the following standard results
|
||
from the Galois field theory <a href="#references">[7]</a>:</p>
|
||
<ul>
|
||
<li>
|
||
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> is isomorphic to the residue class ring of
|
||
univariate polynomials in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span></span></span> modulo <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>,
|
||
because at this point <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> is known to be irreducible,
|
||
and some root of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> is mapped by this isomorphism to
|
||
the residue class the polynomial <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span></span></span> in this ring.</p>
|
||
</li>
|
||
<li>
|
||
<p>All elements of a finite field <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>w</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^w)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em">w</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> and only they
|
||
are roots of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><msup><mi>q</mi><mi>w</mi></msup></msup><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">y^{q^w} - y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0744em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.88em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7385em"><span style="top:-2.931em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em">w</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span></span></span></span>.</p>
|
||
</li>
|
||
</ul>
|
||
<p>The expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>y</mi><mi>i</mi></msup><msup><mo stretchy="false">)</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mo>−</mo><msup><mi>y</mi><mi>i</mi></msup><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((y^i)^{q^{n \: / \: p_j}} - y^i) \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.318em;vertical-align:-0.25em"></span><span class="mopen">((</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>
|
||
can be rewritten as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>y</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><msup><mo stretchy="false">)</mo><mi>i</mi></msup><mo>−</mo><msup><mi>y</mi><mi>i</mi></msup><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((y^{q^{n \: / \: p_j}})^i - y^i) \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.318em;vertical-align:-0.25em"></span><span class="mopen">((</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>,
|
||
which can be computed without exponentiation as the product of
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msup><mo stretchy="false">)</mo><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(y^{(i \: - \: 1)})^{q^{n \: / \: p_j}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.318em;vertical-align:-0.25em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">i</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span> and
|
||
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><msup><mi>q</mi><mrow><mi>n</mi><mtext> </mtext><mi mathvariant="normal">/</mi><mtext> </mtext><msub><mi>p</mi><mi>j</mi></msub></mrow></msup></msup><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y^{q^{n \: / \: p_j}} \mod f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2624em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.068em"><span style="top:-3.068em;margin-right:0.05em"><span class="pstrut" style="height:2.705em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0071em"><span style="top:-3.0072em;margin-right:0.0714em"><span class="pstrut" style="height:2.5357em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.3271em"></span><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3448em;margin-left:0em;margin-right:0.1em"><span class="pstrut" style="height:2.6595em"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5092em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>,
|
||
which has been saved during the irreducibility check.</p>
|
||
<p>The second distinctive feature can be
|
||
explained and proven to be correct in following way.
|
||
The <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> does not have a minimal polynomial of maximum degree,
|
||
if some Krylov subspace of order <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> for it is not <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-dimensional.
|
||
Indeed, the minimal polynomial of the matrix is divisible
|
||
by the minimal polynomial of the restriction of
|
||
this linear operator to an arbitrary subspace,
|
||
and in the considered case the latter polynomial has degree less than <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>,
|
||
because the degree of the minimal polynomial of a linear operator cannot
|
||
exceed the dimension of the subspace the operator acts on.
|
||
Thus, an unconditional computation of the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>
|
||
is not required to determine
|
||
whether this polynomial is irreducible and has maximum degree.
|
||
Using this computation has been replaced with the Krylov method fragment,
|
||
which consists in choosing any nonzero <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span>-dimensional column vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span>
|
||
and solving the system of linear equations <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>X</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">A X = b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.07847em">X</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">b</span></span></span></span>,
|
||
where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> is an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> x <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">n</span></span></span></span> matrix,
|
||
whose columns are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">M v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mn>2</mn></msup><mi>v</mi></mrow><annotation encoding="application/x-tex">M^2 v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msup><mi>v</mi></mrow><annotation encoding="application/x-tex">M^{n \: - \: 1} v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span>,
|
||
and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal">b</span></span></span></span> is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>M</mi><mi>n</mi></msup><mi>v</mi></mrow><annotation encoding="application/x-tex">M^n v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em">M</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span></span></span>.
|
||
If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal">A</span></span></span></span> is singular,
|
||
the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is reducible or does not have maximum degree,
|
||
so checking <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> has been accomplished;
|
||
otherwise, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mclose">)</span></span></span></span>, which is the minimal and characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>,
|
||
can be expressed as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mi>n</mi></msup><mo>−</mo><msub><mi>X</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msub><msup><mi>y</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow></msup><mo>−</mo><msub><mi>X</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>2</mn></mrow></msub><msup><mi>y</mi><mrow><mi>n</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>2</mn></mrow></msup><mo>−</mo><mo>…</mo><mo>−</mo><msub><mi>X</mi><mn>1</mn></msub><mi>y</mi><mo>−</mo><msub><mi>X</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">y^n - X_{n \: - \: 1} y^{n \: - \: 1} -
|
||
X_{n \: - \: 2} y^{n \: - \: 2} - … - X_1 y - X_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8588em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0224em;vertical-align:-0.2083em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1.0224em;vertical-align:-0.2083em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mbin mtight">−</span><span class="mspace mtight" style="margin-right:0.2602em"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.03588em">y</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span>.</p>
|
||
<p>The steps of MDSECheck method can be summarized as follows:</p>
|
||
<ol>
|
||
<li>
|
||
<p>The square MDS matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mclose">)</span></span></span></span>
|
||
and the unconditional P-SPN security level bound <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span> are received as inputs.</p>
|
||
</li>
|
||
<li>
|
||
<p>The Krylov method fragment is used to compute the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>.
|
||
If the computation fails, then <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is not unconditionally secure,
|
||
so the check of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is complete.
|
||
If it succeeds, then the minimal polynomial has maximum degree
|
||
and, therefore, coincides with the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>.</p>
|
||
</li>
|
||
<li>
|
||
<p>Algorithm 2.2.9 is used
|
||
to check for irreducibility the minimal polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>,
|
||
which is also the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> in this case.
|
||
Some data computed during this step is saved to be reused at the next one.
|
||
If the polynomial is reducible, then the check of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is complete,
|
||
because <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> has been found to be not unconditionally secure.</p>
|
||
</li>
|
||
<li>
|
||
<p>The values of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">α^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">α^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">...</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.1056em"></span><span class="mord">...</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>l</mi></msup></mrow><annotation encoding="application/x-tex">α^l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em">l</span></span></span></span></span></span></span></span></span></span></span>,
|
||
where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">α</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal" style="margin-right:0.0037em">α</span></span></span></span> is a root of the characteristic polynomial of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span>,
|
||
are sequentially checked for non-presence in
|
||
nontrivial subfields of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> as described above.
|
||
If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>α</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">α^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> belongs to some nontrivial subfield of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GF(q^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal" style="margin-right:0.13889em">GF</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,
|
||
then the unconditional P-SPN security level of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">i \: - \: 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.6444em"></span><span class="mord">1</span></span></span></span>,
|
||
so the check of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.10903em">M</span></span></span></span> is complete.
|
||
If all the values do not belong to such a subfield,
|
||
then the unconditional P-SPN security level is at least <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi></mrow><annotation encoding="application/x-tex">l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em"></span><span class="mord mathnormal" style="margin-right:0.01968em">l</span></span></span></span>.</p>
|
||
</li>
|
||
</ol>
|
||
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="mdsecheck-library-crate-implementation-in-rust">MDSECheck library crate: implementation in Rust<a href="#mdsecheck-library-crate-implementation-in-rust" class="hash-link" aria-label="Direct link to MDSECheck library crate: implementation in Rust" title="Direct link to MDSECheck library crate: implementation in Rust"></a></h2>
|
||
<p>The library crate <a href="#references">[3]</a> provides tools for
|
||
generating random square Cauchy MDS matrices over prime finite fields
|
||
and applying the MDSECheck method
|
||
to check such matrices for unconditional security.
|
||
The used data types of field elements and polynomials are provided by
|
||
the crates ark-ff <a href="#references">[9]</a> and ark-poly <a href="#references">[10]</a>.
|
||
The auxiliary tools in the crate modules are accessible as well.</p>
|
||
<p>Generating by means of this crate a 10 x 10 MDS matrix,
|
||
which is defined over the BN254 scalar field <a href="#references">[11]</a>
|
||
and has unconditional P-SPN security level is 1000,
|
||
takes less than 60 milliseconds on average
|
||
for the laptop with the processor Intel® Core™ i9-14900HX,
|
||
whose maximum clock frequency is 5.8 GHz.</p>
|
||
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="conclusion">Conclusion<a href="#conclusion" class="hash-link" aria-label="Direct link to Conclusion" title="Direct link to Conclusion"></a></h2>
|
||
<p>The MDSECheck method proposed in this article is a novel approach
|
||
to checking square MDS matrices for unconditional security
|
||
as the components of affine permutation layers of P-SPNs.
|
||
It has been implemented as a practical library crate
|
||
for generating unconditionally secure square MDS matrices
|
||
for P-SPNs over prime finite fields.</p>
|
||
<p>The future research directions may include
|
||
theoretical and experimental studies of performance of approaches,
|
||
which use the MDSECheck method
|
||
to generate unconditionally secure square MDS matrices for P-SPNs.</p>
|
||
<h2 class="anchor anchorWithHideOnScrollNavbar_WYt5" id="references">References<a href="#references" class="hash-link" aria-label="Direct link to References" title="Direct link to References"></a></h2>
|
||
<ol>
|
||
<li>L. Grassi, D. Khovratovich, C. Rechberger, A. Roy, M. Schofnegger. "<a href="https://eprint.iacr.org/2019/458.pdf" target="_blank" rel="noopener noreferrer">POSEIDON: A New Hash Function for Zero-Knowledge Proof Systems (Updated Version)</a>".</li>
|
||
<li>L. Grassi, C. Rechberger, M. Schofnegger. "<a href="https://eprint.iacr.org/2020/500.pdf" target="_blank" rel="noopener noreferrer">Proving Resistance Against Infinitely Long Subspace Trails: How to Choose the Linear Layer</a>".</li>
|
||
<li>The page "<a href="https://crates.io/crates/mdsecheck" target="_blank" rel="noopener noreferrer">mdsecheck</a>" on crates.io.</li>
|
||
<li>Y. Kumar, P. Mishra, S. Samanta, K. Chand Gupta, A. Gaur. "<a href="https://arxiv.org/pdf/2403.10372" target="_blank" rel="noopener noreferrer">Construction of all MDS and involutory MDS matrices</a>".</li>
|
||
<li>The page "<a href="https://proofwiki.org/wiki/Value_of_Cauchy_Determinant" target="_blank" rel="noopener noreferrer">Value of Cauchy Determinant</a>" on proofwiki.org.</li>
|
||
<li>T. Silva, R. Dahab "<a href="https://www.ic.unicamp.br/~reltech/PFG/2021/PFG-21-43.pdf" target="_blank" rel="noopener noreferrer">MDS Matrices for Cryptography</a>".</li>
|
||
<li>S. Huczynska, M. Neunhöffer. "<a href="http://www.math.rwth-aachen.de/~Max.Neunhoeffer/Teaching/ff2012/ff2012.pdf" target="_blank" rel="noopener noreferrer">Finite Fields</a>"</li>
|
||
<li>R. Crandall, C. Pomerance. "<a href="http://thales.doa.fmph.uniba.sk/macaj/skola/teoriapoli/primes.pdf" target="_blank" rel="noopener noreferrer">Prime Numbers: A Computational Perspective</a>" (2nd edition).</li>
|
||
<li>The page "<a href="https://crates.io/crates/ark-ff" target="_blank" rel="noopener noreferrer">ark-ff</a>" on crates.io.</li>
|
||
<li>The page "<a href="https://crates.io/crates/ark-poly" target="_blank" rel="noopener noreferrer">ark-poly</a>" on crates.io.</li>
|
||
<li>The page "<a href="https://crates.io/crates/ark-bn254" target="_blank" rel="noopener noreferrer">ark-bn254</a>" on crates.io.</li>
|
||
</ol></div></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Blog post page navigation"><a class="pagination-nav__link pagination-nav__link--prev" href="/rlog/nim-in-logos-01"><div class="icon_S7Kx m_thRi"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" fill="none" viewBox="0 0 14 14"><path fill="#fff" d="M11.667 6.417h-7.1L7.83 3.156 7 2.333 2.334 7 7 11.667l.823-.823-3.255-3.26h7.099z"></path></svg></div><span class="lsd-typography lsd-typography--body2 pagination-nav__label">Nim in Logos - 1st Edition</span></a><a class="pagination-nav__link pagination-nav__link--next" href="/rlog/2024-recap"><span class="lsd-typography lsd-typography--body2 pagination-nav__label">Vac 2024 Year in Review</span><div class="icon_S7Kx m_thRi"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="14" fill="none" viewBox="0 0 14 14"><path fill="#fff" d="m7 2.334-.823.822 3.255 3.26H2.333v1.167h7.1l-3.256 3.261.823.823L11.667 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class="table-of-contents__link toc-highlight">MDSECheck library crate: implementation in Rust</a></li><li><a href="#conclusion" class="table-of-contents__link toc-highlight">Conclusion</a></li><li><a href="#references" class="table-of-contents__link toc-highlight">References</a></li></ul></div></div></div></div></div><footer class="footer"><div class="container container-fluid firstRow_ar1q"><div class="footer__bottom text--center"><div class="margin-bottom--sm"><a class="footerLogoLink_BH7S" href="/"><img src="/theme/image/logo.svg" alt="Vac Research" class="themedImage_kfRS themedImage--light_BL8e footer__logo" width="22"><img src="/theme/image/logo.svg" alt="Vac Research" class="themedImage_kfRS themedImage--dark_OvIx footer__logo" width="22"></a></div><div class="footer__copyright">Vac Research © 2026<br>All rights reserved.</div></div><div class="row footer__links"><div class="col footer__col"><div class="footer__title"></div><ul class="footer__items clean-list"><li 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