research.logos.co/assets/js/c8454468.7672b46d.js

"use strict";(self.webpackChunkresearch_logos_co=self.webpackChunkresearch_logos_co||[]).push([[5989],{25871:(s,a,e)=>{e.r(a),e.d(a,{assets:()=>h,contentTitle:()=>t,default:()=>x,frontMatter:()=>m,metadata:()=>n,toc:()=>c});var n=e(99722),l=e(74848),i=e(28453);const m={title:"The MDSECheck method: choosing secure square MDS matrices for P-SP-networks",date:new Date("2025-02-28T23:00:00.000Z"),authors:"aleksei",published:!0,slug:"mdsecheck-method",categories:"research",toc_min_heading_level:2,toc_max_heading_level:4},t=void 0,h={authorsImageUrls:[void 0]},c=[{value:"Introduction",id:"introduction",level:2},{value:"MDS matrix: how to define and construct",id:"mds-matrix-how-to-define-and-construct",level:2},{value:"Partial substitution-permutation networks",id:"partial-substitution-permutation-networks",level:2},{value:"Square MDS matrix security check in the context of P-SPNs",id:"square-mds-matrix-security-check-in-the-context-of-p-spns",level:2},{value:"MDSECheck method: getting rid of the matrix powers",id:"mdsecheck-method-getting-rid-of-the-matrix-powers",level:2},{value:"MDSECheck library crate: implementation in Rust",id:"mdsecheck-library-crate-implementation-in-rust",level:2},{value:"Conclusion",id:"conclusion",level:2},{value:"References",id:"references",level:2}];function r(s){const a={a:"a",annotation:"annotation",code:"code",h2:"h2",li:"li",math:"math",mi:"mi",mn:"mn",mo:"mo",mrow:"mrow",mspace:"mspace",msub:"msub",msup:"msup",mtext:"mtext",ol:"ol",p:"p",pre:"pre",semantics:"semantics",span:"span",ul:"ul",...(0,i.R)(),...s.components};return(0,l.jsxs)(l.Fragment,{children:[(0,l.jsx)(a.p,{children:"This article introduces MDSECheck method \u2014 a novel approach\nto checking square MDS matrices for unconditional security\nas the components of affine permutation layers of P-SP-networks."}),"\n","\n",(0,l.jsx)(a.h2,{id:"introduction",children:"Introduction"}),"\n",(0,l.jsxs)(a.p,{children:["Maximum distance separable (MDS) matrices play a significant role\nin algebraic coding theory and symmetric cryptography.\nIn particular, square MDS matrices are commonly used in\naffine permutation layers of\npartial substitution-permutation networks (P-SPNs).\nThese are widespread designs of\nthe modern symmetric ciphers and hash functions.\nA classic example of the latter is Poseidon ",(0,l.jsx)(a.a,{href:"#references",children:"[1]"}),",\na well-known hash function used in zk-SNARK proving systems."]}),"\n",(0,l.jsxs)(a.p,{children:["Square MDS matrices differ in terms of security\nthat they are able to provide for P-SPNs.\nThe use of some such matrices in certain P-SPNs may result in existence\nof infinitely long subspace trails of small period for the latter,\nwhich make them vulnerable to differential cryptanalysis ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"}),"."]}),"\n",(0,l.jsxs)(a.p,{children:["Two methods for security checking of square MDS matrices for P-SPNs\nhave been proposed in ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"}),".\nThe first one, which is referred to as the three tests method\nin the rest of the article, is aimed at security checking for\na specified structure of the substitution layer of a P-SPN.\nThe second method, which is referred here as the sufficient test method,\nhas been designed to determine whether a square MDS matrix satisfies\na sufficient condition of being secure regardless of the structure of\na P-SPN substitution layer, i.e. to check whether the matrix belongs to\nthe class of square MDS matrices, which are referred to\nas unconditionally secure in the current article."]}),"\n",(0,l.jsxs)(a.p,{children:["This article aims to introduce MDSECheck method \u2014\na novel approach to checking square MDS matrices for unconditional security,\nwhich has already been implemented in the Rust programming language as\nthe library crate ",(0,l.jsx)(a.a,{href:"#references",children:"[3]"}),".\nThe next sections explain the notions mentioned above,\ndescribe the MDSECheck method as well as its mathematical foundations,\nprovide a brief overview of the MDSECheck library crate\nand outline possible future research directions."]}),"\n",(0,l.jsx)(a.h2,{id:"mds-matrix-how-to-define-and-construct",children:"MDS matrix: how to define and construct"}),"\n",(0,l.jsxs)(a.p,{children:["An ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"m"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"m"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"m"})]})})]})," x ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]})," matrix ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," over a finite field is called MDS,\nif and only if for distinct ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]}),"-dimensional column vectors ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mn,{children:"1"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"v_1"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"}),(0,l.jsx)(a.span,{className:"msupsub",children:(0,l.jsxs)(a.span,{className:"vlist-t 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mn,{children:"2"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"v_2"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"}),(0,l.jsx)(a.span,{className:"msupsub",children:(0,l.jsxs)(a.span,{className:"vlist-t 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mn,{children:"1"})]}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"\u2223"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mn,{children:"1"})]})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"v_1 \\: | \\: M v_1"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mn,{children:"2"})]}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"\u2223"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mn,{children:"2"})]})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"v_2 \\: | \\: M v_2"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{mathvariant:"normal",children:"\u2223"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"|"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord",children:"\u2223"})]})})]})," stands for vertical concatenation,\ndo not coincide in ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]})," or more components.\nThe set of all possible column vectors ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"v"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"\u2223"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsx)(a.mi,{children:"v"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"v \\: | \\: M v"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(a.span,{className:"mord",children:"\u2223"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"v"})]})})]})," for\nsome fixed matrix ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," is a systematic MDS code, i.e.\na linear code, which contains input symbols on their original positions\nand achieves the Singleton bound.\nThe latter property results in good error-correction capability."]}),"\n",(0,l.jsx)(a.p,{children:"There are several equivalent definitions of MDS matrices,\nbut the next one is especially useful for constructing them\ndirectly by means of algebraic methods.\nA matrix over a finite field is called MDS,\nif and only if all its square submatrices are nonsingular.\nThe matrix entries and the matrix itself are also considered submatrices."}),"\n",(0,l.jsxs)(a.p,{children:["One of the most efficient and straightforward methods to directly construct\nan MDS matrix is generating a Cauchy matrix ",(0,l.jsx)(a.a,{href:"#references",children:"[4]"}),".\nSuch an ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"m"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"m"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord 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using\n",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"m"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"m"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"m"})]})})]}),"-dimensional vector ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"x"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"x"})]})})]})," and ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]}),"-dimensional vector ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"y"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"y"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"})]})})]}),",\nfor which all entries in the concatenation of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"x"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"x"})]})})]})," and ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"y"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"y"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"})]})})]})," are distinct.\nThe entries of the Cauchy matrix are described by the formula\n",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"i"}),(0,l.jsx)(a.mo,{separator:"true",children:","}),(0,l.jsx)(a.mi,{children:"j"})]})]}),(0,l.jsx)(a.mo,{children:"="}),(0,l.jsx)(a.mn,{children:"1"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"/"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"x"}),(0,l.jsx)(a.mi,{children:"i"})]}),(0,l.jsx)(a.mo,{children:"\u2212"}),(0,l.jsxs)(a.msub,{children:[(0,l.jsx)(a.mi,{children:"y"}),(0,l.jsx)(a.mi,{children:"j"})]}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M_{i, 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satisfy the second definition of MDS matrices."]}),"\n",(0,l.jsx)(a.h2,{id:"partial-substitution-permutation-networks",children:"Partial substitution-permutation networks"}),"\n",(0,l.jsxs)(a.p,{children:["Describing SPNs in algebraic terms is convenient,\nso this approach has been chosen for this article.\nSPNs are designs of the symmetric cryptoprimitives,\nwhich operate on an internal state, which is represented\nas an ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]}),"-dimensional vector over some finite field,\nand update this state iteratively by means of\nthe round transformations described below."]}),"\n",(0,l.jsxs)(a.p,{children:["Each round begins with an optional update of the internal state by\nadding to its components some input data or extraction of\nsome of these components as the output data.\nThis optional step depends on the specific cryptoprimitive\nand the current round number.\nThe next step is called the nonlinear substitution layer\nand lies in replacing the 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"i"}),(0,l.jsx)(a.mo,{children:"\u2208"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"["}),(0,l.jsx)(a.mn,{children:"1.."}),(0,l.jsx)(a.mi,{children:"n"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"]"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"i \\in [1..n]"})]})})}),(0,l.jsxs)(a.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6986em",verticalAlign:"-0.0391em"}}),(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"c"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"c"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"c"})]})})]})," is the component value\nand 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mathnormal",style:{marginRight:"0.07847em"},children:"X"})]})})]})," is the current internal state,\n",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," is a nonsingular square matrix and\n",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"c"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"c"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"c"})]})})]})," is the vector of the round constants.\nThe value of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"c"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"c"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"c"})]})})]})," is specific to the cryptoprimitive\nand the current round number,\nwhile ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," typically depends only on the cryptoprimitive.\nThe data flow diagram for an SPN is given below."]}),"\n",(0,l.jsx)(a.pre,{children:(0,l.jsx)(a.code,{children:"..................................                         \n   \u2502        \u2502        \u2502        \u2502                           \n   \u25bc        \u25bc        \u25bc        \u25bc                           \n\u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510                         \n\u2502 Optional addition / extraction \u2502 <\u2500\u2500\u2500\u2500\u2500> Input / output\n\u2514\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2518                         \n   \u25bc        \u25bc        \u25bc        \u25bc                            \n\u250c\u2500\u2500\u2500\u2500\u2500\u2510  \u250c\u2500\u2500\u2500\u2500\u2500\u2510  \u250c\u2500\u2500\u2500\u2500\u2500\u2510  \u250c\u2500\u2500\u2500\u2500\u2500\u2510                         \n\u2502S\u2081(x)\u2502  \u2502S\u2082(x)\u2502  \u2502 ... \u2502  \u2502S\u2099(x)\u2502                         \n\u2514\u2500\u2500\u252c\u2500\u2500\u2518  \u2514\u2500\u2500\u252c\u2500\u2500\u2518  \u2514\u2500\u2500\u252c\u2500\u2500\u2518  \u2514\u2500\u2500\u252c\u2500\u2500\u2518                         \n   \u25bc        \u25bc        \u25bc        \u25bc                            \n\u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510                         \n\u2502       Affine permutation       \u2502                         \n\u2514\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2518                         \n   \u25bc        \u25bc        \u25bc        \u25bc                            \n\u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510                         \n\u2502 Optional addition / extraction \u2502 <\u2500\u2500\u2500\u2500\u2500> Input / output\n\u2514\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2518                         \n   \u25bc        \u25bc        \u25bc        \u25bc                            \n\u250c\u2500\u2500\u2500\u2500\u2500\u2510  \u250c\u2500\u2500\u2500\u2500\u2500\u2510  \u250c\u2500\u2500\u2500\u2500\u2500\u2510  \u250c\u2500\u2500\u2500\u2500\u2500\u2510                         \n\u2502S\u2081(x)\u2502  \u2502S\u2082(x)\u2502  \u2502 ... \u2502  \u2502S\u2099(x)\u2502                         \n\u2514\u2500\u2500\u252c\u2500\u2500\u2518  \u2514\u2500\u2500\u252c\u2500\u2500\u2518  \u2514\u2500\u2500\u252c\u2500\u2500\u2518  \u2514\u2500\u2500\u252c\u2500\u2500\u2518                         \n   \u25bc        \u25bc        \u25bc        \u25bc                            \n\u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510                         \n\u2502       Affine permutation       \u2502                         \n\u2514\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2518                         \n   \u25bc        \u25bc        \u25bc        \u25bc                            \n..................................                         \n"})}),"\n",(0,l.jsxs)(a.p,{children:["Partial SPNs are modifications of SPNs,\nwhere for certain rounds some S-Boxes are replaced with\nthe identity functions to reduce computational efforts ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"}),".\nFor example, the nonlinear substitution layers of the partial rounds of\nPoseidon update only the first internal state component ",(0,l.jsx)(a.a,{href:"#references",children:"[1]"}),".\nIn the case of P-SPNs, security considerations commonly demand to choose ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]}),"\nas a square MDS matrix, because these matrices provide\nperfect diffusion property for the affine permutation layer ",(0,l.jsx)(a.a,{href:"#references",children:"[6]"}),".\nPossessing this property means ensuring that\nany two ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]}),"-dimensional internal states,\nwhich differ in exactly ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"t"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"t"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6151em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"t"})]})})]})," components,\nare mapped by the affine permutation layer to\ntwo new internal states that differ in at least ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"n"}),(0,l.jsx)(a.mo,{children:"\u2212"}),(0,l.jsx)(a.mi,{children:"t"}),(0,l.jsx)(a.mo,{children:"+"}),(0,l.jsx)(a.mn,{children:"1"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n - t + 1"})]})})}),(0,l.jsxs)(a.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(a.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6984em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"t"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(a.span,{className:"mbin",children:"+"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(a.span,{className:"mord",children:"1"})]})]})]})," components."]}),"\n",(0,l.jsx)(a.h2,{id:"square-mds-matrix-security-check-in-the-context-of-p-spns",children:"Square MDS matrix security check in the context of P-SPNs"}),"\n",(0,l.jsxs)(a.p,{children:["Certain square MDS matrices should not be used in certain P-SPNs\nto avoid making them vulnerable to differential cryptanalysis,\nsince it may exploit the existence of infinitely long subspace trails\nof small period for vulnerable P-SPNs. ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"}),".\nSuch matrices are called insecure with respect to particular P-SPNs."]}),"\n",(0,l.jsxs)(a.p,{children:["An infinitely long subspace trail of period ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"l"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"})]})})]})," exists for a P-SPN,\nif and only if there is a proper subspace\nof differences of internal state vectors,\nsuch that if for a pair of initial internal states\nthe difference belongs to this subspace,\nthen the difference for the new internal states,\nwhich are obtained from the initial ones\nby means of the same ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"l"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"})]})})]}),"-round transformation,\nalso belongs to this subspace ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"}),"."]}),"\n",(0,l.jsxs)(a.p,{children:["Two methods for checking square MDS matrices for suitability for P-SPNs\nin terms of existence of infinitely long subspace trails\nhave been proposed in ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"}),".\nThe three tests method is aimed at checking\nwhether using a specified matrix for a P-SPN\nwith a specified structure of the substitution layer\nleads to existence of infinitely long subspace trails of period ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"p"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"p"})]})})]}),"\nfor this P-SPN for all ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"p"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"p"})]})})]})," no larger than a given ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"l"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"})]})})]}),".\nThe sufficient test method has been designed to determine\nwhether a square MDS matrix satisfies a sufficient condition\nof non-existence of infinitely long subspace trails of period ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"p"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"p"})]})})]}),"\nfor P-SPNs using this matrix for all ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"p"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"p"})]})})]})," no larger than a specified ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"l"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"})]})})]}),"."]}),"\n",(0,l.jsxs)(a.p,{children:["The sufficient test method is a direct consequence of\nTheorem 8 in ",(0,l.jsx)(a.a,{href:"#references",children:"[2]"})," and consists in checking that\nthe minimal polynomial of the ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"p"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"p"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"p"})]})})]}),"-th power of the tested matrix\nhas maximum degree and is irreducible for all ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"p"}),(0,l.jsx)(a.mo,{children:"\u2208"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"["}),(0,l.jsx)(a.mn,{children:"1.."}),(0,l.jsx)(a.mi,{children:"l"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"]"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"p \\in [1..l]"})]})})}),(0,l.jsxs)(a.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.7335em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"l"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"})]})})]})," for the matrix ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]}),",\nif and only if the minimal polynomials of 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definition, the purpose of the sufficient test method\ncan be described as checking whether\nthe unconditional P-SPN security level of the specified matrix\nis no less than a given bound."]}),"\n",(0,l.jsx)(a.h2,{id:"mdsecheck-method-getting-rid-of-the-matrix-powers",children:"MDSECheck method: getting rid of the matrix powers"}),"\n",(0,l.jsx)(a.p,{children:'The MDSECheck method, whose name is derived from\nthe words "MDS", "security", "elaborated" and "check",\nhas the same purpose as the sufficient test method,\nbut achieves it differently.\nThe differences of the first method from the latter\nand approaches to implementing them can be described as follows:'}),"\n",(0,l.jsxs)(a.ol,{children:["\n",(0,l.jsxs)(a.li,{children:["\n",(0,l.jsxs)(a.p,{children:["Computation and verification of minimal polynomials\nof 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divides the characteristic one.\nAlso, it is trivially proven that for a matrix with such a minimal polynomial\nit is equal to the characteristic polynomial.\nThus, the required checks for the matrices ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsx)(a.mn,{children:"2"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M^2"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.8141em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{mathvariant:"normal",children:"."}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"."}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"."})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"..."})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.1056em"}}),(0,l.jsx)(a.span,{className:"mord",children:"..."})]})})]}),", ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsx)(a.mi,{children:"l"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M^l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.8491em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]})," x ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]})," matrix over ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"G"}),(0,l.jsx)(a.mi,{children:"F"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"GF(q)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"GF"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"q"}),(0,l.jsx)(a.span,{className:"mclose",children:")"})]})})]}),",\nwhose minimal polynomial is of maximum degree and irreducible.\nThe statements in the previous paragraph imply that ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"f"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsx)(a.mi,{children:"y"}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"f(y)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(a.span,{className:"mclose",children:")"})]})})]}),",\nwhich is the ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]}),"-degree characteristic polynomial of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]}),", is irreducible.\nConsider ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," over the extension field ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"G"}),(0,l.jsx)(a.mi,{children:"F"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsx)(a.mi,{children:"n"})]}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"GF(q^n)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"GF"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"f"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsx)(a.mi,{children:"y"}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"f(y)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(a.span,{className:"mclose",children:")"})]})})]}),".\nLet ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsx)(a.mo,{children:"\u2208"}),(0,l.jsx)(a.mi,{children:"G"}),(0,l.jsx)(a.mi,{children:"F"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsx)(a.mi,{children:"n"})]}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1 \\in GF(q^n)"})]})})}),(0,l.jsxs)(a.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.5782em",verticalAlign:"-0.0391em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03b1"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,l.jsx)(a.span,{className:"mrel",children:"\u2208"}),(0,l.jsx)(a.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"GF"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"q"}),(0,l.jsx)(a.span,{className:"msupsub",children:(0,l.jsx)(a.span,{className:"vlist-t",children:(0,l.jsx)(a.span,{className:"vlist-r",children:(0,l.jsx)(a.span,{className:"vlist",style:{height:"0.6644em"},children:(0,l.jsxs)(a.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(a.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(a.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(a.span,{className:"mord mathnormal mtight",children:"n"})})]})})})})})]}),(0,l.jsx)(a.span,{className:"mclose",children:")"})]})]})]})," be a root of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"f"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsx)(a.mi,{children:"y"}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"f(y)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(a.span,{className:"mclose",children:")"})]})})]}),".\nAccording to standard results from the Galois field theory,\n",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"\u03b1"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03b1"})]})})]}),", ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsx)(a.mi,{children:"q"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1^q"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6644em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{mathvariant:"normal",children:"."}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"."}),(0,l.jsx)(a.mi,{mathvariant:"normal",children:"."})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"..."})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.1056em"}}),(0,l.jsx)(a.span,{className:"mord",children:"..."})]})})]}),", ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"n"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mo,{children:"\u2212"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mn,{children:"1"})]})]})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1^{q^{n \\: - \\: 1}}"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.9869em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"f"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsx)(a.mi,{children:"y"}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"f(y)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10764em"},children:"f"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"}),(0,l.jsx)(a.span,{className:"mclose",children:")"})]})})]})," ",(0,l.jsx)(a.a,{href:"#references",children:"[7]"}),".\nThus, these powers of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"\u03b1"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03b1"})]})})]})," are ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"n"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"n"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",children:"n"})]})})]})," distinct eigenvalues of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]}),".\nHence, due to matrix similarity properties, there is some matrix ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"S"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"S"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"})]})})]}),"\nsuch that ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"S"}),(0,l.jsx)(a.mi,{children:"M"}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"S"}),(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mo,{children:"\u2212"}),(0,l.jsx)(a.mn,{children:"1"})]})]}),(0,l.jsx)(a.mo,{children:"="}),(0,l.jsx)(a.mi,{children:"D"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"S M S^{-1} = D"})]})})}),(0,l.jsxs)(a.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.8141em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"SM"}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"D"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"D"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.02778em"},children:"D"})]})})]})," is the diagonal matrix,\nwhose nonzero elements are\n",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"\u03b1"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03b1"})]})})]}),", ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsx)(a.mi,{children:"q"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1^q"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6644em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsx)(a.mn,{children:"2"})]})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1^{q^2}"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.9869em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," in this case.\nSome data computed during this step is saved to be reused at the next one.\nIf the polynomial is reducible, then the check of 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsx)(a.mi,{children:"l"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1^l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.8491em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"\u03b1"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03b1"})]})})]})," is a root of the characteristic polynomial of 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"G"}),(0,l.jsx)(a.mi,{children:"F"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsx)(a.mi,{children:"n"})]}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"GF(q^n)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"GF"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"\u03b1"}),(0,l.jsx)(a.mi,{children:"i"})]})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"\u03b1^i"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.8247em"}}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.0037em"},children:"\u03b1"}),(0,l.jsx)(a.span,{className:"msupsub",children:(0,l.jsx)(a.span,{className:"vlist-t",children:(0,l.jsx)(a.span,{className:"vlist-r",children:(0,l.jsx)(a.span,{className:"vlist",style:{height:"0.8247em"},children:(0,l.jsxs)(a.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,l.jsx)(a.span,{className:"pstrut",style:{height:"2.7em"}}),(0,l.jsx)(a.span,{className:"sizing reset-size6 size3 mtight",children:(0,l.jsx)(a.span,{className:"mord mathnormal mtight",children:"i"})})]})})})})})]})]})})]})," belongs to some nontrivial subfield of ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"G"}),(0,l.jsx)(a.mi,{children:"F"}),(0,l.jsx)(a.mo,{stretchy:"false",children:"("}),(0,l.jsxs)(a.msup,{children:[(0,l.jsx)(a.mi,{children:"q"}),(0,l.jsx)(a.mi,{children:"n"})]}),(0,l.jsx)(a.mo,{stretchy:"false",children:")"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"GF(q^n)"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"GF"}),(0,l.jsx)(a.span,{className:"mopen",children:"("}),(0,l.jsxs)(a.span,{className:"mord",children:[(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," is ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsxs)(a.mrow,{children:[(0,l.jsx)(a.mi,{children:"i"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mo,{children:"\u2212"}),(0,l.jsx)(a.mtext,{children:"\u2005"}),(0,l.jsx)(a.mn,{children:"1"})]}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"i \\: - \\: 1"})]})})}),(0,l.jsxs)(a.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.7429em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(a.span,{className:"mord 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",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"M"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," is complete.\nIf all the values do not belong to such a subfield,\nthen the unconditional P-SPN security level is at least ",(0,l.jsxs)(a.span,{className:"katex",children:[(0,l.jsx)(a.span,{className:"katex-mathml",children:(0,l.jsx)(a.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(a.semantics,{children:[(0,l.jsx)(a.mrow,{children:(0,l.jsx)(a.mi,{children:"l"})}),(0,l.jsx)(a.annotation,{encoding:"application/x-tex",children:"l"})]})})}),(0,l.jsx)(a.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(a.span,{className:"base",children:[(0,l.jsx)(a.span,{className:"strut",style:{height:"0.6944em"}}),(0,l.jsx)(a.span,{className:"mord mathnormal",style:{marginRight:"0.01968em"},children:"l"})]})})]}),"."]}),"\n"]}),"\n"]}),"\n",(0,l.jsx)(a.h2,{id:"mdsecheck-library-crate-implementation-in-rust",children:"MDSECheck library crate: implementation in Rust"}),"\n",(0,l.jsxs)(a.p,{children:["The library crate ",(0,l.jsx)(a.a,{href:"#references",children:"[3]"})," provides tools for\ngenerating random square Cauchy MDS matrices over prime finite fields\nand applying the MDSECheck method\nto check such matrices for unconditional security.\nThe used data types of field elements and polynomials are provided by\nthe crates ark-ff ",(0,l.jsx)(a.a,{href:"#references",children:"[9]"})," and ark-poly ",(0,l.jsx)(a.a,{href:"#references",children:"[10]"}),".\nThe auxiliary tools in the crate modules are accessible as well."]}),"\n",(0,l.jsxs)(a.p,{children:["Generating by means of this crate a 10 x 10 MDS matrix,\nwhich is defined over the BN254 scalar field ",(0,l.jsx)(a.a,{href:"#references",children:"[11]"}),"\nand has unconditional P-SPN security level is 1000,\ntakes less than 60 milliseconds on average\nfor the laptop with the processor Intel\xae Core\u2122 i9-14900HX,\nwhose maximum clock frequency is 5.8 GHz."]}),"\n",(0,l.jsx)(a.h2,{id:"conclusion",children:"Conclusion"}),"\n",(0,l.jsx)(a.p,{children:"The MDSECheck method proposed in this article is a novel approach\nto checking square MDS matrices for unconditional security\nas the components of affine permutation layers of P-SPNs.\nIt has been implemented as a practical library crate\nfor generating unconditionally secure square MDS matrices\nfor P-SPNs over prime finite fields."}),"\n",(0,l.jsx)(a.p,{children:"The future research directions may include\ntheoretical and experimental studies of performance of approaches,\nwhich use the MDSECheck method\nto generate unconditionally secure square MDS matrices for P-SPNs."}),"\n",(0,l.jsx)(a.h2,{id:"references",children:"References"}),"\n",(0,l.jsxs)(a.ol,{children:["\n",(0,l.jsxs)(a.li,{children:['L. Grassi, D. Khovratovich, C. Rechberger, A. Roy, M. Schofnegger. "',(0,l.jsx)(a.a,{href:"https://eprint.iacr.org/2019/458.pdf",children:"POSEIDON: A New Hash Function for Zero-Knowledge Proof Systems (Updated Version)"}),'".']}),"\n",(0,l.jsxs)(a.li,{children:['L. Grassi, C. Rechberger, M. Schofnegger. "',(0,l.jsx)(a.a,{href:"https://eprint.iacr.org/2020/500.pdf",children:"Proving Resistance Against Infinitely Long Subspace Trails: How to Choose the Linear Layer"}),'".']}),"\n",(0,l.jsxs)(a.li,{children:['The page "',(0,l.jsx)(a.a,{href:"https://crates.io/crates/mdsecheck",children:"mdsecheck"}),'" on crates.io.']}),"\n",(0,l.jsxs)(a.li,{children:['Y. Kumar, P. Mishra, S. Samanta, K. Chand Gupta, A. Gaur. "',(0,l.jsx)(a.a,{href:"https://arxiv.org/pdf/2403.10372",children:"Construction of all MDS and involutory MDS matrices"}),'".']}),"\n",(0,l.jsxs)(a.li,{children:['The page "',(0,l.jsx)(a.a,{href:"https://proofwiki.org/wiki/Value_of_Cauchy_Determinant",children:"Value of Cauchy Determinant"}),'" on proofwiki.org.']}),"\n",(0,l.jsxs)(a.li,{children:['T. Silva, R. Dahab "',(0,l.jsx)(a.a,{href:"https://www.ic.unicamp.br/~reltech/PFG/2021/PFG-21-43.pdf",children:"MDS Matrices for Cryptography"}),'".']}),"\n",(0,l.jsxs)(a.li,{children:['S. Huczynska, M. Neunh\xf6ffer. "',(0,l.jsx)(a.a,{href:"http://www.math.rwth-aachen.de/~Max.Neunhoeffer/Teaching/ff2012/ff2012.pdf",children:"Finite Fields"}),'"']}),"\n",(0,l.jsxs)(a.li,{children:['R. Crandall, C. Pomerance. "',(0,l.jsx)(a.a,{href:"http://thales.doa.fmph.uniba.sk/macaj/skola/teoriapoli/primes.pdf",children:"Prime Numbers: A Computational Perspective"}),'" (2nd edition).']}),"\n",(0,l.jsxs)(a.li,{children:['The page "',(0,l.jsx)(a.a,{href:"https://crates.io/crates/ark-ff",children:"ark-ff"}),'" on crates.io.']}),"\n",(0,l.jsxs)(a.li,{children:['The page "',(0,l.jsx)(a.a,{href:"https://crates.io/crates/ark-poly",children:"ark-poly"}),'" on crates.io.']}),"\n",(0,l.jsxs)(a.li,{children:['The page "',(0,l.jsx)(a.a,{href:"https://crates.io/crates/ark-bn254",children:"ark-bn254"}),'" on crates.io.']}),"\n"]})]})}function x(s={}){const{wrapper:a}={...(0,i.R)(),...s.components};return a?(0,l.jsx)(a,{...s,children:(0,l.jsx)(r,{...s})}):r(s)}},28453:(s,a,e)=>{e.d(a,{R:()=>m,x:()=>t});var n=e(96540);const l={},i=n.createContext(l);function m(s){const a=n.useContext(i);return n.useMemo((function(){return"function"==typeof s?s(a):{...a,...s}}),[a,s])}function t(s){let a;return a=s.disableParentContext?"function"==typeof s.components?s.components(l):s.components||l:m(s.components),n.createElement(i.Provider,{value:a},s.children)}},99722:s=>{s.exports=JSON.parse('{"permalink":"/rlog/mdsecheck-method","source":"@site/rlog/2025-02-28-mdsecheck-method.mdx","title":"The MDSECheck method: choosing secure square MDS matrices for P-SP-networks","description":"This article introduces MDSECheck method \u2014 a novel approach","date":"2025-02-28T23:00:00.000Z","tags":[],"readingTime":17.12,"hasTruncateMarker":true,"authors":[{"name":"Aleksei Vambol","github":"AlekseiVambol","key":"aleksei","page":null}],"frontMatter":{"title":"The MDSECheck method: choosing secure square MDS matrices for P-SP-networks","date":"2025-02-28T23:00:00.000Z","authors":"aleksei","published":true,"slug":"mdsecheck-method","categories":"research","toc_min_heading_level":2,"toc_max_heading_level":4},"unlisted":false,"prevItem":{"title":"Nim in Logos - 1st Edition","permalink":"/rlog/nim-in-logos-01"},"nextItem":{"title":"Vac 2024 Year in Review","permalink":"/rlog/2024-recap"}}')}}]);