diff --git a/404.html b/404.html index f2cef6f..6f490d4 100644 --- a/404.html +++ b/404.html @@ -6,7 +6,6 @@ Page not found - Rate-Limiting Nullifier - @@ -27,17 +26,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/appendix.html b/appendix.html index 82db1aa..f5b111b 100644 --- a/appendix.html +++ b/appendix.html @@ -5,7 +5,6 @@ Appendix - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/book.js b/book.js index d40440c..627a368 100644 --- a/book.js +++ b/book.js @@ -11,7 +11,7 @@ function playground_text(playground) { let editor = window.ace.edit(code_block); return editor.getValue(); } else { - return code_block.textContent; + return code_block.innerText; } } @@ -300,6 +300,13 @@ function playground_text(playground) { themePopup.querySelector("button#" + get_theme()).focus(); } + function updateThemeSelected() { + themePopup.querySelectorAll('.theme-selected').forEach(function (el) { + el.classList.remove('theme-selected'); + }); + themePopup.querySelector("button#" + get_theme()).classList.add('theme-selected'); + } + function hideThemes() { themePopup.style.display = 'none'; themeToggleButton.setAttribute('aria-expanded', false); @@ -355,6 +362,7 @@ function playground_text(playground) { html.classList.remove(previousTheme); html.classList.add(theme); + updateThemeSelected(); } // Set theme diff --git a/circuits.html b/circuits.html index d16b4a8..7e8862a 100644 --- a/circuits.html +++ b/circuits.html @@ -5,7 +5,6 @@ Circuits - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/css/chrome.css b/css/chrome.css index 10fa4b3..59eae11 100644 --- a/css/chrome.css +++ b/css/chrome.css @@ -507,6 +507,8 @@ ul#searchresults span.teaser em { padding: 0; list-style: none; display: none; + /* Don't let the children's background extend past the rounded corners. */ + overflow: hidden; } .theme-popup .default { color: var(--icons); @@ -515,7 +517,7 @@ ul#searchresults span.teaser em { width: 100%; border: 0; margin: 0; - padding: 2px 10px; + padding: 2px 20px; line-height: 25px; white-space: nowrap; text-align: left; @@ -527,8 +529,10 @@ ul#searchresults span.teaser em { .theme-popup .theme:hover { background-color: var(--theme-hover); } -.theme-popup .theme:hover:first-child, -.theme-popup .theme:hover:last-child { - border-top-left-radius: inherit; - border-top-right-radius: inherit; + +.theme-selected::before { + display: inline-block; + content: "✓"; + margin-left: -14px; + width: 14px; } diff --git a/css/general.css b/css/general.css index 0e4f07a..344b53e 100644 --- a/css/general.css +++ b/css/general.css @@ -22,8 +22,8 @@ body { } code { - font-family: "Source Code Pro", Consolas, "Ubuntu Mono", Menlo, "DejaVu Sans Mono", monospace, monospace !important; - font-size: 0.875em; /* please adjust the ace font size accordingly in editor.js */ + font-family: var(--mono-font) !important; + font-size: var(--code-font-size); } /* make long words/inline code not x overflow */ @@ -148,6 +148,18 @@ blockquote { border-bottom: .1em solid var(--quote-border); } +kbd { + background-color: var(--table-border-color); + border-radius: 4px; + border: solid 1px var(--theme-popup-border); + box-shadow: inset 0 -1px 0 var(--theme-hover); + display: inline-block; + font-size: var(--code-font-size); + font-family: var(--mono-font); + line-height: 10px; + padding: 4px 5px; + vertical-align: middle; +} :not(.footnote-definition) + .footnote-definition, .footnote-definition + :not(.footnote-definition) { diff --git a/css/variables.css b/css/variables.css index 56b634b..21bf8e5 100644 --- a/css/variables.css +++ b/css/variables.css @@ -6,6 +6,8 @@ --page-padding: 15px; --content-max-width: 750px; --menu-bar-height: 50px; + --mono-font: "Source Code Pro", Consolas, "Ubuntu Mono", Menlo, "DejaVu Sans Mono", monospace, monospace; + --code-font-size: 0.875em /* please adjust the ace font size accordingly in editor.js */ } /* Themes */ diff --git a/how_to_use.html b/how_to_use.html index cee2fd9..4f9c512 100644 --- a/how_to_use.html +++ b/how_to_use.html @@ -5,7 +5,6 @@ How to use - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/images/rln-circuit.png b/images/rln-circuit.png index 5f64f0d..f9fc1e4 100644 Binary files a/images/rln-circuit.png and b/images/rln-circuit.png differ diff --git a/index.html b/index.html index 1757fea..af6bc9f 100644 --- a/index.html +++ b/index.html @@ -5,7 +5,6 @@ RLN - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/overview.html b/overview.html index 4841887..ab1436e 100644 --- a/overview.html +++ b/overview.html @@ -5,7 +5,6 @@ Overview - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/print.html b/print.html index 1558587..c818cbe 100644 --- a/print.html +++ b/print.html @@ -6,7 +6,6 @@ Rate-Limiting Nullifier - @@ -27,17 +26,17 @@ - + - - - - - - - - - - + + + + + + - - - + + + - - - - - - - - - - + + + + + + - - + + diff --git a/references.html b/references.html index ef63306..26bf954 100644 --- a/references.html +++ b/references.html @@ -5,7 +5,6 @@ B - References - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/rln.html b/rln.html index 1757fea..af6bc9f 100644 --- a/rln.html +++ b/rln.html @@ -5,7 +5,6 @@ RLN - Rate-Limiting Nullifier - @@ -26,17 +25,17 @@ - + - - - - - - - - - - + + + + + + - - + + diff --git a/searchindex.js b/searchindex.js index 9729bf2..ece1b72 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Object.assign(window.search, {"doc_urls":["rln.html#rln","overview.html#overview","what_is_rln.html#what-is-rate-limiting-nullifier","what_is_rln.html#how-it-works","what_is_rln.html#user-registration","what_is_rln.html#user-interaction","what_is_rln.html#user-removal-slashing","under_the_hood.html#under-the-hood","protocol_spec.html#technical-side-of-rln","protocol_spec.html#user-registration","protocol_spec.html#signalling","protocol_spec.html#slashing","protocol_spec.html#some-important-notes","circuits.html#circuits","circuits.html#merkle-tree-circuit","circuits.html#rln-core","circuits.html#membership-in-merkle-tree","circuits.html#correctness-of-secret-share","circuits.html#main-runner-of-the-circuits","uses.html#uses","how_to_use.html#how-to-use","theory.html#theory","sss.html#shamirs-secret-sharing-scheme","sss.html#overview","sss.html#polynomial-lagrange-interpolation","sss.html#shamirs-secret-sharing","sss.html#sharing","sss.html#recovering","sss.html#important-notes","appendix.html#appendix","terminology.html#terminology","references.html#references"],"index":{"documentStore":{"docInfo":{"0":{"body":35,"breadcrumbs":2,"title":1},"1":{"body":19,"breadcrumbs":2,"title":1},"10":{"body":64,"breadcrumbs":6,"title":1},"11":{"body":12,"breadcrumbs":6,"title":1},"12":{"body":163,"breadcrumbs":7,"title":2},"13":{"body":35,"breadcrumbs":5,"title":1},"14":{"body":209,"breadcrumbs":7,"title":3},"15":{"body":171,"breadcrumbs":6,"title":2},"16":{"body":56,"breadcrumbs":7,"title":3},"17":{"body":37,"breadcrumbs":7,"title":3},"18":{"body":55,"breadcrumbs":7,"title":3},"19":{"body":34,"breadcrumbs":3,"title":1},"2":{"body":74,"breadcrumbs":5,"title":3},"20":{"body":14,"breadcrumbs":2,"title":1},"21":{"body":12,"breadcrumbs":2,"title":1},"22":{"body":19,"breadcrumbs":8,"title":4},"23":{"body":78,"breadcrumbs":5,"title":1},"24":{"body":135,"breadcrumbs":7,"title":3},"25":{"body":30,"breadcrumbs":7,"title":3},"26":{"body":79,"breadcrumbs":5,"title":1},"27":{"body":16,"breadcrumbs":5,"title":1},"28":{"body":32,"breadcrumbs":6,"title":2},"29":{"body":9,"breadcrumbs":2,"title":1},"3":{"body":43,"breadcrumbs":3,"title":1},"30":{"body":156,"breadcrumbs":3,"title":1},"31":{"body":26,"breadcrumbs":4,"title":1},"4":{"body":69,"breadcrumbs":4,"title":2},"5":{"body":136,"breadcrumbs":4,"title":2},"6":{"body":64,"breadcrumbs":5,"title":3},"7":{"body":15,"breadcrumbs":5,"title":2},"8":{"body":44,"breadcrumbs":8,"title":3},"9":{"body":119,"breadcrumbs":7,"title":2}},"docs":{"0":{"body":"RLN (Rate-Limiting Nullifier) is a zk-gadget/protocol that enables spam prevention mechanism for anonymous environments. RLN is part of ( PSE ) Privacy & Scaling Explorations , a multidisciplinary team supported by the Ethereum Foundation. PSE explores new use cases for zero-knowledge proofs and other cryptographic primitives. alt text","breadcrumbs":"RLN » RLN","id":"0","title":"RLN"},"1":{"body":"This section is a starting point for understanding the concepts of RLN . Here we'll discuss: Basic explanation of the RLN protocol RLN protocol under the hood RLN uses","breadcrumbs":"Overview » Overview","id":"1","title":"Overview"},"10":{"body":"Now that the user is registered, he wants to interact with the system. Imagine that the system is an anonymous chat and the interaction is the sending of messages. So, to send a message user have to come up with share - the point \\((x, y)\\) on her polynomial. We denote: \\(x = Poseidon(message), y = A(x)\\). Thus, if the same epoch user sends more than one message, their polynomial and, therefore, their secret (\\(a_0\\)) can be recovered. Of course, we somehow must prove that our share = \\((x, y)\\) is valid (that this is really a point on our polynomial = A(x)), as well as we must prove other things are valid too, that's why we use zkSNARK. An explanation of the zk-circuits can be found in the next topic.","breadcrumbs":"Overview » Under the hood » Protocol spec » Signalling","id":"10","title":"Signalling"},"11":{"body":"As it's been said, if a user sends more than one message, everyone else will be able to recover his secret, slash them and take their stake.","breadcrumbs":"Overview » Under the hood » Protocol spec » Slashing","id":"11","title":"Slashing"},"12":{"body":"There are also nullifier and rln-identifier, which can be found in the RLN protocol/circuits. So, rln-identifier is just a random value that's unique per RLN app. It's used for additional cross-application security - to protect the user secrets from being compromised if they use the same credentials across different RLN apps. If rln-identifier is not present, the user uses the same credentials and sends a message in two different RLN apps using the same epoch, then their secret key can be revealed. Adding the rln-identifier field, we obscure the nullifier, so this kind of attack cannot happen. The only kind of attack that is possible is if we have an entity with a global view of all messages, and they try to brute-force different combinations of x and y shares for different nullifiers. Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received shares and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public nullifier (\\(Poseidon(a_1, rln-identifier)\\)), so if a user sends more than one message, it will be immediately visible to everyone. Also, in our example (and zk-chat implementation), we use linear polynomial, but SSS allows us to use various degree polynomials; therefore we can implement a protocol, where more than one signal (message) can be sent in per epoch. To learn more, check out the specification ; there are also circuits implemented for various degree polynomials too.","breadcrumbs":"Overview » Under the hood » Protocol spec » Some important notes","id":"12","title":"Some important notes"},"13":{"body":"zkSNARK is used in the RLN core. Therefore, we must represent the protocol in R1CS (as we use Groth16 ). Circom was chosen for this. This section explains RLN circuits for the linear polynomial case (one message per epoch). You can find implementation for the general case here RLN circuits implement the logic described in previous topic .","breadcrumbs":"Overview » Under the hood » Circuits » Circuits","id":"13","title":"Circuits"},"14":{"body":"One of the critical components of RLN is the Incremental Merkle Tree for the membership tree. Any Merkle tree can be used, but we have chosen the Incremental Merkle Tree for gas efficiency. Let's look at the implementation . At the beginning of the file, we denote that we use Circom 2.0 and include two helper zk-gadgets : pragma circom 2.0.0; include \"../node_modules/circomlib/circuits/poseidon.circom\";\ninclude \"../node_modules/circomlib/circuits/mux1.circom\"; Poseidon gadget is just the implementation of the Poseidon hash function; the mux1 gadget will be described later. Next, we can see two implemented gadgets: template PoseidonHashT3() { var nInputs = 2; signal input inputs[nInputs]; signal output out; component hasher = Poseidon(nInputs); for (var i = 0; i < nInputs; i ++) { hasher.inputs[i] <== inputs[i]; } out <== hasher.out;\n} template HashLeftRight() { signal input left; signal input right; signal output hash; component hasher = PoseidonHashT3(); left ==> hasher.inputs[0]; right ==> hasher.inputs[1]; hash <== hasher.out;\n} These are helper gadgets to make the code more clean. Poseidon gadget is implemented with the ability to take a different number of arguments. We use PoseidonHashT3() to initialize it like a function with two arguments. And HashLeftRight use PoseidonHashT3 in a more \"readable\" way: it takes two inputs, left and right, and outputs the result of the calculation. Next comes the core of the Merkle Tree gadget: template MerkleTreeInclusionProof(n_levels) { signal input leaf; signal input path_index[n_levels]; signal input path_elements[n_levels][1]; signal output root; component hashers[n_levels]; component mux[n_levels]; signal levelHashes[n_levels + 1]; levelHashes[0] <== leaf; ... root <== levelHashes[n_levels];\n} Here we have three inputs: leaf, path_index, and path_elements. path_index is the position of the leaf represented in binary. We need the binary representation of the position in the Merkle tree to understand the hashing path from the leaf to the root (more on that \"3. Recursive Incremental Merkle Tree Algorithm, page 4\" ). path_elements are sibling leaves that are part of Merkle Proof. leaf = Poseidon(identity_secret), so it's just identity commitment . There is a Merkle Tree hashing algorithm in the omitted part, no more than that.","breadcrumbs":"Overview » Under the hood » Circuits » Merkle Tree circuit","id":"14","title":"Merkle Tree circuit"},"15":{"body":"RLN circuit is the implementation of RLN logic itself (which in turn uses the Merkle Tree gadget). You can find the implementation here . So, let's start with helper gadgets: template CalculateIdentityCommitment() { signal input identity_secret; signal output out; component hasher = Poseidon(1); hasher.inputs[0] <== identity_secret; out <== hasher.out;\n} template CalculateA1() { signal input a_0; signal input epoch; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== a_0; hasher.inputs[1] <== epoch; out <== hasher.out;\n} template CalculateNullifier() { signal input a_1; signal input rln_identifier; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== a_1; hasher.inputs[1] <== rln_identifier; out <== hasher.out;\n} It's easy to understand these samples: CalculateIdentityCommitment() is used to calculate the identity commitment. It takes secret and outputs the commitment. CalculateA1() and CalculateNullifier() are used to calculate a_1 and nullifier (internal nullifier); they are implemented as it's described in previous topic . Now, let's look at the core logic of the RLN circuit. ... signal input identity_secret; signal input path_elements[n_levels][LEAVES_PER_PATH_LEVEL]; signal input identity_path_index[n_levels]; signal input x; signal input epoch; signal input rln_identifier; signal output y; signal output root; signal output nullifier; ... So, here we have many inputs. Private inputs are: identity_secret (basically a_0 from the polynomial), path_elements[][], identity_path_index[]. Public inputs are: x (actually just the hash of a signal), epoch, rln_identifier. Outputs are: y' (share of the secret), rootof a Merkle Tree, andnullifier.` RLN circuit consists of two checks: Membership in Merkle Tree Correctness of secret share","breadcrumbs":"Overview » Under the hood » Circuits » RLN core","id":"15","title":"RLN core"},"16":{"body":"To check membership in a Merkle Tree, we can simply use the previously described Merkle Tree gadget: ... component identity_commitment = CalculateIdentityCommitment(); identity_commitment.identity_secret <== identity_secret; var i; var j; component inclusionProof = MerkleTreeInclusionProof(n_levels); inclusionProof.leaf <== identity_commitment.out; for (i = 0; i < n_levels; i++) { for (j = 0; j < LEAVES_PER_PATH_LEVEL; j++) { inclusionProof.path_elements[i][j] <== path_elements[i][j]; } inclusionProof.path_index[i] <== identity_path_index[i]; } ... Here we are calculating the identity_commitment and passing it along with sibling leaves and binary representation of the position to a Merkle Tree gadget. It gives us the calculated root as an output, and we can put the constraint on that: root <== inclusionProof.root;","breadcrumbs":"Overview » Under the hood » Circuits » Membership in Merkle Tree","id":"16","title":"Membership in Merkle Tree"},"17":{"body":"As we use linear polynomial we need to check that y = a_1 * x + a_0 (a_0 is identity secret). For that, we need these constraints: ... component a_1 = CalculateA1(); a_1.a_0 <== identity_secret; a_1.epoch <== epoch; y <== identity_secret + a_1.out * x; ... To calculate and reveal the nullifier: ... component calculateNullifier = CalculateNullifier(); calculateNullifier.a_1 <== a_1.out; calculateNullifier.rln_identifier <== rln_identifier; nullifier <== calculateNullifier.out; ...","breadcrumbs":"Overview » Under the hood » Circuits » Correctness of secret share","id":"17","title":"Correctness of secret share"},"18":{"body":"Now the Circuits can be used as gadgets. If we want to use it in our app, we need to initialize it and have a main - starting point function. It can be found here . The implementation is super basic: pragma circom 2.0.0; include \"./rln-base.circom\"; component main {public [x, epoch, rln_identifier ]} = RLN(15); That's the whole RLN Circom Circuit :) Here we just need to list all public inputs (x, epoch, rln_identifier; the rest of the inputs are private). Also, we set the depth of the Merkle Tree = 15 (max of 32768 members).","breadcrumbs":"Overview » Under the hood » Circuits » Main runner of the circuits","id":"18","title":"Main runner of the circuits"},"19":{"body":"This section contains list of apps that use RLN : zk-chat - A spam resistant instant messaging application for private and anonymous communication rln-chat-app - PoC app, created using rln-js waku-rln-relay - Extension of waku-relay (spam protection with RLN )","breadcrumbs":"Overview » Uses » Uses","id":"19","title":"Uses"},"2":{"body":"This topic is a part of complete overview by Blagoj . RLN is a zero-knowledge gadget that enables spam prevention for decentralized, anonymous environments. The anonymity property opens up the possibility for spam and Sybil attack vectors for certain applications, which could seriously degrade the user experience and the overall functioning of the application. For example, imagine a chat application where users are anonymous. Now, everyone can write an unlimited number of spam messages, but we don't have the ability to kick this member because the spammer is anonymous. RLN helps us identify and \"kick\" the spammer. Moreover, RLN can be useful not only to prevent spam attacks but, in general, to limit users (in anonymous environments) in the number of actions (f.e. to vote or to make a bid).","breadcrumbs":"Overview » What is RLN » What is Rate-Limiting Nullifier?","id":"2","title":"What is Rate-Limiting Nullifier?"},"20":{"body":"This section provides information on how to use RLN in your project: JavaScript RLN (for rln-js ) Rust RLN (for zerokit-rln )","breadcrumbs":"How to use » How to use","id":"20","title":"How to use"},"21":{"body":"This section provides theoretical information that underpins RLN . Here we'll discuss: Shamir's Secret Sharing","breadcrumbs":"Theory » Theory","id":"21","title":"Theory"},"22":{"body":"This topic is an explanation of Shamir's Secret Sharing scheme ( SSS ), also known as \\((k, n)\\) threshold secret sharing scheme. SSS is one of the critical parts of RLN .","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"22","title":"Shamir's Secret Sharing Scheme"},"23":{"body":"Imagine if you have some important secret (secret key) and you don't want to store it anywhere. For that, you can use the SSS scheme. It allows you to split this secret into \\(n\\) parts (each individual part doesn't give any information about the secret) and restore this secret upon presentation of \\(k\\) \\((k <= n)\\) parts. For example, you have a secret that you want to split into \\(n\\) parts/shares. You can divide these shares between your friends (1 share to 1 friend). Now when \\(k\\) of your friends reveal their share, you can restore the secret. This scheme is also called \\((k, n)\\) threshold secret sharing scheme . This scheme is possible due to polynomial interpolation (especially Lagrange interpolation). Let's describe how Lagrange interpolation works and how it's used in a SSS scheme.","breadcrumbs":"Theory » Shamir's Secret Sharing » Overview","id":"23","title":"Overview"},"24":{"body":"Interpolation is a method of constructing (or restoring) new points/values (or function) based on the range of a set of known points/values (f.e. we can restore the line (linear function) from two points that are from this line). The previous example describes how that works. An unlimited number of parabolas (second-degree polynomials) can be drawn through two points. To choose the only one, you need a third point. Thus, if we have a polynomial \\(f(x) = 3x + 2\\), we only need two points from this polynomial to restore it. Let's peek two random \\(x\\) values and calculate \\(f(x)\\): For \\(x = 1\\) we have \\(f(1) = 3 * 1 + 2 = 5\\) For \\(x = 10\\) we have \\(f(10) = 32\\) Now we have to shares: \\((1, 5)\\) and \\((10, 32)\\). If we draw a graph based on these two shares, we can easily see that this is the same line (function): We also can \"restore\" the function analytically. For that let's denote: \\[f(x) = y_1 * \\frac{x - x_2}{x_1 - x_2} + y_2 * \\frac{x - x_1}{x_2 - x_1}\\] where \\(x_1 = 5, x_2 = 10, y_1 = 5, y_2 = 32\\). If we make substitution we got: \\[f(x) = 3x + 2 \\] which is the same polynomial. The same technique can be made with every polynomial. Main thing to remember is that we need \\(n + 1\\) points to interpolate \\(n\\)-degree polynomial. Now that we know how interpolation works, we can learn how it is used in SSS.","breadcrumbs":"Theory » Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"24","title":"Polynomial (Lagrange) interpolation"},"25":{"body":"To create the SSS construct, we must choose \\((k, n)\\), where \\(n\\) is the number of shares we want to get from the secret and \\(k\\) is the number of shares required to restore the secret. The degree of the \"secret\" polynomial is \\(k - 1\\) (covered in the previous section). Let's try to construct SSS with an example.","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing","id":"25","title":"Shamir's Secret Sharing"},"26":{"body":"Our secret = \\(S = 30\\) As the linear polynomial used in current RLN implementations, let's set \\(k = 2\\) (2 points are enough to recover the polynomial); \\(n\\) is not that important, but we can make it any number, f.e. 3 The secret polynomial is: \\[f(x) = a_1 * x + a_0 \\] where zero coefficient \\(a_0 = S\\), and \\(a_1\\) is some random number (f.e. 5); We must pick \\(n = 3\\) different points (shares) on that polynomial, for that we can pick three random \\(x\\) values (f.e. 5, 8, 16) and calculate \\(f(x)\\): \\[f(5) = 5 * 5 + 30 = 55\\] \\[f(8) = 5 * 8 + 30 = 70 \\] \\[f(16) = 5 * 16 + 30 = 110 \\] So, the shares are: \\((5, 55), (8, 70), (16, 110)\\)","breadcrumbs":"Theory » Shamir's Secret Sharing » Sharing","id":"26","title":"Sharing"},"27":{"body":"We can take any two shares to recover (as described in the interpolation section) the \"secret\" polynomial. Zero coefficient (\\(a_0\\)) in the recovered polynomial is the secret \\(S\\).","breadcrumbs":"Theory » Shamir's Secret Sharing » Recovering","id":"27","title":"Recovering"},"28":{"body":"Arithmetic in this topic is usual for us. However, in real life, SSS arithmetic is defined over some finite field. This means that all calculations are carried out modulo some big prime field. In fact, it happens by itself in Circom because the arithmetic there is defined over the finite field, too, so we don't need to do anything extra).","breadcrumbs":"Theory » Shamir's Secret Sharing » Important notes","id":"28","title":"Important notes"},"29":{"body":"The following sections contain reference material you may find useful: Terminology References","breadcrumbs":"Appendix » Appendix","id":"29","title":"Appendix"},"3":{"body":"The RLN construct's functionality consists of three parts, which, when integrated together, provide spam and Sybil attack protection. These parts should be integrated by the upstream applications, which require anonymity and spam protection. The applications can be centralized or decentralized. For decentralized applications, each user maintains separate storage and compute resources for the application. The three parts are: User registration User interaction User removal (slashing)","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"Term Description zkSNARK Proof construction where one can prove possession of certain information, e.g. a secret key, without revealing that information, and without any interaction between the prover and verifier. Circuit A program, that describes constraints for the prover in zkSNARK (for more information read this ). zk-Gadget Circuit, that can be used as a building block for another circuit, e.g. Poseidon hash function gadget. Stake Financial or social stake required for registering in the RLN applications. Common stake examples are: locking cryptocurrency (financial), linking reputable social identity. Identity secret Random number, which must be kept private by the user. Identity commitment The result of Poseidon(Identity secret) calculation. It is used by the users for registering in the protocol. Signal The message generated by a user. It is an arbitrary bit string that may represent a chat message, a URL request, protobuf message, etc. Signal hash Hash of the signal, used as an input in the RLN circuit. RLN Identifier Random finite field value unique per RLN app. It is used for additional cross-application security. The role of the RLN identifier is protection of the user secrets being compromised if signals are being generated with the same credentials at different apps. RLN membership tree Merkle tree data structure, filled with identity commitments of the users. Serves as a data structure that ensures user registrations. Merkle proof Proof that a user is member of the RLN membership tree.","breadcrumbs":"Appendix » A - Terminology » Terminology","id":"30","title":"Terminology"},"31":{"body":"First Proposal/Idea of RLN by Barry WhiteHat RLN Overview by Blagoj Demo RLN Spec VAC RLN Spec Understand zkSNARK Circom Docs rln-js zerokit-rln Incremental Merkle Tree paper","breadcrumbs":"Appendix » B - References » References","id":"31","title":"References"},"4":{"body":"Before registering to the application, the user needs to generate a secret key and derive an identity commitment from the secret key using the Poseidon hash function identityCommitment = posseidonHash(secretKey). The user registers to the application by providing a form of stake and their identity commitment, which is derived from the secret key. The application maintains a Merkle tree data structure (in the latest iteration of RLN , we use an Incremental Merkle Tree algorithm for gas efficiency, but the Merkle tree does not have to be on-chain), which stores the identity commitments of the registered users. Upon successful registration, the user's identity commitment is stored in a leaf of the Merkle tree, and an index is given to them, representing their position in the tree.","breadcrumbs":"Overview » What is RLN » User registration","id":"4","title":"User registration"},"5":{"body":"For each interaction that the user wants to make with the application, the user must generate a zero-knowledge proof ensuring that their identity commitment is part of the membership Merkle tree. There are a number of use-cases for RLN , such as voting applications (1 vote per election), chat (one message per second), and rate-limiting cache access (CDN denial of service protection). The verifier can be a server for centralized applications or the other users for decentralized applications. The general anti-spam rule is usually in the form of: Users must not make more than X interactions per epoch. The epoch can be translated as a time interval of Y units of time unit Z. For simplicity's sake, let's transform the rule into: `Users must not send more than one message per second. We can implement this using Shamir's Secret Sharing scheme ( read more ), which allows you to split a secret (f.e. to n parts) and recover it when any m of n parts (m <= n) are presented. Thus, users have to split their secret_key into n parts, and for every interaction, they have to reveal the new part of the secret_key. So, in addition to proving the membership in the Merkle Tree , users have to prove that the revealed part is truly the part of their secret_key. If they make more interactions than allowed per epoch, their secret key can be fully reconstructed.","breadcrumbs":"Overview » What is RLN » User interaction","id":"5","title":"User interaction"},"6":{"body":"The final property of the RLN mechanism is that it allows for the users to be removed from the membership tree by anyone that knows their secret key. The membership tree contains the identity commitments of all registered users. Users' identity commitment is derived from their secret key, and the secret key of the user is only revealed in a spam event (except for the scenarios where the original users want to remove themselves, which they can always do because they know their secret key). When an economic stake is present, the RLN mechanism can be implemented in a way that the spammer's stake is sent to the first user that correctly reports the spammer by providing the reconstructed secret key of the spammer as proof.","breadcrumbs":"Overview » What is RLN » User removal (slashing)","id":"6","title":"User removal (slashing)"},"7":{"body":"This section provides deep and technical RLN overview. We'll discuss: Technical side of RLN (specification demo) How circuits are implemented","breadcrumbs":"Overview » Under the hood » Under the hood","id":"7","title":"Under the hood"},"8":{"body":"This topic is a less strict version of specifications. If you want a more formal description, you can find specs in the references . Also, if you're unfamiliar with Shamir's Secret Sharing scheme, you can read it here . alt text Under the hood: The RLN Circom Circuit RLN consists of three parts: User registration User interaction (signaling) User removal (slashing) - additional part Well, let's discuss them.","breadcrumbs":"Overview » Under the hood » Protocol spec » Technical side of RLN","id":"8","title":"Technical side of RLN"},"9":{"body":"The first part of RLN is registration. There is nothing special in RLN registration; it's almost the same process as in other protocols/apps with anonymous environments: we need to create a Merkle Tree, and every participant must submit a commitment and place it in the Merkle Tree, and after that to interact with the app every participant will create a zkProof's, that they are a member of the tree (we use an Incremental Merkle Tree , as it more GAS efficient ). So, each member generates a secret key, denoted by \\(a_0\\). Identity commitment \\(q\\) is the hash (Poseidon) of the secret key: \\(q = Poseidon(a_0)\\). RLN wouldn't work if there were no punishment for spam; that's why to become a member, a user has to register and provide something at stake. So, whoever has our \\(a_0\\) can \"slash\" us. The slight difference is that we must enable a secret sharing scheme (to split the commitment into parts). We need to come up with a polynomial. For simplicity we use linear polynomial (e.g. \\(f(x) = kx + b\\). Therefore, with two points, we can reconstruct the polynomial and recover the secret. Our polynomial will be: \\(A(x) = a_1 * x + a_0\\), where \\(a_1 = Poseidon(a_0, epoch)\\). epoch is a simple identifier (also called external nullifier ). And each epoch, there is a polynomial with new \\(a_1\\) and the same \\(a_0\\).","breadcrumbs":"Overview » Under the hood » Protocol spec » User registration","id":"9","title":"User 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{"doc_urls":["rln.html#rln","overview.html#overview","what_is_rln.html#what-is-rate-limiting-nullifier","what_is_rln.html#how-it-works","what_is_rln.html#user-registration","what_is_rln.html#user-interaction","what_is_rln.html#user-removal-slashing","under_the_hood.html#under-the-hood","protocol_spec.html#technical-side-of-rln","protocol_spec.html#user-registration","protocol_spec.html#signalling","protocol_spec.html#slashing","protocol_spec.html#nullifiers","protocol_spec.html#some-important-notes","circuits.html#circuits","circuits.html#merkle-tree-circuit","circuits.html#rln-core","circuits.html#membership-in-merkle-tree","circuits.html#correctness-of-secret-share","circuits.html#main-runner-of-the-circuits","uses.html#uses","how_to_use.html#how-to-use","theory.html#theory","sss.html#shamirs-secret-sharing-scheme","sss.html#overview","sss.html#polynomial-lagrange-interpolation","sss.html#shamirs-secret-sharing","sss.html#sharing","sss.html#recovering","sss.html#important-notes","appendix.html#appendix","terminology.html#terminology","references.html#references"],"index":{"documentStore":{"docInfo":{"0":{"body":35,"breadcrumbs":2,"title":1},"1":{"body":19,"breadcrumbs":2,"title":1},"10":{"body":64,"breadcrumbs":6,"title":1},"11":{"body":12,"breadcrumbs":6,"title":1},"12":{"body":92,"breadcrumbs":6,"title":1},"13":{"body":33,"breadcrumbs":7,"title":2},"14":{"body":35,"breadcrumbs":5,"title":1},"15":{"body":209,"breadcrumbs":7,"title":3},"16":{"body":178,"breadcrumbs":6,"title":2},"17":{"body":56,"breadcrumbs":7,"title":3},"18":{"body":47,"breadcrumbs":7,"title":3},"19":{"body":55,"breadcrumbs":7,"title":3},"2":{"body":69,"breadcrumbs":5,"title":3},"20":{"body":34,"breadcrumbs":3,"title":1},"21":{"body":14,"breadcrumbs":2,"title":1},"22":{"body":12,"breadcrumbs":2,"title":1},"23":{"body":19,"breadcrumbs":8,"title":4},"24":{"body":78,"breadcrumbs":5,"title":1},"25":{"body":135,"breadcrumbs":7,"title":3},"26":{"body":30,"breadcrumbs":7,"title":3},"27":{"body":79,"breadcrumbs":5,"title":1},"28":{"body":16,"breadcrumbs":5,"title":1},"29":{"body":32,"breadcrumbs":6,"title":2},"3":{"body":43,"breadcrumbs":3,"title":1},"30":{"body":9,"breadcrumbs":2,"title":1},"31":{"body":156,"breadcrumbs":3,"title":1},"32":{"body":26,"breadcrumbs":4,"title":1},"4":{"body":69,"breadcrumbs":4,"title":2},"5":{"body":136,"breadcrumbs":4,"title":2},"6":{"body":64,"breadcrumbs":5,"title":3},"7":{"body":15,"breadcrumbs":5,"title":2},"8":{"body":44,"breadcrumbs":8,"title":3},"9":{"body":110,"breadcrumbs":7,"title":2}},"docs":{"0":{"body":"RLN (Rate-Limiting Nullifier) is a zk-gadget/protocol that enables spam prevention mechanism for anonymous environments. RLN is part of ( PSE ) Privacy & Scaling Explorations , a multidisciplinary team supported by the Ethereum Foundation. PSE explores new use cases for zero-knowledge proofs and other cryptographic primitives. alt text","breadcrumbs":"RLN » RLN","id":"0","title":"RLN"},"1":{"body":"This section is a starting point for understanding the concepts of RLN . Here we'll discuss: Basic explanation of the RLN protocol RLN protocol under the hood RLN uses","breadcrumbs":"Overview » Overview","id":"1","title":"Overview"},"10":{"body":"Now that the user is registered, he wants to interact with the system. Imagine that the system is an anonymous chat and the interaction is the sending of messages. So, to send a message user have to come up with share - the point \\((x, y)\\) on her polynomial. We denote: \\(x = Poseidon(message), y = A(x)\\). Thus, if the same epoch user sends more than one message, their polynomial and, therefore, their secret (\\(a_0\\)) can be recovered. Of course, we somehow must prove that our share = \\((x, y)\\) is valid (that this is really a point on our polynomial = A(x)), as well as we must prove other things are valid too, that's why we use zkSNARK. An explanation of the zk-circuits can be found in the next topic.","breadcrumbs":"Overview » Under the hood » Protocol spec » Signalling","id":"10","title":"Signalling"},"11":{"body":"As it's been said, if a user sends more than one message, everyone else will be able to recover his secret, slash them and take their stake.","breadcrumbs":"Overview » Under the hood » Protocol spec » Slashing","id":"11","title":"Slashing"},"12":{"body":"There are also \\(internal\\_nullifier\\) and \\(external\\_nullifier\\), which can be found in the RLN protocol/circuits. \\(external\\_nullifier = Poseidon(epoch, rln\\_identifier)\\), where \\(rln\\_identifier\\) is a random finite field value, unique per RLN app. The \\(external\\_nullifier\\) is required so that the user can securely use the same private key \\(a_0\\) across different RLN apps - in different applications (and in different eras) with the same secret key, the user will have different values ​​of the coefficient \\(a_1\\). Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received shares and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public \\(internal\\_nullifier = Poseidon(a_1)\\), so if a user sends more than one message, it will be immediately visible to everyone.","breadcrumbs":"Overview » Under the hood » Protocol spec » Nullifiers","id":"12","title":"Nullifiers"},"13":{"body":"Also, in our example (and zk-chat implementation), we use linear polynomial, but SSS allows us to use various degree polynomials; therefore we can implement a protocol, where more than one signal (message) can be sent in per epoch. To learn more, check out the specification ; there are also circuits implemented for various degree polynomials too.","breadcrumbs":"Overview » Under the hood » Protocol spec » Some important notes","id":"13","title":"Some important notes"},"14":{"body":"zkSNARK is used in the RLN core. Therefore, we must represent the protocol in R1CS (as we use Groth16 ). Circom was chosen for this. This section explains RLN circuits for the linear polynomial case (one message per epoch). You can find implementation for the general case here RLN circuits implement the logic described in previous topic .","breadcrumbs":"Overview » Under the hood » Circuits » Circuits","id":"14","title":"Circuits"},"15":{"body":"One of the critical components of RLN is the Incremental Merkle Tree for the membership tree. Any Merkle tree can be used, but we have chosen the Incremental Merkle Tree for gas efficiency. Let's look at the implementation . At the beginning of the file, we denote that we use Circom 2.0 and include two helper zk-gadgets : pragma circom 2.0.0; include \"../node_modules/circomlib/circuits/poseidon.circom\";\ninclude \"../node_modules/circomlib/circuits/mux1.circom\"; Poseidon gadget is just the implementation of the Poseidon hash function; the mux1 gadget will be described later. Next, we can see two implemented gadgets: template PoseidonHashT3() { var nInputs = 2; signal input inputs[nInputs]; signal output out; component hasher = Poseidon(nInputs); for (var i = 0; i < nInputs; i ++) { hasher.inputs[i] <== inputs[i]; } out <== hasher.out;\n} template HashLeftRight() { signal input left; signal input right; signal output hash; component hasher = PoseidonHashT3(); left ==> hasher.inputs[0]; right ==> hasher.inputs[1]; hash <== hasher.out;\n} These are helper gadgets to make the code more clean. Poseidon gadget is implemented with the ability to take a different number of arguments. We use PoseidonHashT3() to initialize it like a function with two arguments. And HashLeftRight use PoseidonHashT3 in a more \"readable\" way: it takes two inputs, left and right, and outputs the result of the calculation. Next comes the core of the Merkle Tree gadget: template MerkleTreeInclusionProof(n_levels) { signal input leaf; signal input path_index[n_levels]; signal input path_elements[n_levels][1]; signal output root; component hashers[n_levels]; component mux[n_levels]; signal levelHashes[n_levels + 1]; levelHashes[0] <== leaf; ... root <== levelHashes[n_levels];\n} Here we have three inputs: leaf, path_index, and path_elements. path_index is the position of the leaf represented in binary. We need the binary representation of the position in the Merkle tree to understand the hashing path from the leaf to the root (more on that \"3. Recursive Incremental Merkle Tree Algorithm, page 4\" ). path_elements are sibling leaves that are part of Merkle Proof. leaf = Poseidon(identity_secret), so it's just identity commitment . There is a Merkle Tree hashing algorithm in the omitted part, no more than that.","breadcrumbs":"Overview » Under the hood » Circuits » Merkle Tree circuit","id":"15","title":"Merkle Tree circuit"},"16":{"body":"RLN circuit is the implementation of RLN logic itself (which in turn uses the Merkle Tree gadget). You can find the implementation here . So, let's start with helper gadgets: template CalculateIdentityCommitment() { signal input identity_secret; signal output out; component hasher = Poseidon(1); hasher.inputs[0] <== identity_secret; out <== hasher.out;\n} template CalculateExternalNullifier() { signal input epoch; signal input rln_identifier; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== epoch; hasher.inputs[1] <== rln_identifier; out <== hasher.out;\n} template CalculateA1() { signal input a_0; signal input external_nullifier; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== a_0; hasher.inputs[1] <== external_nullifier; out <== hasher.out;\n} template CalculateInternalNullifier() { signal input a_1; signal output out; component hasher = Poseidon(1); hasher.inputs[0] <== a_1; out <== hasher.out;\n} It's easy to understand these samples: CalculateIdentityCommitment(), CalculateA1(), CalculateInternalNullifier(), CalculateExternalNullifier() - they do exactly what their name says; they are implemented as it's described in previous topic . Now, let's look at the core logic of the RLN circuit. ... signal input identity_secret; signal input path_elements[n_levels][LEAVES_PER_PATH_LEVEL]; signal input identity_path_index[n_levels]; signal input x; signal input epoch; signal input rln_identifier; signal output y; signal output root; signal output nullifier; ... So, here we have many inputs. Private inputs are: identity_secret (basically a_0 from the polynomial), path_elements[][], identity_path_index[]. Public inputs are: x (actually just the hash of a signal), epoch, rln_identifier. Outputs are: y (polynomial share/secret share), root of a Merkle Tree, and nullifier (which is basically internal_nullifier). RLN circuit consists of two checks: Membership in Merkle Tree Correctness of secret share","breadcrumbs":"Overview » Under the hood » Circuits » RLN core","id":"16","title":"RLN core"},"17":{"body":"To check membership in a Merkle Tree, we can simply use the previously described Merkle Tree gadget: ... component identity_commitment = CalculateIdentityCommitment(); identity_commitment.identity_secret <== identity_secret; var i; var j; component inclusionProof = MerkleTreeInclusionProof(n_levels); inclusionProof.leaf <== identity_commitment.out; for (i = 0; i < n_levels; i++) { for (j = 0; j < LEAVES_PER_PATH_LEVEL; j++) { inclusionProof.path_elements[i][j] <== path_elements[i][j]; } inclusionProof.path_index[i] <== identity_path_index[i]; } ... Here we are calculating the identity_commitment and passing it along with sibling leaves and binary representation of the position to a Merkle Tree gadget. It gives us the calculated root as an output, and we can put the constraint on that: root <== inclusionProof.root;","breadcrumbs":"Overview » Under the hood » Circuits » Membership in Merkle Tree","id":"17","title":"Membership in Merkle Tree"},"18":{"body":"As we use linear polynomial we need to check that y = a_1 * x + a_0 (a_0 is identity secret). For that, we need to calculate external_nullifier and constraints on a_1 and secret share: ... component external_nullifier = CalculateExternalNullifier(); external_nullifier.epoch <== epoch; external_nullifier.rln_identifier <== rln_identifier; component a_1 = CalculateA1(); a_1.a_0 <== identity_secret; a_1.external_nullifier <== external_nullifier.out; y <== identity_secret + a_1.out * x; ... To calculate and reveal the nullifier: ... component calculateNullifier = CalculateInternalNullifier(); calculateNullifier.a_1 <== a_1.out; nullifier <== calculateNullifier.out; ...","breadcrumbs":"Overview » Under the hood » Circuits » Correctness of secret share","id":"18","title":"Correctness of secret share"},"19":{"body":"Now the Circuits can be used as gadgets. If we want to use it in our app, we need to initialize it and have a main - starting point function. It can be found here . The implementation is super basic: pragma circom 2.0.0; include \"./rln-base.circom\"; component main { public [x, epoch, rln_identifier] } = RLN(15); That's the whole RLN Circom Circuit :) Here we just need to list all public inputs (x, epoch, rln_identifier; the rest of the inputs are private). Also, we set the depth of the Merkle Tree = 15 (max of 32768 members).","breadcrumbs":"Overview » Under the hood » Circuits » Main runner of the circuits","id":"19","title":"Main runner of the circuits"},"2":{"body":"RLN is a zero-knowledge gadget that enables spam prevention for decentralized, anonymous environments. The anonymity property opens up the possibility for spam and Sybil attack vectors for certain applications, which could seriously degrade the user experience and the overall functioning of the application. For example, imagine a chat application where users are anonymous. Now, everyone can write an unlimited number of spam messages, but we don't have the ability to kick this member because the spammer is anonymous. RLN helps us identify and \"kick\" the spammer. Moreover, RLN can be useful not only to prevent spam attacks but, in general, to limit users (in anonymous environments) in the number of actions (f.e. to vote or to make a bid).","breadcrumbs":"Overview » What is RLN » What is Rate-Limiting Nullifier?","id":"2","title":"What is Rate-Limiting Nullifier?"},"20":{"body":"This section contains list of apps that use RLN : zk-chat - A spam resistant instant messaging application for private and anonymous communication rln-chat-app - PoC app, created using rln-js waku-rln-relay - Extension of waku-relay (spam protection with RLN )","breadcrumbs":"Overview » Uses » Uses","id":"20","title":"Uses"},"21":{"body":"This section provides information on how to use RLN in your project: JavaScript RLN (for rln-js ) Rust RLN (for zerokit-rln )","breadcrumbs":"How to use » How to use","id":"21","title":"How to use"},"22":{"body":"This section provides theoretical information that underpins RLN . Here we'll discuss: Shamir's Secret Sharing","breadcrumbs":"Theory » Theory","id":"22","title":"Theory"},"23":{"body":"This topic is an explanation of Shamir's Secret Sharing scheme ( SSS ), also known as \\((k, n)\\) threshold secret sharing scheme. SSS is one of the critical parts of RLN .","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"23","title":"Shamir's Secret Sharing Scheme"},"24":{"body":"Imagine if you have some important secret (secret key) and you don't want to store it anywhere. For that, you can use the SSS scheme. It allows you to split this secret into \\(n\\) parts (each individual part doesn't give any information about the secret) and restore this secret upon presentation of \\(k\\) \\((k <= n)\\) parts. For example, you have a secret that you want to split into \\(n\\) parts/shares. You can divide these shares between your friends (1 share to 1 friend). Now when \\(k\\) of your friends reveal their share, you can restore the secret. This scheme is also called \\((k, n)\\) threshold secret sharing scheme . This scheme is possible due to polynomial interpolation (especially Lagrange interpolation). Let's describe how Lagrange interpolation works and how it's used in a SSS scheme.","breadcrumbs":"Theory » Shamir's Secret Sharing » Overview","id":"24","title":"Overview"},"25":{"body":"Interpolation is a method of constructing (or restoring) new points/values (or function) based on the range of a set of known points/values (f.e. we can restore the line (linear function) from two points that are from this line). The previous example describes how that works. An unlimited number of parabolas (second-degree polynomials) can be drawn through two points. To choose the only one, you need a third point. Thus, if we have a polynomial \\(f(x) = 3x + 2\\), we only need two points from this polynomial to restore it. Let's peek two random \\(x\\) values and calculate \\(f(x)\\): For \\(x = 1\\) we have \\(f(1) = 3 * 1 + 2 = 5\\) For \\(x = 10\\) we have \\(f(10) = 32\\) Now we have to shares: \\((1, 5)\\) and \\((10, 32)\\). If we draw a graph based on these two shares, we can easily see that this is the same line (function): We also can \"restore\" the function analytically. For that let's denote: \\[f(x) = y_1 * \\frac{x - x_2}{x_1 - x_2} + y_2 * \\frac{x - x_1}{x_2 - x_1}\\] where \\(x_1 = 5, x_2 = 10, y_1 = 5, y_2 = 32\\). If we make substitution we got: \\[f(x) = 3x + 2 \\] which is the same polynomial. The same technique can be made with every polynomial. Main thing to remember is that we need \\(n + 1\\) points to interpolate \\(n\\)-degree polynomial. Now that we know how interpolation works, we can learn how it is used in SSS.","breadcrumbs":"Theory » Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"25","title":"Polynomial (Lagrange) interpolation"},"26":{"body":"To create the SSS construct, we must choose \\((k, n)\\), where \\(n\\) is the number of shares we want to get from the secret and \\(k\\) is the number of shares required to restore the secret. The degree of the \"secret\" polynomial is \\(k - 1\\) (covered in the previous section). Let's try to construct SSS with an example.","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing","id":"26","title":"Shamir's Secret Sharing"},"27":{"body":"Our secret = \\(S = 30\\) As the linear polynomial used in current RLN implementations, let's set \\(k = 2\\) (2 points are enough to recover the polynomial); \\(n\\) is not that important, but we can make it any number, f.e. 3 The secret polynomial is: \\[f(x) = a_1 * x + a_0 \\] where zero coefficient \\(a_0 = S\\), and \\(a_1\\) is some random number (f.e. 5); We must pick \\(n = 3\\) different points (shares) on that polynomial, for that we can pick three random \\(x\\) values (f.e. 5, 8, 16) and calculate \\(f(x)\\): \\[f(5) = 5 * 5 + 30 = 55\\] \\[f(8) = 5 * 8 + 30 = 70 \\] \\[f(16) = 5 * 16 + 30 = 110 \\] So, the shares are: \\((5, 55), (8, 70), (16, 110)\\)","breadcrumbs":"Theory » Shamir's Secret Sharing » Sharing","id":"27","title":"Sharing"},"28":{"body":"We can take any two shares to recover (as described in the interpolation section) the \"secret\" polynomial. Zero coefficient (\\(a_0\\)) in the recovered polynomial is the secret \\(S\\).","breadcrumbs":"Theory » Shamir's Secret Sharing » Recovering","id":"28","title":"Recovering"},"29":{"body":"Arithmetic in this topic is usual for us. However, in real life, SSS arithmetic is defined over some finite field. This means that all calculations are carried out modulo some big prime field. In fact, it happens by itself in Circom because the arithmetic there is defined over the finite field, too, so we don't need to do anything extra).","breadcrumbs":"Theory » Shamir's Secret Sharing » Important notes","id":"29","title":"Important notes"},"3":{"body":"The RLN construct's functionality consists of three parts, which, when integrated together, provide spam and Sybil attack protection. These parts should be integrated by the upstream applications, which require anonymity and spam protection. The applications can be centralized or decentralized. For decentralized applications, each user maintains separate storage and compute resources for the application. The three parts are: User registration User interaction User removal (slashing)","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"The following sections contain reference material you may find useful: Terminology References","breadcrumbs":"Appendix » Appendix","id":"30","title":"Appendix"},"31":{"body":"Term Description zkSNARK Proof construction where one can prove possession of certain information, e.g. a secret key, without revealing that information, and without any interaction between the prover and verifier. Circuit A program, that describes constraints for the prover in zkSNARK (for more information read this ). zk-Gadget Circuit, that can be used as a building block for another circuit, e.g. Poseidon hash function gadget. Stake Financial or social stake required for registering in the RLN applications. Common stake examples are: locking cryptocurrency (financial), linking reputable social identity. Identity secret Random number, which must be kept private by the user. Identity commitment The result of Poseidon(Identity secret) calculation. It is used by the users for registering in the protocol. Signal The message generated by a user. It is an arbitrary bit string that may represent a chat message, a URL request, protobuf message, etc. Signal hash Hash of the signal, used as an input in the RLN circuit. RLN Identifier Random finite field value unique per RLN app. It is used for additional cross-application security. The role of the RLN identifier is protection of the user secrets being compromised if signals are being generated with the same credentials at different apps. RLN membership tree Merkle tree data structure, filled with identity commitments of the users. Serves as a data structure that ensures user registrations. Merkle proof Proof that a user is member of the RLN membership tree.","breadcrumbs":"Appendix » A - Terminology » Terminology","id":"31","title":"Terminology"},"32":{"body":"First Proposal/Idea of RLN by Barry WhiteHat RLN Overview by Blagoj Demo RLN Spec VAC RLN Spec Understand zkSNARK Circom Docs rln-js zerokit-rln Incremental Merkle Tree paper","breadcrumbs":"Appendix » B - References » References","id":"32","title":"References"},"4":{"body":"Before registering to the application, the user needs to generate a secret key and derive an identity commitment from the secret key using the Poseidon hash function identityCommitment = posseidonHash(secretKey). The user registers to the application by providing a form of stake and their identity commitment, which is derived from the secret key. The application maintains a Merkle tree data structure (in the latest iteration of RLN , we use an Incremental Merkle Tree algorithm for gas efficiency, but the Merkle tree does not have to be on-chain), which stores the identity commitments of the registered users. Upon successful registration, the user's identity commitment is stored in a leaf of the Merkle tree, and an index is given to them, representing their position in the tree.","breadcrumbs":"Overview » What is RLN » User registration","id":"4","title":"User registration"},"5":{"body":"For each interaction that the user wants to make with the application, the user must generate a zero-knowledge proof ensuring that their identity commitment is part of the membership Merkle tree. There are a number of use-cases for RLN , such as voting applications (1 vote per election), chat (one message per second), and rate-limiting cache access (CDN denial of service protection). The verifier can be a server for centralized applications or the other users for decentralized applications. The general anti-spam rule is usually in the form of: Users must not make more than X interactions per epoch. The epoch can be translated as a time interval of Y units of time unit Z. For simplicity's sake, let's transform the rule into: `Users must not send more than one message per second. We can implement this using Shamir's Secret Sharing scheme ( read more ), which allows you to split a secret (f.e. to n parts) and recover it when any m of n parts (m <= n) are presented. Thus, users have to split their secret_key into n parts, and for every interaction, they have to reveal the new part of the secret_key. So, in addition to proving the membership in the Merkle Tree , users have to prove that the revealed part is truly the part of their secret_key. If they make more interactions than allowed per epoch, their secret key can be fully reconstructed.","breadcrumbs":"Overview » What is RLN » User interaction","id":"5","title":"User interaction"},"6":{"body":"The final property of the RLN mechanism is that it allows for the users to be removed from the membership tree by anyone that knows their secret key. The membership tree contains the identity commitments of all registered users. Users' identity commitment is derived from their secret key, and the secret key of the user is only revealed in a spam event (except for the scenarios where the original users want to remove themselves, which they can always do because they know their secret key). When an economic stake is present, the RLN mechanism can be implemented in a way that the spammer's stake is sent to the first user that correctly reports the spammer by providing the reconstructed secret key of the spammer as proof.","breadcrumbs":"Overview » What is RLN » User removal (slashing)","id":"6","title":"User removal (slashing)"},"7":{"body":"This section provides deep and technical RLN overview. We'll discuss: Technical side of RLN (specification demo) How circuits are implemented","breadcrumbs":"Overview » Under the hood » Under the hood","id":"7","title":"Under the hood"},"8":{"body":"This topic is a less strict version of specifications. If you want a more formal description, you can find specs in the references . Also, if you're unfamiliar with Shamir's Secret Sharing scheme, you can read it here . alt text Under the hood: The RLN Circom Circuit RLN consists of three parts: User registration User interaction (signaling) User removal (slashing) - additional part Well, let's discuss them.","breadcrumbs":"Overview » Under the hood » Protocol spec » Technical side of RLN","id":"8","title":"Technical side of RLN"},"9":{"body":"The first part of RLN is registration. There is nothing special in RLN registration; it's almost the same process as in other protocols/apps with anonymous environments: we need to create a Merkle Tree, and every participant must submit a commitment and place it in the Merkle Tree, and after that to interact with the app every participant will create a zkProof's, that they are a member of the tree (we use an Incremental Merkle Tree , as it more GAS efficient ). So, each member generates a secret key, denoted by \\(a_0\\). 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-{"doc_urls":["rln.html#rln","overview.html#overview","what_is_rln.html#what-is-rate-limiting-nullifier","what_is_rln.html#how-it-works","what_is_rln.html#user-registration","what_is_rln.html#user-interaction","what_is_rln.html#user-removal-slashing","under_the_hood.html#under-the-hood","protocol_spec.html#technical-side-of-rln","protocol_spec.html#user-registration","protocol_spec.html#signalling","protocol_spec.html#slashing","protocol_spec.html#some-important-notes","circuits.html#circuits","circuits.html#merkle-tree-circuit","circuits.html#rln-core","circuits.html#membership-in-merkle-tree","circuits.html#correctness-of-secret-share","circuits.html#main-runner-of-the-circuits","uses.html#uses","how_to_use.html#how-to-use","theory.html#theory","sss.html#shamirs-secret-sharing-scheme","sss.html#overview","sss.html#polynomial-lagrange-interpolation","sss.html#shamirs-secret-sharing","sss.html#sharing","sss.html#recovering","sss.html#important-notes","appendix.html#appendix","terminology.html#terminology","references.html#references"],"index":{"documentStore":{"docInfo":{"0":{"body":35,"breadcrumbs":2,"title":1},"1":{"body":19,"breadcrumbs":2,"title":1},"10":{"body":64,"breadcrumbs":6,"title":1},"11":{"body":12,"breadcrumbs":6,"title":1},"12":{"body":163,"breadcrumbs":7,"title":2},"13":{"body":35,"breadcrumbs":5,"title":1},"14":{"body":209,"breadcrumbs":7,"title":3},"15":{"body":171,"breadcrumbs":6,"title":2},"16":{"body":56,"breadcrumbs":7,"title":3},"17":{"body":37,"breadcrumbs":7,"title":3},"18":{"body":55,"breadcrumbs":7,"title":3},"19":{"body":34,"breadcrumbs":3,"title":1},"2":{"body":74,"breadcrumbs":5,"title":3},"20":{"body":14,"breadcrumbs":2,"title":1},"21":{"body":12,"breadcrumbs":2,"title":1},"22":{"body":19,"breadcrumbs":8,"title":4},"23":{"body":78,"breadcrumbs":5,"title":1},"24":{"body":135,"breadcrumbs":7,"title":3},"25":{"body":30,"breadcrumbs":7,"title":3},"26":{"body":79,"breadcrumbs":5,"title":1},"27":{"body":16,"breadcrumbs":5,"title":1},"28":{"body":32,"breadcrumbs":6,"title":2},"29":{"body":9,"breadcrumbs":2,"title":1},"3":{"body":43,"breadcrumbs":3,"title":1},"30":{"body":156,"breadcrumbs":3,"title":1},"31":{"body":26,"breadcrumbs":4,"title":1},"4":{"body":69,"breadcrumbs":4,"title":2},"5":{"body":136,"breadcrumbs":4,"title":2},"6":{"body":64,"breadcrumbs":5,"title":3},"7":{"body":15,"breadcrumbs":5,"title":2},"8":{"body":44,"breadcrumbs":8,"title":3},"9":{"body":119,"breadcrumbs":7,"title":2}},"docs":{"0":{"body":"RLN (Rate-Limiting Nullifier) is a zk-gadget/protocol that enables spam prevention mechanism for anonymous environments. RLN is part of ( PSE ) Privacy & Scaling Explorations , a multidisciplinary team supported by the Ethereum Foundation. PSE explores new use cases for zero-knowledge proofs and other cryptographic primitives. alt text","breadcrumbs":"RLN » RLN","id":"0","title":"RLN"},"1":{"body":"This section is a starting point for understanding the concepts of RLN . Here we'll discuss: Basic explanation of the RLN protocol RLN protocol under the hood RLN uses","breadcrumbs":"Overview » Overview","id":"1","title":"Overview"},"10":{"body":"Now that the user is registered, he wants to interact with the system. Imagine that the system is an anonymous chat and the interaction is the sending of messages. So, to send a message user have to come up with share - the point \\((x, y)\\) on her polynomial. We denote: \\(x = Poseidon(message), y = A(x)\\). Thus, if the same epoch user sends more than one message, their polynomial and, therefore, their secret (\\(a_0\\)) can be recovered. Of course, we somehow must prove that our share = \\((x, y)\\) is valid (that this is really a point on our polynomial = A(x)), as well as we must prove other things are valid too, that's why we use zkSNARK. An explanation of the zk-circuits can be found in the next topic.","breadcrumbs":"Overview » Under the hood » Protocol spec » Signalling","id":"10","title":"Signalling"},"11":{"body":"As it's been said, if a user sends more than one message, everyone else will be able to recover his secret, slash them and take their stake.","breadcrumbs":"Overview » Under the hood » Protocol spec » Slashing","id":"11","title":"Slashing"},"12":{"body":"There are also nullifier and rln-identifier, which can be found in the RLN protocol/circuits. So, rln-identifier is just a random value that's unique per RLN app. It's used for additional cross-application security - to protect the user secrets from being compromised if they use the same credentials across different RLN apps. If rln-identifier is not present, the user uses the same credentials and sends a message in two different RLN apps using the same epoch, then their secret key can be revealed. Adding the rln-identifier field, we obscure the nullifier, so this kind of attack cannot happen. The only kind of attack that is possible is if we have an entity with a global view of all messages, and they try to brute-force different combinations of x and y shares for different nullifiers. Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received shares and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public nullifier (\\(Poseidon(a_1, rln-identifier)\\)), so if a user sends more than one message, it will be immediately visible to everyone. Also, in our example (and zk-chat implementation), we use linear polynomial, but SSS allows us to use various degree polynomials; therefore we can implement a protocol, where more than one signal (message) can be sent in per epoch. To learn more, check out the specification ; there are also circuits implemented for various degree polynomials too.","breadcrumbs":"Overview » Under the hood » Protocol spec » Some important notes","id":"12","title":"Some important notes"},"13":{"body":"zkSNARK is used in the RLN core. Therefore, we must represent the protocol in R1CS (as we use Groth16 ). Circom was chosen for this. This section explains RLN circuits for the linear polynomial case (one message per epoch). You can find implementation for the general case here RLN circuits implement the logic described in previous topic .","breadcrumbs":"Overview » Under the hood » Circuits » Circuits","id":"13","title":"Circuits"},"14":{"body":"One of the critical components of RLN is the Incremental Merkle Tree for the membership tree. Any Merkle tree can be used, but we have chosen the Incremental Merkle Tree for gas efficiency. Let's look at the implementation . At the beginning of the file, we denote that we use Circom 2.0 and include two helper zk-gadgets : pragma circom 2.0.0; include \"../node_modules/circomlib/circuits/poseidon.circom\";\ninclude \"../node_modules/circomlib/circuits/mux1.circom\"; Poseidon gadget is just the implementation of the Poseidon hash function; the mux1 gadget will be described later. Next, we can see two implemented gadgets: template PoseidonHashT3() { var nInputs = 2; signal input inputs[nInputs]; signal output out; component hasher = Poseidon(nInputs); for (var i = 0; i < nInputs; i ++) { hasher.inputs[i] <== inputs[i]; } out <== hasher.out;\n} template HashLeftRight() { signal input left; signal input right; signal output hash; component hasher = PoseidonHashT3(); left ==> hasher.inputs[0]; right ==> hasher.inputs[1]; hash <== hasher.out;\n} These are helper gadgets to make the code more clean. Poseidon gadget is implemented with the ability to take a different number of arguments. We use PoseidonHashT3() to initialize it like a function with two arguments. And HashLeftRight use PoseidonHashT3 in a more \"readable\" way: it takes two inputs, left and right, and outputs the result of the calculation. Next comes the core of the Merkle Tree gadget: template MerkleTreeInclusionProof(n_levels) { signal input leaf; signal input path_index[n_levels]; signal input path_elements[n_levels][1]; signal output root; component hashers[n_levels]; component mux[n_levels]; signal levelHashes[n_levels + 1]; levelHashes[0] <== leaf; ... root <== levelHashes[n_levels];\n} Here we have three inputs: leaf, path_index, and path_elements. path_index is the position of the leaf represented in binary. We need the binary representation of the position in the Merkle tree to understand the hashing path from the leaf to the root (more on that \"3. Recursive Incremental Merkle Tree Algorithm, page 4\" ). path_elements are sibling leaves that are part of Merkle Proof. leaf = Poseidon(identity_secret), so it's just identity commitment . There is a Merkle Tree hashing algorithm in the omitted part, no more than that.","breadcrumbs":"Overview » Under the hood » Circuits » Merkle Tree circuit","id":"14","title":"Merkle Tree circuit"},"15":{"body":"RLN circuit is the implementation of RLN logic itself (which in turn uses the Merkle Tree gadget). You can find the implementation here . So, let's start with helper gadgets: template CalculateIdentityCommitment() { signal input identity_secret; signal output out; component hasher = Poseidon(1); hasher.inputs[0] <== identity_secret; out <== hasher.out;\n} template CalculateA1() { signal input a_0; signal input epoch; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== a_0; hasher.inputs[1] <== epoch; out <== hasher.out;\n} template CalculateNullifier() { signal input a_1; signal input rln_identifier; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== a_1; hasher.inputs[1] <== rln_identifier; out <== hasher.out;\n} It's easy to understand these samples: CalculateIdentityCommitment() is used to calculate the identity commitment. It takes secret and outputs the commitment. CalculateA1() and CalculateNullifier() are used to calculate a_1 and nullifier (internal nullifier); they are implemented as it's described in previous topic . Now, let's look at the core logic of the RLN circuit. ... signal input identity_secret; signal input path_elements[n_levels][LEAVES_PER_PATH_LEVEL]; signal input identity_path_index[n_levels]; signal input x; signal input epoch; signal input rln_identifier; signal output y; signal output root; signal output nullifier; ... So, here we have many inputs. Private inputs are: identity_secret (basically a_0 from the polynomial), path_elements[][], identity_path_index[]. Public inputs are: x (actually just the hash of a signal), epoch, rln_identifier. Outputs are: y' (share of the secret), rootof a Merkle Tree, andnullifier.` RLN circuit consists of two checks: Membership in Merkle Tree Correctness of secret share","breadcrumbs":"Overview » Under the hood » Circuits » RLN core","id":"15","title":"RLN core"},"16":{"body":"To check membership in a Merkle Tree, we can simply use the previously described Merkle Tree gadget: ... component identity_commitment = CalculateIdentityCommitment(); identity_commitment.identity_secret <== identity_secret; var i; var j; component inclusionProof = MerkleTreeInclusionProof(n_levels); inclusionProof.leaf <== identity_commitment.out; for (i = 0; i < n_levels; i++) { for (j = 0; j < LEAVES_PER_PATH_LEVEL; j++) { inclusionProof.path_elements[i][j] <== path_elements[i][j]; } inclusionProof.path_index[i] <== identity_path_index[i]; } ... Here we are calculating the identity_commitment and passing it along with sibling leaves and binary representation of the position to a Merkle Tree gadget. It gives us the calculated root as an output, and we can put the constraint on that: root <== inclusionProof.root;","breadcrumbs":"Overview » Under the hood » Circuits » Membership in Merkle Tree","id":"16","title":"Membership in Merkle Tree"},"17":{"body":"As we use linear polynomial we need to check that y = a_1 * x + a_0 (a_0 is identity secret). For that, we need these constraints: ... component a_1 = CalculateA1(); a_1.a_0 <== identity_secret; a_1.epoch <== epoch; y <== identity_secret + a_1.out * x; ... To calculate and reveal the nullifier: ... component calculateNullifier = CalculateNullifier(); calculateNullifier.a_1 <== a_1.out; calculateNullifier.rln_identifier <== rln_identifier; nullifier <== calculateNullifier.out; ...","breadcrumbs":"Overview » Under the hood » Circuits » Correctness of secret share","id":"17","title":"Correctness of secret share"},"18":{"body":"Now the Circuits can be used as gadgets. If we want to use it in our app, we need to initialize it and have a main - starting point function. It can be found here . The implementation is super basic: pragma circom 2.0.0; include \"./rln-base.circom\"; component main {public [x, epoch, rln_identifier ]} = RLN(15); That's the whole RLN Circom Circuit :) Here we just need to list all public inputs (x, epoch, rln_identifier; the rest of the inputs are private). Also, we set the depth of the Merkle Tree = 15 (max of 32768 members).","breadcrumbs":"Overview » Under the hood » Circuits » Main runner of the circuits","id":"18","title":"Main runner of the circuits"},"19":{"body":"This section contains list of apps that use RLN : zk-chat - A spam resistant instant messaging application for private and anonymous communication rln-chat-app - PoC app, created using rln-js waku-rln-relay - Extension of waku-relay (spam protection with RLN )","breadcrumbs":"Overview » Uses » Uses","id":"19","title":"Uses"},"2":{"body":"This topic is a part of complete overview by Blagoj . RLN is a zero-knowledge gadget that enables spam prevention for decentralized, anonymous environments. The anonymity property opens up the possibility for spam and Sybil attack vectors for certain applications, which could seriously degrade the user experience and the overall functioning of the application. For example, imagine a chat application where users are anonymous. Now, everyone can write an unlimited number of spam messages, but we don't have the ability to kick this member because the spammer is anonymous. RLN helps us identify and \"kick\" the spammer. Moreover, RLN can be useful not only to prevent spam attacks but, in general, to limit users (in anonymous environments) in the number of actions (f.e. to vote or to make a bid).","breadcrumbs":"Overview » What is RLN » What is Rate-Limiting Nullifier?","id":"2","title":"What is Rate-Limiting Nullifier?"},"20":{"body":"This section provides information on how to use RLN in your project: JavaScript RLN (for rln-js ) Rust RLN (for zerokit-rln )","breadcrumbs":"How to use » How to use","id":"20","title":"How to use"},"21":{"body":"This section provides theoretical information that underpins RLN . Here we'll discuss: Shamir's Secret Sharing","breadcrumbs":"Theory » Theory","id":"21","title":"Theory"},"22":{"body":"This topic is an explanation of Shamir's Secret Sharing scheme ( SSS ), also known as \\((k, n)\\) threshold secret sharing scheme. SSS is one of the critical parts of RLN .","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"22","title":"Shamir's Secret Sharing Scheme"},"23":{"body":"Imagine if you have some important secret (secret key) and you don't want to store it anywhere. For that, you can use the SSS scheme. It allows you to split this secret into \\(n\\) parts (each individual part doesn't give any information about the secret) and restore this secret upon presentation of \\(k\\) \\((k <= n)\\) parts. For example, you have a secret that you want to split into \\(n\\) parts/shares. You can divide these shares between your friends (1 share to 1 friend). Now when \\(k\\) of your friends reveal their share, you can restore the secret. This scheme is also called \\((k, n)\\) threshold secret sharing scheme . This scheme is possible due to polynomial interpolation (especially Lagrange interpolation). Let's describe how Lagrange interpolation works and how it's used in a SSS scheme.","breadcrumbs":"Theory » Shamir's Secret Sharing » Overview","id":"23","title":"Overview"},"24":{"body":"Interpolation is a method of constructing (or restoring) new points/values (or function) based on the range of a set of known points/values (f.e. we can restore the line (linear function) from two points that are from this line). The previous example describes how that works. An unlimited number of parabolas (second-degree polynomials) can be drawn through two points. To choose the only one, you need a third point. Thus, if we have a polynomial \\(f(x) = 3x + 2\\), we only need two points from this polynomial to restore it. Let's peek two random \\(x\\) values and calculate \\(f(x)\\): For \\(x = 1\\) we have \\(f(1) = 3 * 1 + 2 = 5\\) For \\(x = 10\\) we have \\(f(10) = 32\\) Now we have to shares: \\((1, 5)\\) and \\((10, 32)\\). If we draw a graph based on these two shares, we can easily see that this is the same line (function): We also can \"restore\" the function analytically. For that let's denote: \\[f(x) = y_1 * \\frac{x - x_2}{x_1 - x_2} + y_2 * \\frac{x - x_1}{x_2 - x_1}\\] where \\(x_1 = 5, x_2 = 10, y_1 = 5, y_2 = 32\\). If we make substitution we got: \\[f(x) = 3x + 2 \\] which is the same polynomial. The same technique can be made with every polynomial. Main thing to remember is that we need \\(n + 1\\) points to interpolate \\(n\\)-degree polynomial. Now that we know how interpolation works, we can learn how it is used in SSS.","breadcrumbs":"Theory » Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"24","title":"Polynomial (Lagrange) interpolation"},"25":{"body":"To create the SSS construct, we must choose \\((k, n)\\), where \\(n\\) is the number of shares we want to get from the secret and \\(k\\) is the number of shares required to restore the secret. The degree of the \"secret\" polynomial is \\(k - 1\\) (covered in the previous section). Let's try to construct SSS with an example.","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing","id":"25","title":"Shamir's Secret Sharing"},"26":{"body":"Our secret = \\(S = 30\\) As the linear polynomial used in current RLN implementations, let's set \\(k = 2\\) (2 points are enough to recover the polynomial); \\(n\\) is not that important, but we can make it any number, f.e. 3 The secret polynomial is: \\[f(x) = a_1 * x + a_0 \\] where zero coefficient \\(a_0 = S\\), and \\(a_1\\) is some random number (f.e. 5); We must pick \\(n = 3\\) different points (shares) on that polynomial, for that we can pick three random \\(x\\) values (f.e. 5, 8, 16) and calculate \\(f(x)\\): \\[f(5) = 5 * 5 + 30 = 55\\] \\[f(8) = 5 * 8 + 30 = 70 \\] \\[f(16) = 5 * 16 + 30 = 110 \\] So, the shares are: \\((5, 55), (8, 70), (16, 110)\\)","breadcrumbs":"Theory » Shamir's Secret Sharing » Sharing","id":"26","title":"Sharing"},"27":{"body":"We can take any two shares to recover (as described in the interpolation section) the \"secret\" polynomial. Zero coefficient (\\(a_0\\)) in the recovered polynomial is the secret \\(S\\).","breadcrumbs":"Theory » Shamir's Secret Sharing » Recovering","id":"27","title":"Recovering"},"28":{"body":"Arithmetic in this topic is usual for us. However, in real life, SSS arithmetic is defined over some finite field. This means that all calculations are carried out modulo some big prime field. In fact, it happens by itself in Circom because the arithmetic there is defined over the finite field, too, so we don't need to do anything extra).","breadcrumbs":"Theory » Shamir's Secret Sharing » Important notes","id":"28","title":"Important notes"},"29":{"body":"The following sections contain reference material you may find useful: Terminology References","breadcrumbs":"Appendix » Appendix","id":"29","title":"Appendix"},"3":{"body":"The RLN construct's functionality consists of three parts, which, when integrated together, provide spam and Sybil attack protection. These parts should be integrated by the upstream applications, which require anonymity and spam protection. The applications can be centralized or decentralized. For decentralized applications, each user maintains separate storage and compute resources for the application. The three parts are: User registration User interaction User removal (slashing)","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"Term Description zkSNARK Proof construction where one can prove possession of certain information, e.g. a secret key, without revealing that information, and without any interaction between the prover and verifier. Circuit A program, that describes constraints for the prover in zkSNARK (for more information read this ). zk-Gadget Circuit, that can be used as a building block for another circuit, e.g. Poseidon hash function gadget. Stake Financial or social stake required for registering in the RLN applications. Common stake examples are: locking cryptocurrency (financial), linking reputable social identity. Identity secret Random number, which must be kept private by the user. Identity commitment The result of Poseidon(Identity secret) calculation. It is used by the users for registering in the protocol. Signal The message generated by a user. It is an arbitrary bit string that may represent a chat message, a URL request, protobuf message, etc. Signal hash Hash of the signal, used as an input in the RLN circuit. RLN Identifier Random finite field value unique per RLN app. It is used for additional cross-application security. The role of the RLN identifier is protection of the user secrets being compromised if signals are being generated with the same credentials at different apps. RLN membership tree Merkle tree data structure, filled with identity commitments of the users. Serves as a data structure that ensures user registrations. Merkle proof Proof that a user is member of the RLN membership tree.","breadcrumbs":"Appendix » A - Terminology » Terminology","id":"30","title":"Terminology"},"31":{"body":"First Proposal/Idea of RLN by Barry WhiteHat RLN Overview by Blagoj Demo RLN Spec VAC RLN Spec Understand zkSNARK Circom Docs rln-js zerokit-rln Incremental Merkle Tree paper","breadcrumbs":"Appendix » B - References » References","id":"31","title":"References"},"4":{"body":"Before registering to the application, the user needs to generate a secret key and derive an identity commitment from the secret key using the Poseidon hash function identityCommitment = posseidonHash(secretKey). The user registers to the application by providing a form of stake and their identity commitment, which is derived from the secret key. The application maintains a Merkle tree data structure (in the latest iteration of RLN , we use an Incremental Merkle Tree algorithm for gas efficiency, but the Merkle tree does not have to be on-chain), which stores the identity commitments of the registered users. Upon successful registration, the user's identity commitment is stored in a leaf of the Merkle tree, and an index is given to them, representing their position in the tree.","breadcrumbs":"Overview » What is RLN » User registration","id":"4","title":"User registration"},"5":{"body":"For each interaction that the user wants to make with the application, the user must generate a zero-knowledge proof ensuring that their identity commitment is part of the membership Merkle tree. There are a number of use-cases for RLN , such as voting applications (1 vote per election), chat (one message per second), and rate-limiting cache access (CDN denial of service protection). The verifier can be a server for centralized applications or the other users for decentralized applications. The general anti-spam rule is usually in the form of: Users must not make more than X interactions per epoch. The epoch can be translated as a time interval of Y units of time unit Z. For simplicity's sake, let's transform the rule into: `Users must not send more than one message per second. We can implement this using Shamir's Secret Sharing scheme ( read more ), which allows you to split a secret (f.e. to n parts) and recover it when any m of n parts (m <= n) are presented. Thus, users have to split their secret_key into n parts, and for every interaction, they have to reveal the new part of the secret_key. So, in addition to proving the membership in the Merkle Tree , users have to prove that the revealed part is truly the part of their secret_key. If they make more interactions than allowed per epoch, their secret key can be fully reconstructed.","breadcrumbs":"Overview » What is RLN » User interaction","id":"5","title":"User interaction"},"6":{"body":"The final property of the RLN mechanism is that it allows for the users to be removed from the membership tree by anyone that knows their secret key. The membership tree contains the identity commitments of all registered users. Users' identity commitment is derived from their secret key, and the secret key of the user is only revealed in a spam event (except for the scenarios where the original users want to remove themselves, which they can always do because they know their secret key). When an economic stake is present, the RLN mechanism can be implemented in a way that the spammer's stake is sent to the first user that correctly reports the spammer by providing the reconstructed secret key of the spammer as proof.","breadcrumbs":"Overview » What is RLN » User removal (slashing)","id":"6","title":"User removal (slashing)"},"7":{"body":"This section provides deep and technical RLN overview. We'll discuss: Technical side of RLN (specification demo) How circuits are implemented","breadcrumbs":"Overview » Under the hood » Under the hood","id":"7","title":"Under the hood"},"8":{"body":"This topic is a less strict version of specifications. If you want a more formal description, you can find specs in the references . Also, if you're unfamiliar with Shamir's Secret Sharing scheme, you can read it here . alt text Under the hood: The RLN Circom Circuit RLN consists of three parts: User registration User interaction (signaling) User removal (slashing) - additional part Well, let's discuss them.","breadcrumbs":"Overview » Under the hood » Protocol spec » Technical side of RLN","id":"8","title":"Technical side of RLN"},"9":{"body":"The first part of RLN is registration. There is nothing special in RLN registration; it's almost the same process as in other protocols/apps with anonymous environments: we need to create a Merkle Tree, and every participant must submit a commitment and place it in the Merkle Tree, and after that to interact with the app every participant will create a zkProof's, that they are a member of the tree (we use an Incremental Merkle Tree , as it more GAS efficient ). So, each member generates a secret key, denoted by \\(a_0\\). Identity commitment \\(q\\) is the hash (Poseidon) of the secret key: \\(q = Poseidon(a_0)\\). RLN wouldn't work if there were no punishment for spam; that's why to become a member, a user has to register and provide something at stake. So, whoever has our \\(a_0\\) can \"slash\" us. The slight difference is that we must enable a secret sharing scheme (to split the commitment into parts). We need to come up with a polynomial. For simplicity we use linear polynomial (e.g. \\(f(x) = kx + b\\). Therefore, with two points, we can reconstruct the polynomial and recover the secret. Our polynomial will be: \\(A(x) = a_1 * x + a_0\\), where \\(a_1 = Poseidon(a_0, epoch)\\). epoch is a simple identifier (also called external nullifier ). 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+{"doc_urls":["rln.html#rln","overview.html#overview","what_is_rln.html#what-is-rate-limiting-nullifier","what_is_rln.html#how-it-works","what_is_rln.html#user-registration","what_is_rln.html#user-interaction","what_is_rln.html#user-removal-slashing","under_the_hood.html#under-the-hood","protocol_spec.html#technical-side-of-rln","protocol_spec.html#user-registration","protocol_spec.html#signalling","protocol_spec.html#slashing","protocol_spec.html#nullifiers","protocol_spec.html#some-important-notes","circuits.html#circuits","circuits.html#merkle-tree-circuit","circuits.html#rln-core","circuits.html#membership-in-merkle-tree","circuits.html#correctness-of-secret-share","circuits.html#main-runner-of-the-circuits","uses.html#uses","how_to_use.html#how-to-use","theory.html#theory","sss.html#shamirs-secret-sharing-scheme","sss.html#overview","sss.html#polynomial-lagrange-interpolation","sss.html#shamirs-secret-sharing","sss.html#sharing","sss.html#recovering","sss.html#important-notes","appendix.html#appendix","terminology.html#terminology","references.html#references"],"index":{"documentStore":{"docInfo":{"0":{"body":35,"breadcrumbs":2,"title":1},"1":{"body":19,"breadcrumbs":2,"title":1},"10":{"body":64,"breadcrumbs":6,"title":1},"11":{"body":12,"breadcrumbs":6,"title":1},"12":{"body":92,"breadcrumbs":6,"title":1},"13":{"body":33,"breadcrumbs":7,"title":2},"14":{"body":35,"breadcrumbs":5,"title":1},"15":{"body":209,"breadcrumbs":7,"title":3},"16":{"body":178,"breadcrumbs":6,"title":2},"17":{"body":56,"breadcrumbs":7,"title":3},"18":{"body":47,"breadcrumbs":7,"title":3},"19":{"body":55,"breadcrumbs":7,"title":3},"2":{"body":69,"breadcrumbs":5,"title":3},"20":{"body":34,"breadcrumbs":3,"title":1},"21":{"body":14,"breadcrumbs":2,"title":1},"22":{"body":12,"breadcrumbs":2,"title":1},"23":{"body":19,"breadcrumbs":8,"title":4},"24":{"body":78,"breadcrumbs":5,"title":1},"25":{"body":135,"breadcrumbs":7,"title":3},"26":{"body":30,"breadcrumbs":7,"title":3},"27":{"body":79,"breadcrumbs":5,"title":1},"28":{"body":16,"breadcrumbs":5,"title":1},"29":{"body":32,"breadcrumbs":6,"title":2},"3":{"body":43,"breadcrumbs":3,"title":1},"30":{"body":9,"breadcrumbs":2,"title":1},"31":{"body":156,"breadcrumbs":3,"title":1},"32":{"body":26,"breadcrumbs":4,"title":1},"4":{"body":69,"breadcrumbs":4,"title":2},"5":{"body":136,"breadcrumbs":4,"title":2},"6":{"body":64,"breadcrumbs":5,"title":3},"7":{"body":15,"breadcrumbs":5,"title":2},"8":{"body":44,"breadcrumbs":8,"title":3},"9":{"body":110,"breadcrumbs":7,"title":2}},"docs":{"0":{"body":"RLN (Rate-Limiting Nullifier) is a zk-gadget/protocol that enables spam prevention mechanism for anonymous environments. RLN is part of ( PSE ) Privacy & Scaling Explorations , a multidisciplinary team supported by the Ethereum Foundation. PSE explores new use cases for zero-knowledge proofs and other cryptographic primitives. alt text","breadcrumbs":"RLN » RLN","id":"0","title":"RLN"},"1":{"body":"This section is a starting point for understanding the concepts of RLN . Here we'll discuss: Basic explanation of the RLN protocol RLN protocol under the hood RLN uses","breadcrumbs":"Overview » Overview","id":"1","title":"Overview"},"10":{"body":"Now that the user is registered, he wants to interact with the system. Imagine that the system is an anonymous chat and the interaction is the sending of messages. So, to send a message user have to come up with share - the point \\((x, y)\\) on her polynomial. We denote: \\(x = Poseidon(message), y = A(x)\\). Thus, if the same epoch user sends more than one message, their polynomial and, therefore, their secret (\\(a_0\\)) can be recovered. Of course, we somehow must prove that our share = \\((x, y)\\) is valid (that this is really a point on our polynomial = A(x)), as well as we must prove other things are valid too, that's why we use zkSNARK. An explanation of the zk-circuits can be found in the next topic.","breadcrumbs":"Overview » Under the hood » Protocol spec » Signalling","id":"10","title":"Signalling"},"11":{"body":"As it's been said, if a user sends more than one message, everyone else will be able to recover his secret, slash them and take their stake.","breadcrumbs":"Overview » Under the hood » Protocol spec » Slashing","id":"11","title":"Slashing"},"12":{"body":"There are also \\(internal\\_nullifier\\) and \\(external\\_nullifier\\), which can be found in the RLN protocol/circuits. \\(external\\_nullifier = Poseidon(epoch, rln\\_identifier)\\), where \\(rln\\_identifier\\) is a random finite field value, unique per RLN app. The \\(external\\_nullifier\\) is required so that the user can securely use the same private key \\(a_0\\) across different RLN apps - in different applications (and in different eras) with the same secret key, the user will have different values ​​of the coefficient \\(a_1\\). Now, imagine there are a lot of users sending messages, and after each received message, we need to check if any member can be slashed. To do this, we can use all combinations of received shares and try to recover the polynomial, but this is a naive and non-optimal approach. Suppose we have a mechanism that will tell us about the connection between a person and their messages while not revealing their identity. In that case, we can solve this without brute-forcing all possibilities by using a public \\(internal\\_nullifier = Poseidon(a_1)\\), so if a user sends more than one message, it will be immediately visible to everyone.","breadcrumbs":"Overview » Under the hood » Protocol spec » Nullifiers","id":"12","title":"Nullifiers"},"13":{"body":"Also, in our example (and zk-chat implementation), we use linear polynomial, but SSS allows us to use various degree polynomials; therefore we can implement a protocol, where more than one signal (message) can be sent in per epoch. To learn more, check out the specification ; there are also circuits implemented for various degree polynomials too.","breadcrumbs":"Overview » Under the hood » Protocol spec » Some important notes","id":"13","title":"Some important notes"},"14":{"body":"zkSNARK is used in the RLN core. Therefore, we must represent the protocol in R1CS (as we use Groth16 ). Circom was chosen for this. This section explains RLN circuits for the linear polynomial case (one message per epoch). You can find implementation for the general case here RLN circuits implement the logic described in previous topic .","breadcrumbs":"Overview » Under the hood » Circuits » Circuits","id":"14","title":"Circuits"},"15":{"body":"One of the critical components of RLN is the Incremental Merkle Tree for the membership tree. Any Merkle tree can be used, but we have chosen the Incremental Merkle Tree for gas efficiency. Let's look at the implementation . At the beginning of the file, we denote that we use Circom 2.0 and include two helper zk-gadgets : pragma circom 2.0.0; include \"../node_modules/circomlib/circuits/poseidon.circom\";\ninclude \"../node_modules/circomlib/circuits/mux1.circom\"; Poseidon gadget is just the implementation of the Poseidon hash function; the mux1 gadget will be described later. Next, we can see two implemented gadgets: template PoseidonHashT3() { var nInputs = 2; signal input inputs[nInputs]; signal output out; component hasher = Poseidon(nInputs); for (var i = 0; i < nInputs; i ++) { hasher.inputs[i] <== inputs[i]; } out <== hasher.out;\n} template HashLeftRight() { signal input left; signal input right; signal output hash; component hasher = PoseidonHashT3(); left ==> hasher.inputs[0]; right ==> hasher.inputs[1]; hash <== hasher.out;\n} These are helper gadgets to make the code more clean. Poseidon gadget is implemented with the ability to take a different number of arguments. We use PoseidonHashT3() to initialize it like a function with two arguments. And HashLeftRight use PoseidonHashT3 in a more \"readable\" way: it takes two inputs, left and right, and outputs the result of the calculation. Next comes the core of the Merkle Tree gadget: template MerkleTreeInclusionProof(n_levels) { signal input leaf; signal input path_index[n_levels]; signal input path_elements[n_levels][1]; signal output root; component hashers[n_levels]; component mux[n_levels]; signal levelHashes[n_levels + 1]; levelHashes[0] <== leaf; ... root <== levelHashes[n_levels];\n} Here we have three inputs: leaf, path_index, and path_elements. path_index is the position of the leaf represented in binary. We need the binary representation of the position in the Merkle tree to understand the hashing path from the leaf to the root (more on that \"3. Recursive Incremental Merkle Tree Algorithm, page 4\" ). path_elements are sibling leaves that are part of Merkle Proof. leaf = Poseidon(identity_secret), so it's just identity commitment . There is a Merkle Tree hashing algorithm in the omitted part, no more than that.","breadcrumbs":"Overview » Under the hood » Circuits » Merkle Tree circuit","id":"15","title":"Merkle Tree circuit"},"16":{"body":"RLN circuit is the implementation of RLN logic itself (which in turn uses the Merkle Tree gadget). You can find the implementation here . So, let's start with helper gadgets: template CalculateIdentityCommitment() { signal input identity_secret; signal output out; component hasher = Poseidon(1); hasher.inputs[0] <== identity_secret; out <== hasher.out;\n} template CalculateExternalNullifier() { signal input epoch; signal input rln_identifier; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== epoch; hasher.inputs[1] <== rln_identifier; out <== hasher.out;\n} template CalculateA1() { signal input a_0; signal input external_nullifier; signal output out; component hasher = Poseidon(2); hasher.inputs[0] <== a_0; hasher.inputs[1] <== external_nullifier; out <== hasher.out;\n} template CalculateInternalNullifier() { signal input a_1; signal output out; component hasher = Poseidon(1); hasher.inputs[0] <== a_1; out <== hasher.out;\n} It's easy to understand these samples: CalculateIdentityCommitment(), CalculateA1(), CalculateInternalNullifier(), CalculateExternalNullifier() - they do exactly what their name says; they are implemented as it's described in previous topic . Now, let's look at the core logic of the RLN circuit. ... signal input identity_secret; signal input path_elements[n_levels][LEAVES_PER_PATH_LEVEL]; signal input identity_path_index[n_levels]; signal input x; signal input epoch; signal input rln_identifier; signal output y; signal output root; signal output nullifier; ... So, here we have many inputs. Private inputs are: identity_secret (basically a_0 from the polynomial), path_elements[][], identity_path_index[]. Public inputs are: x (actually just the hash of a signal), epoch, rln_identifier. Outputs are: y (polynomial share/secret share), root of a Merkle Tree, and nullifier (which is basically internal_nullifier). RLN circuit consists of two checks: Membership in Merkle Tree Correctness of secret share","breadcrumbs":"Overview » Under the hood » Circuits » RLN core","id":"16","title":"RLN core"},"17":{"body":"To check membership in a Merkle Tree, we can simply use the previously described Merkle Tree gadget: ... component identity_commitment = CalculateIdentityCommitment(); identity_commitment.identity_secret <== identity_secret; var i; var j; component inclusionProof = MerkleTreeInclusionProof(n_levels); inclusionProof.leaf <== identity_commitment.out; for (i = 0; i < n_levels; i++) { for (j = 0; j < LEAVES_PER_PATH_LEVEL; j++) { inclusionProof.path_elements[i][j] <== path_elements[i][j]; } inclusionProof.path_index[i] <== identity_path_index[i]; } ... Here we are calculating the identity_commitment and passing it along with sibling leaves and binary representation of the position to a Merkle Tree gadget. It gives us the calculated root as an output, and we can put the constraint on that: root <== inclusionProof.root;","breadcrumbs":"Overview » Under the hood » Circuits » Membership in Merkle Tree","id":"17","title":"Membership in Merkle Tree"},"18":{"body":"As we use linear polynomial we need to check that y = a_1 * x + a_0 (a_0 is identity secret). For that, we need to calculate external_nullifier and constraints on a_1 and secret share: ... component external_nullifier = CalculateExternalNullifier(); external_nullifier.epoch <== epoch; external_nullifier.rln_identifier <== rln_identifier; component a_1 = CalculateA1(); a_1.a_0 <== identity_secret; a_1.external_nullifier <== external_nullifier.out; y <== identity_secret + a_1.out * x; ... To calculate and reveal the nullifier: ... component calculateNullifier = CalculateInternalNullifier(); calculateNullifier.a_1 <== a_1.out; nullifier <== calculateNullifier.out; ...","breadcrumbs":"Overview » Under the hood » Circuits » Correctness of secret share","id":"18","title":"Correctness of secret share"},"19":{"body":"Now the Circuits can be used as gadgets. If we want to use it in our app, we need to initialize it and have a main - starting point function. It can be found here . The implementation is super basic: pragma circom 2.0.0; include \"./rln-base.circom\"; component main { public [x, epoch, rln_identifier] } = RLN(15); That's the whole RLN Circom Circuit :) Here we just need to list all public inputs (x, epoch, rln_identifier; the rest of the inputs are private). Also, we set the depth of the Merkle Tree = 15 (max of 32768 members).","breadcrumbs":"Overview » Under the hood » Circuits » Main runner of the circuits","id":"19","title":"Main runner of the circuits"},"2":{"body":"RLN is a zero-knowledge gadget that enables spam prevention for decentralized, anonymous environments. The anonymity property opens up the possibility for spam and Sybil attack vectors for certain applications, which could seriously degrade the user experience and the overall functioning of the application. For example, imagine a chat application where users are anonymous. Now, everyone can write an unlimited number of spam messages, but we don't have the ability to kick this member because the spammer is anonymous. RLN helps us identify and \"kick\" the spammer. Moreover, RLN can be useful not only to prevent spam attacks but, in general, to limit users (in anonymous environments) in the number of actions (f.e. to vote or to make a bid).","breadcrumbs":"Overview » What is RLN » What is Rate-Limiting Nullifier?","id":"2","title":"What is Rate-Limiting Nullifier?"},"20":{"body":"This section contains list of apps that use RLN : zk-chat - A spam resistant instant messaging application for private and anonymous communication rln-chat-app - PoC app, created using rln-js waku-rln-relay - Extension of waku-relay (spam protection with RLN )","breadcrumbs":"Overview » Uses » Uses","id":"20","title":"Uses"},"21":{"body":"This section provides information on how to use RLN in your project: JavaScript RLN (for rln-js ) Rust RLN (for zerokit-rln )","breadcrumbs":"How to use » How to use","id":"21","title":"How to use"},"22":{"body":"This section provides theoretical information that underpins RLN . Here we'll discuss: Shamir's Secret Sharing","breadcrumbs":"Theory » Theory","id":"22","title":"Theory"},"23":{"body":"This topic is an explanation of Shamir's Secret Sharing scheme ( SSS ), also known as \\((k, n)\\) threshold secret sharing scheme. SSS is one of the critical parts of RLN .","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"23","title":"Shamir's Secret Sharing Scheme"},"24":{"body":"Imagine if you have some important secret (secret key) and you don't want to store it anywhere. For that, you can use the SSS scheme. It allows you to split this secret into \\(n\\) parts (each individual part doesn't give any information about the secret) and restore this secret upon presentation of \\(k\\) \\((k <= n)\\) parts. For example, you have a secret that you want to split into \\(n\\) parts/shares. You can divide these shares between your friends (1 share to 1 friend). Now when \\(k\\) of your friends reveal their share, you can restore the secret. This scheme is also called \\((k, n)\\) threshold secret sharing scheme . This scheme is possible due to polynomial interpolation (especially Lagrange interpolation). Let's describe how Lagrange interpolation works and how it's used in a SSS scheme.","breadcrumbs":"Theory » Shamir's Secret Sharing » Overview","id":"24","title":"Overview"},"25":{"body":"Interpolation is a method of constructing (or restoring) new points/values (or function) based on the range of a set of known points/values (f.e. we can restore the line (linear function) from two points that are from this line). The previous example describes how that works. An unlimited number of parabolas (second-degree polynomials) can be drawn through two points. To choose the only one, you need a third point. Thus, if we have a polynomial \\(f(x) = 3x + 2\\), we only need two points from this polynomial to restore it. Let's peek two random \\(x\\) values and calculate \\(f(x)\\): For \\(x = 1\\) we have \\(f(1) = 3 * 1 + 2 = 5\\) For \\(x = 10\\) we have \\(f(10) = 32\\) Now we have to shares: \\((1, 5)\\) and \\((10, 32)\\). If we draw a graph based on these two shares, we can easily see that this is the same line (function): We also can \"restore\" the function analytically. For that let's denote: \\[f(x) = y_1 * \\frac{x - x_2}{x_1 - x_2} + y_2 * \\frac{x - x_1}{x_2 - x_1}\\] where \\(x_1 = 5, x_2 = 10, y_1 = 5, y_2 = 32\\). If we make substitution we got: \\[f(x) = 3x + 2 \\] which is the same polynomial. The same technique can be made with every polynomial. Main thing to remember is that we need \\(n + 1\\) points to interpolate \\(n\\)-degree polynomial. Now that we know how interpolation works, we can learn how it is used in SSS.","breadcrumbs":"Theory » Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"25","title":"Polynomial (Lagrange) interpolation"},"26":{"body":"To create the SSS construct, we must choose \\((k, n)\\), where \\(n\\) is the number of shares we want to get from the secret and \\(k\\) is the number of shares required to restore the secret. The degree of the \"secret\" polynomial is \\(k - 1\\) (covered in the previous section). Let's try to construct SSS with an example.","breadcrumbs":"Theory » Shamir's Secret Sharing » Shamir's Secret Sharing","id":"26","title":"Shamir's Secret Sharing"},"27":{"body":"Our secret = \\(S = 30\\) As the linear polynomial used in current RLN implementations, let's set \\(k = 2\\) (2 points are enough to recover the polynomial); \\(n\\) is not that important, but we can make it any number, f.e. 3 The secret polynomial is: \\[f(x) = a_1 * x + a_0 \\] where zero coefficient \\(a_0 = S\\), and \\(a_1\\) is some random number (f.e. 5); We must pick \\(n = 3\\) different points (shares) on that polynomial, for that we can pick three random \\(x\\) values (f.e. 5, 8, 16) and calculate \\(f(x)\\): \\[f(5) = 5 * 5 + 30 = 55\\] \\[f(8) = 5 * 8 + 30 = 70 \\] \\[f(16) = 5 * 16 + 30 = 110 \\] So, the shares are: \\((5, 55), (8, 70), (16, 110)\\)","breadcrumbs":"Theory » Shamir's Secret Sharing » Sharing","id":"27","title":"Sharing"},"28":{"body":"We can take any two shares to recover (as described in the interpolation section) the \"secret\" polynomial. Zero coefficient (\\(a_0\\)) in the recovered polynomial is the secret \\(S\\).","breadcrumbs":"Theory » Shamir's Secret Sharing » Recovering","id":"28","title":"Recovering"},"29":{"body":"Arithmetic in this topic is usual for us. However, in real life, SSS arithmetic is defined over some finite field. This means that all calculations are carried out modulo some big prime field. In fact, it happens by itself in Circom because the arithmetic there is defined over the finite field, too, so we don't need to do anything extra).","breadcrumbs":"Theory » Shamir's Secret Sharing » Important notes","id":"29","title":"Important notes"},"3":{"body":"The RLN construct's functionality consists of three parts, which, when integrated together, provide spam and Sybil attack protection. These parts should be integrated by the upstream applications, which require anonymity and spam protection. The applications can be centralized or decentralized. For decentralized applications, each user maintains separate storage and compute resources for the application. The three parts are: User registration User interaction User removal (slashing)","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"The following sections contain reference material you may find useful: Terminology References","breadcrumbs":"Appendix » Appendix","id":"30","title":"Appendix"},"31":{"body":"Term Description zkSNARK Proof construction where one can prove possession of certain information, e.g. a secret key, without revealing that information, and without any interaction between the prover and verifier. Circuit A program, that describes constraints for the prover in zkSNARK (for more information read this ). zk-Gadget Circuit, that can be used as a building block for another circuit, e.g. Poseidon hash function gadget. Stake Financial or social stake required for registering in the RLN applications. Common stake examples are: locking cryptocurrency (financial), linking reputable social identity. Identity secret Random number, which must be kept private by the user. Identity commitment The result of Poseidon(Identity secret) calculation. It is used by the users for registering in the protocol. Signal The message generated by a user. It is an arbitrary bit string that may represent a chat message, a URL request, protobuf message, etc. Signal hash Hash of the signal, used as an input in the RLN circuit. RLN Identifier Random finite field value unique per RLN app. It is used for additional cross-application security. The role of the RLN identifier is protection of the user secrets being compromised if signals are being generated with the same credentials at different apps. RLN membership tree Merkle tree data structure, filled with identity commitments of the users. Serves as a data structure that ensures user registrations. Merkle proof Proof that a user is member of the RLN membership tree.","breadcrumbs":"Appendix » A - Terminology » Terminology","id":"31","title":"Terminology"},"32":{"body":"First Proposal/Idea of RLN by Barry WhiteHat RLN Overview by Blagoj Demo RLN Spec VAC RLN Spec Understand zkSNARK Circom Docs rln-js zerokit-rln Incremental Merkle Tree paper","breadcrumbs":"Appendix » B - References » References","id":"32","title":"References"},"4":{"body":"Before registering to the application, the user needs to generate a secret key and derive an identity commitment from the secret key using the Poseidon hash function identityCommitment = posseidonHash(secretKey). The user registers to the application by providing a form of stake and their identity commitment, which is derived from the secret key. The application maintains a Merkle tree data structure (in the latest iteration of RLN , we use an Incremental Merkle Tree algorithm for gas efficiency, but the Merkle tree does not have to be on-chain), which stores the identity commitments of the registered users. Upon successful registration, the user's identity commitment is stored in a leaf of the Merkle tree, and an index is given to them, representing their position in the tree.","breadcrumbs":"Overview » What is RLN » User registration","id":"4","title":"User registration"},"5":{"body":"For each interaction that the user wants to make with the application, the user must generate a zero-knowledge proof ensuring that their identity commitment is part of the membership Merkle tree. There are a number of use-cases for RLN , such as voting applications (1 vote per election), chat (one message per second), and rate-limiting cache access (CDN denial of service protection). The verifier can be a server for centralized applications or the other users for decentralized applications. The general anti-spam rule is usually in the form of: Users must not make more than X interactions per epoch. The epoch can be translated as a time interval of Y units of time unit Z. For simplicity's sake, let's transform the rule into: `Users must not send more than one message per second. We can implement this using Shamir's Secret Sharing scheme ( read more ), which allows you to split a secret (f.e. to n parts) and recover it when any m of n parts (m <= n) are presented. Thus, users have to split their secret_key into n parts, and for every interaction, they have to reveal the new part of the secret_key. So, in addition to proving the membership in the Merkle Tree , users have to prove that the revealed part is truly the part of their secret_key. If they make more interactions than allowed per epoch, their secret key can be fully reconstructed.","breadcrumbs":"Overview » What is RLN » User interaction","id":"5","title":"User interaction"},"6":{"body":"The final property of the RLN mechanism is that it allows for the users to be removed from the membership tree by anyone that knows their secret key. The membership tree contains the identity commitments of all registered users. Users' identity commitment is derived from their secret key, and the secret key of the user is only revealed in a spam event (except for the scenarios where the original users want to remove themselves, which they can always do because they know their secret key). When an economic stake is present, the RLN mechanism can be implemented in a way that the spammer's stake is sent to the first user that correctly reports the spammer by providing the reconstructed secret key of the spammer as proof.","breadcrumbs":"Overview » What is RLN » User removal (slashing)","id":"6","title":"User removal (slashing)"},"7":{"body":"This section provides deep and technical RLN overview. We'll discuss: Technical side of RLN (specification demo) How circuits are implemented","breadcrumbs":"Overview » Under the hood » Under the hood","id":"7","title":"Under the hood"},"8":{"body":"This topic is a less strict version of specifications. If you want a more formal description, you can find specs in the references . Also, if you're unfamiliar with Shamir's Secret Sharing scheme, you can read it here . alt text Under the hood: The RLN Circom Circuit RLN consists of three parts: User registration User interaction (signaling) User removal (slashing) - additional part Well, let's discuss them.","breadcrumbs":"Overview » Under the hood » Protocol spec » Technical side of RLN","id":"8","title":"Technical side of RLN"},"9":{"body":"The first part of RLN is registration. There is nothing special in RLN registration; it's almost the same process as in other protocols/apps with anonymous environments: we need to create a Merkle Tree, and every participant must submit a commitment and place it in the Merkle Tree, and after that to interact with the app every participant will create a zkProof's, that they are a member of the tree (we use an Incremental Merkle Tree , as it more GAS efficient ). So, each member generates a secret key, denoted by \\(a_0\\). 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