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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).
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+In addition to production-ready circom-rln version of RLN protocol, there is also wip on RLN R&D to reduce proving time and have bigger choice of implementations.
+For now you can follow the work on:
+
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-Object.assign(window.search, {"doc_urls":["rln.html","overview.html#overview","what_is_rln.html#what-is-rate-limiting-nullifier","what_is_rln.html#how-it-works","what_is_rln.html#registration","what_is_rln.html#interaction","what_is_rln.html#withdrawal-or-slashing","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","formal_spec.html#formal-spec-of-circom-rln","formal_spec.html#utils","formal_spec.html#merkletreeinclusionproof","formal_spec.html#rangecheck","formal_spec.html#rln","formal_spec.html#withdrawal","smart_contract.html#smart-contract","uses.html#uses","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","research.html#research","kzg_rln.html#kzg-rln","noir_rln.html#noir-rln"],"index":{"documentStore":{"docInfo":{"0":{"body":35,"breadcrumbs":1,"title":1},"1":{"body":19,"breadcrumbs":2,"title":1},"10":{"body":12,"breadcrumbs":4,"title":1},"11":{"body":92,"breadcrumbs":4,"title":1},"12":{"body":33,"breadcrumbs":5,"title":2},"13":{"body":5,"breadcrumbs":7,"title":4},"14":{"body":22,"breadcrumbs":4,"title":1},"15":{"body":82,"breadcrumbs":4,"title":1},"16":{"body":52,"breadcrumbs":4,"title":1},"17":{"body":124,"breadcrumbs":4,"title":1},"18":{"body":38,"breadcrumbs":4,"title":1},"19":{"body":0,"breadcrumbs":5,"title":2},"2":{"body":63,"breadcrumbs":5,"title":3},"20":{"body":39,"breadcrumbs":4,"title":1},"21":{"body":19,"breadcrumbs":7,"title":4},"22":{"body":78,"breadcrumbs":4,"title":1},"23":{"body":135,"breadcrumbs":6,"title":3},"24":{"body":30,"breadcrumbs":6,"title":3},"25":{"body":79,"breadcrumbs":4,"title":1},"26":{"body":16,"breadcrumbs":4,"title":1},"27":{"body":32,"breadcrumbs":5,"title":2},"28":{"body":0,"breadcrumbs":2,"title":1},"29":{"body":0,"breadcrumbs":5,"title":2},"3":{"body":33,"breadcrumbs":3,"title":1},"30":{"body":0,"breadcrumbs":5,"title":2},"4":{"body":71,"breadcrumbs":3,"title":1},"5":{"body":140,"breadcrumbs":3,"title":1},"6":{"body":31,"breadcrumbs":4,"title":2},"7":{"body":44,"breadcrumbs":6,"title":3},"8":{"body":110,"breadcrumbs":5,"title":2},"9":{"body":64,"breadcrumbs":4,"title":1}},"docs":{"0":{"body":"RLN Logo 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.","breadcrumbs":"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":"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 » RLN in details » Slashing","id":"10","title":"Slashing"},"11":{"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 » RLN in details » Nullifiers","id":"11","title":"Nullifiers"},"12":{"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 » RLN in details » Some important notes","id":"12","title":"Some important notes"},"13":{"body":"Utils MerkleTreeInclusionProof RangeCheck RLN Withdrawal","breadcrumbs":"Overview » Formal spec » Formal spec of circom-rln","id":"13","title":"Formal spec of circom-rln"},"14":{"body":"utils.circom is a set of templates/gadgets that the RLN circuit uses. These are: MerkleTreeInclusionProof - Merkle tree inclusion check, used like set membership check; RangeCheck - used for range check. Their description is given below.","breadcrumbs":"Overview » Formal spec » Utils","id":"14","title":"Utils"},"15":{"body":"MerkleTreeInclusionProof(DEPTH) template used for verification of inclusion in full binary incremental merkle tree. The implementation is a fork of https://github.com/privacy-scaling-explorations/incrementalquintree, and changed to binary tree and refactored to Circom 2.1.0 . Parameters : DEPTH - depth of the Merkle Tree. Inputs : \\(leaf\\) - \\(Poseidon(elem)\\), where \\(elem\\) is the element that's checked for inclusion; \\(pathIndex[DEPTH]\\) - array of length = \\(DEPTH\\), consists of \\(0 | 1\\), represents Merkle proof path. Basically, it says how to calculate Poseidon hash, e.g. for two inputs \\(input1\\), \\(input2\\), if the \\(pathIndex[i] = 0\\) it shoud be calculated as \\(Poseidon(input1, input2)\\), otherwise \\(Poseidon(input2, input1)\\); \\(pathElements[DEPTH]\\) - array of length = \\(DEPTH\\), represents elements of the Merkle proof. Outputs : \\(root\\) - Root of the merkle tree. Templates used : mux1.circom from circomlib; poseidon.circom from circomlib.","breadcrumbs":"Overview » Formal spec » MerkleTreeInclusionProof","id":"15","title":"MerkleTreeInclusionProof"},"16":{"body":"RangeCheck(LIMIT_BIT_SIZE) template used for range check, e.g. \\(x \\le y \\le z\\). Parameters : \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check, f.e. for the \\(LIMIT\\_BIT\\_SIZE = 16\\), input numbers allowed to be in the interval \\([0, 65536)\\). Inputs : \\(messageId\\) - denotes counter value, that'll be described further; \\(limit\\) - maximum value. Templates used : LessThan(n) from circomlib; Num2Bits(n) from circomlib. Logic/Constraints : Checked that \\(0 \\le messageId < limit\\).","breadcrumbs":"Overview » Formal spec » RangeCheck","id":"16","title":"RangeCheck"},"17":{"body":"rln.circom is a template that's used for RLN protocol. Parameters : \\(DEPTH\\) - depth of a Merkle Tree. Described here ; \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check. Described here . Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as a private key; \\(userMessageLimit\\) - message limit of the user; \\(messageId\\) - id of the message; \\(pathElements[DEPTH]\\) - pathElements[DEPTH], described here ; \\(identityPathIndex[DEPTH]\\) - pathIndex[DEPTH], described here . Public inputs : \\(x\\) - \\(Hash(signal)\\), where \\(signal\\) is for example message, that was sent by user; \\(externalNullifier\\) - \\(Hash(epoch, rln_identifier)\\). Outputs : \\(y\\) - calculated first-degree linear polynomial \\((y = kx + b)\\); \\(root\\) - root of the Merkle Tree; \\(nullifier\\) - internal nullifier/pseudonym of the user in anonyomus environment. Logic/Constraints : Merkle tree membership check: \\(identityCommitment = Poseidon(identitySecret)\\) calculation; \\(rateCommitment = Poseidon(identityCommitment, userMessageLimit)\\) calculation; Merkle tree inclusion check for the \\(rateCommitment\\). Range check: Range check that \\(0 \\le messageId < limit\\). Polynomial share calculation: \\(a_1 = Poseidon(identitySecret, externalNullifier, messageId)\\); \\(y = identitySecret + a_1 * x\\). Output of calculated \\(root\\), \\(y = share\\) and \\(nullifier = Poseidon(a_1)\\) values.","breadcrumbs":"Overview » Formal spec » RLN","id":"17","title":"RLN"},"18":{"body":"withdraw.circom is a circuit that's used for the withdrawal/slashing and is needed to prevent frontrun while withdrawing the stake from the smart-contract/registry. Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as private key. Public inputs : \\(address\\) - \\(\\mathbb{F_p}\\) scalar field element; denotes ETH address that'll receive stake. Outputs : \\(identityCommitment = Poseidon(identitySecret)\\).","breadcrumbs":"Overview » Formal spec » Withdrawal","id":"18","title":"Withdrawal"},"19":{"body":"","breadcrumbs":"Overview » Smart-contract » Smart-contract","id":"19","title":"Smart-contract"},"2":{"body":"RLN is a zero-knowledge gadget that enables spam prevention in anonymous environments. The anonymity property opens up the possibility for spam, 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; waku-rln-relay - extension of waku-relay (spam protection with RLN ); lambdadelta - P2P event deed secured by RLN proofs; bernkastel - decentralized event feed, based on lambdadelta library.","breadcrumbs":"Overview » RLN Uses » Uses","id":"20","title":"Uses"},"21":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"21","title":"Shamir's Secret Sharing Scheme"},"22":{"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":"Shamir's Secret Sharing » Overview","id":"22","title":"Overview"},"23":{"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 = 1, 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":"Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"23","title":"Polynomial (Lagrange) interpolation"},"24":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing","id":"24","title":"Shamir's Secret Sharing"},"25":{"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":"Shamir's Secret Sharing » Sharing","id":"25","title":"Sharing"},"26":{"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":"Shamir's Secret Sharing » Recovering","id":"26","title":"Recovering"},"27":{"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":"Shamir's Secret Sharing » Important notes","id":"27","title":"Important notes"},"28":{"body":"","breadcrumbs":"Research » Research","id":"28","title":"Research"},"29":{"body":"","breadcrumbs":"Research » KZG-RLN » KZG-RLN","id":"29","title":"KZG-RLN"},"3":{"body":"The RLN construct's functionality consists of three parts These parts should be integrated by the upstream applications, that 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: registration; interaction; withdrawal (or slashing);","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"","breadcrumbs":"Research » Noir-RLN » Noir-RLN","id":"30","title":"Noir-RLN"},"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 = Poseidon(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. Based on the stake amount apps can derive what's the messageLimit (\\(userMessageLimit\\)) for a user. Then the rateCommitment: \\[rateCommitment = Poseidon(identitytCommitment, userMessageLimit)\\] will be stored in the membership Merkle tree.","breadcrumbs":"Overview » What is RLN » Registration","id":"4","title":"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 (or specifically rate commitment) is the 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 (SSS) scheme , which allows you to split a secret to \\(n\\) parts and recover it when any \\(m\\) of \\(n\\) parts \\(m \\le n\\) are presented. Thus, users have to split their secret key into \\(n\\) parts, and for each interaction, they have to reveal the new part of the secret key. So, in addition to proving the membership, 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 » Interaction","id":"5","title":"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. Thus, if someone spams, it'll be possible to recover the secret key and withdraw the stake (or slash ) of a spammer - that's why it's economically inefficient for users to spam.","breadcrumbs":"Overview » What is RLN » Withdrawal (or slashing)","id":"6","title":"Withdrawal (or slashing)"},"7":{"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 » RLN in details » Technical side of RLN","id":"7","title":"Technical side of RLN"},"8":{"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, external\\_nullifier)\\). The meaning of \\(external\\_nullifier\\) is described below.","breadcrumbs":"Overview » RLN in details » User registration","id":"8","title":"User registration"},"9":{"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 » RLN in details » 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To learn more, check out the specification ; there are also circuits implemented for various degree polynomials too.","breadcrumbs":"Overview » RLN in details » Some important notes","id":"12","title":"Some important notes"},"13":{"body":"Utils MerkleTreeInclusionProof RangeCheck RLN Withdrawal","breadcrumbs":"Overview » Formal spec » Formal spec of circom-rln","id":"13","title":"Formal spec of circom-rln"},"14":{"body":"utils.circom is a set of templates/gadgets that the RLN circuit uses. These are: MerkleTreeInclusionProof - Merkle tree inclusion check, used like set membership check; RangeCheck - used for range check. Their description is given below.","breadcrumbs":"Overview » Formal spec » Utils","id":"14","title":"Utils"},"15":{"body":"MerkleTreeInclusionProof(DEPTH) template used for verification of inclusion in full binary incremental merkle tree. The implementation is a fork of https://github.com/privacy-scaling-explorations/incrementalquintree, and changed to binary tree and refactored to Circom 2.1.0 . Parameters : DEPTH - depth of the Merkle Tree. Inputs : \\(leaf\\) - \\(Poseidon(elem)\\), where \\(elem\\) is the element that's checked for inclusion; \\(pathIndex[DEPTH]\\) - array of length = \\(DEPTH\\), consists of \\(0 | 1\\), represents Merkle proof path. Basically, it says how to calculate Poseidon hash, e.g. for two inputs \\(input1\\), \\(input2\\), if the \\(pathIndex[i] = 0\\) it shoud be calculated as \\(Poseidon(input1, input2)\\), otherwise \\(Poseidon(input2, input1)\\); \\(pathElements[DEPTH]\\) - array of length = \\(DEPTH\\), represents elements of the Merkle proof. Outputs : \\(root\\) - Root of the merkle tree. Templates used : mux1.circom from circomlib; poseidon.circom from circomlib.","breadcrumbs":"Overview » Formal spec » MerkleTreeInclusionProof","id":"15","title":"MerkleTreeInclusionProof"},"16":{"body":"RangeCheck(LIMIT_BIT_SIZE) template used for range check, e.g. \\(x \\le y \\le z\\). Parameters : \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check, f.e. for the \\(LIMIT\\_BIT\\_SIZE = 16\\), input numbers allowed to be in the interval \\([0, 65536)\\). Inputs : \\(messageId\\) - denotes counter value, that'll be described further; \\(limit\\) - maximum value. Templates used : LessThan(n) from circomlib; Num2Bits(n) from circomlib. Logic/Constraints : Checked that \\(0 \\le messageId < limit\\).","breadcrumbs":"Overview » Formal spec » RangeCheck","id":"16","title":"RangeCheck"},"17":{"body":"rln.circom is a template that's used for RLN protocol. Parameters : \\(DEPTH\\) - depth of a Merkle Tree. Described here ; \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check. Described here . Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as a private key; \\(userMessageLimit\\) - message limit of the user; \\(messageId\\) - id of the message; \\(pathElements[DEPTH]\\) - pathElements[DEPTH], described here ; \\(identityPathIndex[DEPTH]\\) - pathIndex[DEPTH], described here . Public inputs : \\(x\\) - \\(Hash(signal)\\), where \\(signal\\) is for example message, that was sent by user; \\(externalNullifier\\) - \\(Hash(epoch, rln_identifier)\\). Outputs : \\(y\\) - calculated first-degree linear polynomial \\((y = kx + b)\\); \\(root\\) - root of the Merkle Tree; \\(nullifier\\) - internal nullifier/pseudonym of the user in anonyomus environment. Logic/Constraints : Merkle tree membership check: \\(identityCommitment = Poseidon(identitySecret)\\) calculation; \\(rateCommitment = Poseidon(identityCommitment, userMessageLimit)\\) calculation; Merkle tree inclusion check for the \\(rateCommitment\\). Range check: Range check that \\(0 \\le messageId < limit\\). Polynomial share calculation: \\(a_1 = Poseidon(identitySecret, externalNullifier, messageId)\\); \\(y = identitySecret + a_1 * x\\). Output of calculated \\(root\\), \\(y = share\\) and \\(nullifier = Poseidon(a_1)\\) values.","breadcrumbs":"Overview » Formal spec » RLN","id":"17","title":"RLN"},"18":{"body":"withdraw.circom is a circuit that's used for the withdrawal/slashing and is needed to prevent frontrun while withdrawing the stake from the smart-contract/registry. Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as private key. Public inputs : \\(address\\) - \\(\\mathbb{F_p}\\) scalar field element; denotes ETH address that'll receive stake. Outputs : \\(identityCommitment = Poseidon(identitySecret)\\).","breadcrumbs":"Overview » Formal spec » Withdrawal","id":"18","title":"Withdrawal"},"19":{"body":"","breadcrumbs":"Overview » Smart-contract » Smart-contract","id":"19","title":"Smart-contract"},"2":{"body":"RLN is a zero-knowledge gadget that enables spam prevention in anonymous environments. The anonymity property opens up the possibility for spam, 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; waku-rln-relay - extension of waku-relay (spam protection with RLN ); lambdadelta - P2P event deed secured by RLN proofs; bernkastel - decentralized event feed, based on lambdadelta library.","breadcrumbs":"Overview » RLN Uses » Uses","id":"20","title":"Uses"},"21":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"21","title":"Shamir's Secret Sharing Scheme"},"22":{"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":"Shamir's Secret Sharing » Overview","id":"22","title":"Overview"},"23":{"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 = 1, 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":"Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"23","title":"Polynomial (Lagrange) interpolation"},"24":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing","id":"24","title":"Shamir's Secret Sharing"},"25":{"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":"Shamir's Secret Sharing » Sharing","id":"25","title":"Sharing"},"26":{"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":"Shamir's Secret Sharing » Recovering","id":"26","title":"Recovering"},"27":{"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":"Shamir's Secret Sharing » Important notes","id":"27","title":"Important notes"},"28":{"body":"In addition to production-ready circom-rln version of RLN protocol, there is also wip on RLN R&D to reduce proving time and have bigger choice of implementations. For now you can follow the work on: kzg-rln ; noir-rln .","breadcrumbs":"Research » Research","id":"28","title":"Research"},"29":{"body":"Read this post on zkresear.ch. WIP in the kzg-rln repo.","breadcrumbs":"Research » KZG-RLN » KZG-RLN","id":"29","title":"KZG-RLN"},"3":{"body":"The RLN construct's functionality consists of three parts These parts should be integrated by the upstream applications, that 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: registration; interaction; withdrawal (or slashing);","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"WIP in the noir-rln repo.","breadcrumbs":"Research » Noir-RLN » Noir-RLN","id":"30","title":"Noir-RLN"},"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 = Poseidon(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. Based on the stake amount apps can derive what's the messageLimit (\\(userMessageLimit\\)) for a user. Then the rateCommitment: \\[rateCommitment = Poseidon(identitytCommitment, userMessageLimit)\\] will be stored in the membership Merkle tree.","breadcrumbs":"Overview » What is RLN » Registration","id":"4","title":"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 (or specifically rate commitment) is the 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 (SSS) scheme , which allows you to split a secret to \\(n\\) parts and recover it when any \\(m\\) of \\(n\\) parts \\(m \\le n\\) are presented. Thus, users have to split their secret key into \\(n\\) parts, and for each interaction, they have to reveal the new part of the secret key. So, in addition to proving the membership, 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 » Interaction","id":"5","title":"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. Thus, if someone spams, it'll be possible to recover the secret key and withdraw the stake (or slash ) of a spammer - that's why it's economically inefficient for users to spam.","breadcrumbs":"Overview » What is RLN » Withdrawal (or slashing)","id":"6","title":"Withdrawal (or slashing)"},"7":{"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 » RLN in details » Technical side of RLN","id":"7","title":"Technical side of RLN"},"8":{"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, external\\_nullifier)\\). The meaning of \\(external\\_nullifier\\) is described below.","breadcrumbs":"Overview » RLN in details » User registration","id":"8","title":"User registration"},"9":{"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. 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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.","breadcrumbs":"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":"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 » RLN in details » Slashing","id":"10","title":"Slashing"},"11":{"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 » RLN in details » Nullifiers","id":"11","title":"Nullifiers"},"12":{"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 » RLN in details » Some important notes","id":"12","title":"Some important notes"},"13":{"body":"Utils MerkleTreeInclusionProof RangeCheck RLN Withdrawal","breadcrumbs":"Overview » Formal spec » Formal spec of circom-rln","id":"13","title":"Formal spec of circom-rln"},"14":{"body":"utils.circom is a set of templates/gadgets that the RLN circuit uses. These are: MerkleTreeInclusionProof - Merkle tree inclusion check, used like set membership check; RangeCheck - used for range check. Their description is given below.","breadcrumbs":"Overview » Formal spec » Utils","id":"14","title":"Utils"},"15":{"body":"MerkleTreeInclusionProof(DEPTH) template used for verification of inclusion in full binary incremental merkle tree. The implementation is a fork of https://github.com/privacy-scaling-explorations/incrementalquintree, and changed to binary tree and refactored to Circom 2.1.0 . Parameters : DEPTH - depth of the Merkle Tree. Inputs : \\(leaf\\) - \\(Poseidon(elem)\\), where \\(elem\\) is the element that's checked for inclusion; \\(pathIndex[DEPTH]\\) - array of length = \\(DEPTH\\), consists of \\(0 | 1\\), represents Merkle proof path. Basically, it says how to calculate Poseidon hash, e.g. for two inputs \\(input1\\), \\(input2\\), if the \\(pathIndex[i] = 0\\) it shoud be calculated as \\(Poseidon(input1, input2)\\), otherwise \\(Poseidon(input2, input1)\\); \\(pathElements[DEPTH]\\) - array of length = \\(DEPTH\\), represents elements of the Merkle proof. Outputs : \\(root\\) - Root of the merkle tree. Templates used : mux1.circom from circomlib; poseidon.circom from circomlib.","breadcrumbs":"Overview » Formal spec » MerkleTreeInclusionProof","id":"15","title":"MerkleTreeInclusionProof"},"16":{"body":"RangeCheck(LIMIT_BIT_SIZE) template used for range check, e.g. \\(x \\le y \\le z\\). Parameters : \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check, f.e. for the \\(LIMIT\\_BIT\\_SIZE = 16\\), input numbers allowed to be in the interval \\([0, 65536)\\). Inputs : \\(messageId\\) - denotes counter value, that'll be described further; \\(limit\\) - maximum value. Templates used : LessThan(n) from circomlib; Num2Bits(n) from circomlib. Logic/Constraints : Checked that \\(0 \\le messageId < limit\\).","breadcrumbs":"Overview » Formal spec » RangeCheck","id":"16","title":"RangeCheck"},"17":{"body":"rln.circom is a template that's used for RLN protocol. Parameters : \\(DEPTH\\) - depth of a Merkle Tree. Described here ; \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check. Described here . Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as a private key; \\(userMessageLimit\\) - message limit of the user; \\(messageId\\) - id of the message; \\(pathElements[DEPTH]\\) - pathElements[DEPTH], described here ; \\(identityPathIndex[DEPTH]\\) - pathIndex[DEPTH], described here . Public inputs : \\(x\\) - \\(Hash(signal)\\), where \\(signal\\) is for example message, that was sent by user; \\(externalNullifier\\) - \\(Hash(epoch, rln_identifier)\\). Outputs : \\(y\\) - calculated first-degree linear polynomial \\((y = kx + b)\\); \\(root\\) - root of the Merkle Tree; \\(nullifier\\) - internal nullifier/pseudonym of the user in anonyomus environment. Logic/Constraints : Merkle tree membership check: \\(identityCommitment = Poseidon(identitySecret)\\) calculation; \\(rateCommitment = Poseidon(identityCommitment, userMessageLimit)\\) calculation; Merkle tree inclusion check for the \\(rateCommitment\\). Range check: Range check that \\(0 \\le messageId < limit\\). Polynomial share calculation: \\(a_1 = Poseidon(identitySecret, externalNullifier, messageId)\\); \\(y = identitySecret + a_1 * x\\). Output of calculated \\(root\\), \\(y = share\\) and \\(nullifier = Poseidon(a_1)\\) values.","breadcrumbs":"Overview » Formal spec » RLN","id":"17","title":"RLN"},"18":{"body":"withdraw.circom is a circuit that's used for the withdrawal/slashing and is needed to prevent frontrun while withdrawing the stake from the smart-contract/registry. Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as private key. Public inputs : \\(address\\) - \\(\\mathbb{F_p}\\) scalar field element; denotes ETH address that'll receive stake. Outputs : \\(identityCommitment = Poseidon(identitySecret)\\).","breadcrumbs":"Overview » Formal spec » Withdrawal","id":"18","title":"Withdrawal"},"19":{"body":"","breadcrumbs":"Overview » Smart-contract » Smart-contract","id":"19","title":"Smart-contract"},"2":{"body":"RLN is a zero-knowledge gadget that enables spam prevention in anonymous environments. The anonymity property opens up the possibility for spam, 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; waku-rln-relay - extension of waku-relay (spam protection with RLN ); lambdadelta - P2P event deed secured by RLN proofs; bernkastel - decentralized event feed, based on lambdadelta library.","breadcrumbs":"Overview » RLN Uses » Uses","id":"20","title":"Uses"},"21":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"21","title":"Shamir's Secret Sharing Scheme"},"22":{"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":"Shamir's Secret Sharing » Overview","id":"22","title":"Overview"},"23":{"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 = 1, 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":"Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"23","title":"Polynomial (Lagrange) interpolation"},"24":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing","id":"24","title":"Shamir's Secret Sharing"},"25":{"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":"Shamir's Secret Sharing » Sharing","id":"25","title":"Sharing"},"26":{"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":"Shamir's Secret Sharing » Recovering","id":"26","title":"Recovering"},"27":{"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":"Shamir's Secret Sharing » Important notes","id":"27","title":"Important notes"},"28":{"body":"","breadcrumbs":"Research » Research","id":"28","title":"Research"},"29":{"body":"","breadcrumbs":"Research » KZG-RLN » KZG-RLN","id":"29","title":"KZG-RLN"},"3":{"body":"The RLN construct's functionality consists of three parts These parts should be integrated by the upstream applications, that 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: registration; interaction; withdrawal (or slashing);","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"","breadcrumbs":"Research » Noir-RLN » Noir-RLN","id":"30","title":"Noir-RLN"},"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 = Poseidon(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. Based on the stake amount apps can derive what's the messageLimit (\\(userMessageLimit\\)) for a user. Then the rateCommitment: \\[rateCommitment = Poseidon(identitytCommitment, userMessageLimit)\\] will be stored in the membership Merkle tree.","breadcrumbs":"Overview » What is RLN » Registration","id":"4","title":"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 (or specifically rate commitment) is the 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 (SSS) scheme , which allows you to split a secret to \\(n\\) parts and recover it when any \\(m\\) of \\(n\\) parts \\(m \\le n\\) are presented. Thus, users have to split their secret key into \\(n\\) parts, and for each interaction, they have to reveal the new part of the secret key. So, in addition to proving the membership, 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 » Interaction","id":"5","title":"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. Thus, if someone spams, it'll be possible to recover the secret key and withdraw the stake (or slash ) of a spammer - that's why it's economically inefficient for users to spam.","breadcrumbs":"Overview » What is RLN » Withdrawal (or slashing)","id":"6","title":"Withdrawal (or slashing)"},"7":{"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 » RLN in details » Technical side of RLN","id":"7","title":"Technical side of RLN"},"8":{"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, external\\_nullifier)\\). The meaning of \\(external\\_nullifier\\) is described below.","breadcrumbs":"Overview » RLN in details » User registration","id":"8","title":"User registration"},"9":{"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 » RLN in details » 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To learn more, check out the specification ; there are also circuits implemented for various degree polynomials too.","breadcrumbs":"Overview » RLN in details » Some important notes","id":"12","title":"Some important notes"},"13":{"body":"Utils MerkleTreeInclusionProof RangeCheck RLN Withdrawal","breadcrumbs":"Overview » Formal spec » Formal spec of circom-rln","id":"13","title":"Formal spec of circom-rln"},"14":{"body":"utils.circom is a set of templates/gadgets that the RLN circuit uses. These are: MerkleTreeInclusionProof - Merkle tree inclusion check, used like set membership check; RangeCheck - used for range check. Their description is given below.","breadcrumbs":"Overview » Formal spec » Utils","id":"14","title":"Utils"},"15":{"body":"MerkleTreeInclusionProof(DEPTH) template used for verification of inclusion in full binary incremental merkle tree. The implementation is a fork of https://github.com/privacy-scaling-explorations/incrementalquintree, and changed to binary tree and refactored to Circom 2.1.0 . Parameters : DEPTH - depth of the Merkle Tree. Inputs : \\(leaf\\) - \\(Poseidon(elem)\\), where \\(elem\\) is the element that's checked for inclusion; \\(pathIndex[DEPTH]\\) - array of length = \\(DEPTH\\), consists of \\(0 | 1\\), represents Merkle proof path. Basically, it says how to calculate Poseidon hash, e.g. for two inputs \\(input1\\), \\(input2\\), if the \\(pathIndex[i] = 0\\) it shoud be calculated as \\(Poseidon(input1, input2)\\), otherwise \\(Poseidon(input2, input1)\\); \\(pathElements[DEPTH]\\) - array of length = \\(DEPTH\\), represents elements of the Merkle proof. Outputs : \\(root\\) - Root of the merkle tree. Templates used : mux1.circom from circomlib; poseidon.circom from circomlib.","breadcrumbs":"Overview » Formal spec » MerkleTreeInclusionProof","id":"15","title":"MerkleTreeInclusionProof"},"16":{"body":"RangeCheck(LIMIT_BIT_SIZE) template used for range check, e.g. \\(x \\le y \\le z\\). Parameters : \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check, f.e. for the \\(LIMIT\\_BIT\\_SIZE = 16\\), input numbers allowed to be in the interval \\([0, 65536)\\). Inputs : \\(messageId\\) - denotes counter value, that'll be described further; \\(limit\\) - maximum value. Templates used : LessThan(n) from circomlib; Num2Bits(n) from circomlib. Logic/Constraints : Checked that \\(0 \\le messageId < limit\\).","breadcrumbs":"Overview » Formal spec » RangeCheck","id":"16","title":"RangeCheck"},"17":{"body":"rln.circom is a template that's used for RLN protocol. Parameters : \\(DEPTH\\) - depth of a Merkle Tree. Described here ; \\(LIMIT\\_BIT\\_SIZE\\) - maximum bit size of numbers that are used in range check. Described here . Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as a private key; \\(userMessageLimit\\) - message limit of the user; \\(messageId\\) - id of the message; \\(pathElements[DEPTH]\\) - pathElements[DEPTH], described here ; \\(identityPathIndex[DEPTH]\\) - pathIndex[DEPTH], described here . Public inputs : \\(x\\) - \\(Hash(signal)\\), where \\(signal\\) is for example message, that was sent by user; \\(externalNullifier\\) - \\(Hash(epoch, rln_identifier)\\). Outputs : \\(y\\) - calculated first-degree linear polynomial \\((y = kx + b)\\); \\(root\\) - root of the Merkle Tree; \\(nullifier\\) - internal nullifier/pseudonym of the user in anonyomus environment. Logic/Constraints : Merkle tree membership check: \\(identityCommitment = Poseidon(identitySecret)\\) calculation; \\(rateCommitment = Poseidon(identityCommitment, userMessageLimit)\\) calculation; Merkle tree inclusion check for the \\(rateCommitment\\). Range check: Range check that \\(0 \\le messageId < limit\\). Polynomial share calculation: \\(a_1 = Poseidon(identitySecret, externalNullifier, messageId)\\); \\(y = identitySecret + a_1 * x\\). Output of calculated \\(root\\), \\(y = share\\) and \\(nullifier = Poseidon(a_1)\\) values.","breadcrumbs":"Overview » Formal spec » RLN","id":"17","title":"RLN"},"18":{"body":"withdraw.circom is a circuit that's used for the withdrawal/slashing and is needed to prevent frontrun while withdrawing the stake from the smart-contract/registry. Private inputs : \\(identitySecret\\) - randomly generated number in \\(\\mathbb{F_p}\\), used as private key. Public inputs : \\(address\\) - \\(\\mathbb{F_p}\\) scalar field element; denotes ETH address that'll receive stake. Outputs : \\(identityCommitment = Poseidon(identitySecret)\\).","breadcrumbs":"Overview » Formal spec » Withdrawal","id":"18","title":"Withdrawal"},"19":{"body":"","breadcrumbs":"Overview » Smart-contract » Smart-contract","id":"19","title":"Smart-contract"},"2":{"body":"RLN is a zero-knowledge gadget that enables spam prevention in anonymous environments. The anonymity property opens up the possibility for spam, 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; waku-rln-relay - extension of waku-relay (spam protection with RLN ); lambdadelta - P2P event deed secured by RLN proofs; bernkastel - decentralized event feed, based on lambdadelta library.","breadcrumbs":"Overview » RLN Uses » Uses","id":"20","title":"Uses"},"21":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing Scheme","id":"21","title":"Shamir's Secret Sharing Scheme"},"22":{"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":"Shamir's Secret Sharing » Overview","id":"22","title":"Overview"},"23":{"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 = 1, 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":"Shamir's Secret Sharing » Polynomial (Lagrange) interpolation","id":"23","title":"Polynomial (Lagrange) interpolation"},"24":{"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":"Shamir's Secret Sharing » Shamir's Secret Sharing","id":"24","title":"Shamir's Secret Sharing"},"25":{"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":"Shamir's Secret Sharing » Sharing","id":"25","title":"Sharing"},"26":{"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":"Shamir's Secret Sharing » Recovering","id":"26","title":"Recovering"},"27":{"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":"Shamir's Secret Sharing » Important notes","id":"27","title":"Important notes"},"28":{"body":"In addition to production-ready circom-rln version of RLN protocol, there is also wip on RLN R&D to reduce proving time and have bigger choice of implementations. For now you can follow the work on: kzg-rln ; noir-rln .","breadcrumbs":"Research » Research","id":"28","title":"Research"},"29":{"body":"Read this post on zkresear.ch. WIP in the kzg-rln repo.","breadcrumbs":"Research » KZG-RLN » KZG-RLN","id":"29","title":"KZG-RLN"},"3":{"body":"The RLN construct's functionality consists of three parts These parts should be integrated by the upstream applications, that 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: registration; interaction; withdrawal (or slashing);","breadcrumbs":"Overview » What is RLN » How it works","id":"3","title":"How it works"},"30":{"body":"WIP in the noir-rln repo.","breadcrumbs":"Research » Noir-RLN » Noir-RLN","id":"30","title":"Noir-RLN"},"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 = Poseidon(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. Based on the stake amount apps can derive what's the messageLimit (\\(userMessageLimit\\)) for a user. Then the rateCommitment: \\[rateCommitment = Poseidon(identitytCommitment, userMessageLimit)\\] will be stored in the membership Merkle tree.","breadcrumbs":"Overview » What is RLN » Registration","id":"4","title":"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 (or specifically rate commitment) is the 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 (SSS) scheme , which allows you to split a secret to \\(n\\) parts and recover it when any \\(m\\) of \\(n\\) parts \\(m \\le n\\) are presented. Thus, users have to split their secret key into \\(n\\) parts, and for each interaction, they have to reveal the new part of the secret key. So, in addition to proving the membership, 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 » Interaction","id":"5","title":"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. Thus, if someone spams, it'll be possible to recover the secret key and withdraw the stake (or slash ) of a spammer - that's why it's economically inefficient for users to spam.","breadcrumbs":"Overview » What is RLN » Withdrawal (or slashing)","id":"6","title":"Withdrawal (or slashing)"},"7":{"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 » RLN in details » Technical side of RLN","id":"7","title":"Technical side of RLN"},"8":{"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, external\\_nullifier)\\). The meaning of \\(external\\_nullifier\\) is described below.","breadcrumbs":"Overview » RLN in details » User registration","id":"8","title":"User registration"},"9":{"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. 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