herta
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Advanced mathematics framework for scientific, engineering, and financial applications
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JavaScript
/**
* Zero-Knowledge Proofs module for herta.js
* Provides mathematical foundations for privacy-preserving blockchain applications
* including ZK-SNARKs, ZK-STARKs, and bulletproofs implementations
*/
const numberTheory = require('./numberTheory');
const matrix = require('../core/matrix');
const arithmetic = require('../core/arithmetic');
const cryptoeconomics = require('./cryptoeconomics');
const zeroKnowledgeProofs = {};
/**
* Implement a simplified Fiat-Shamir ZKP for discrete logarithm knowledge
* @param {Object} params - Parameters for the proof
* @returns {Object} - Functions for creating and verifying proofs
*/
zeroKnowledgeProofs.discreteLogProof = function (params = {}) {
const p = params.p || 2147483647; // Prime modulus (Mersenne prime 2^31 - 1)
const g = params.g || 3; // Generator
return {
/**
* Generate proof of knowledge of x such that h = g^x mod p
* @param {number} x - Secret value
* @returns {Object} - Proof and public values
*/
prove(x) {
// Compute public value h = g^x mod p
const h = numberTheory.modPow(g, x, p);
// Prover selects a random value r
const r = Math.floor(Math.random() * (p - 2)) + 1;
// Compute commitment t = g^r mod p
const t = numberTheory.modPow(g, r, p);
// In a real implementation, the challenge would come from the verifier
// Here we use Fiat-Shamir heuristic to generate the challenge from t and h
const challenge = (t * h) % p;
// Compute response s = r + challenge * x mod (p-1)
let s = (r + challenge * x) % (p - 1);
if (s < 0) s += (p - 1);
return {
publicValue: h,
commitment: t,
challenge,
response: s
};
},
/**
* Verify a discrete logarithm proof
* @param {Object} proof - The proof to verify
* @returns {boolean} - True if the proof is valid
*/
verify(proof) {
const {
publicValue, commitment, challenge, response
} = proof;
// Verify that the challenge is consistent (Fiat-Shamir)
const expectedChallenge = (commitment * publicValue) % p;
if (challenge !== expectedChallenge) {
return false;
}
// Verify g^s = t * h^challenge mod p
const leftSide = numberTheory.modPow(g, response, p);
const rightSide = (commitment * numberTheory.modPow(publicValue, challenge, p)) % p;
return leftSide === rightSide;
}
};
};
/**
* Implement a simplified zero-knowledge range proof
* @param {Object} params - Parameters for the proof
* @returns {Object} - Functions for creating and verifying range proofs
*/
zeroKnowledgeProofs.rangeProof = function (params = {}) {
const p = params.p || 2147483647; // Prime modulus
const g = params.g || 3; // Generator
const h = params.h || 5; // Second generator (independent from g)
return {
/**
* Generate a proof that v is in range [0, upperBound]
* @param {number} v - Secret value to prove is in range
* @param {number} upperBound - Upper bound of the range
* @returns {Object} - Proof and commitments
*/
prove(v, upperBound) {
if (v < 0 || v > upperBound) {
throw new Error(`Value ${v} is outside of range [0, ${upperBound}]`);
}
// Create bit representation of the value
const bits = [];
let temp = v;
while (temp > 0 || bits.length < Math.ceil(Math.log2(upperBound + 1))) {
bits.push(temp % 2);
temp = Math.floor(temp / 2);
}
// Create commitments to each bit
const commitments = [];
const challenges = [];
const responses = [];
for (let i = 0; i < bits.length; i++) {
const bit = bits[i];
// Random blinding factor
const r = Math.floor(Math.random() * (p - 2)) + 1;
// Commit to the bit: C = g^bit * h^r mod p
const commitment = (numberTheory.modPow(g, bit, p)
* numberTheory.modPow(h, r, p)) % p;
// In a real implementation, the challenge would come from the verifier
// Here we use a simplification
const challenge = (commitment * (i + 1)) % p;
// For a bit b, we need to prove that b * (1 - b) = 0
// This means b is either 0 or 1
// Response for the proof
let response;
if (bit === 0) {
response = r; // For bit=0, response is just r
} else {
response = (r + challenge) % (p - 1); // For bit=1, response includes challenge
}
commitments.push(commitment);
challenges.push(challenge);
responses.push(response);
}
// Combine bit commitments to prove the value is correct
// For a real range proof, we'd use a homomorphic commitment scheme
// This is a simplified version
return {
bitCommitments: commitments,
challenges,
responses,
numBits: bits.length
};
},
/**
* Verify a range proof
* @param {Object} proof - The proof to verify
* @param {number} upperBound - Upper bound of the range
* @returns {boolean} - True if the proof is valid
*/
verify(proof, upperBound) {
const {
bitCommitments, challenges, responses, numBits
} = proof;
const expectedNumBits = Math.ceil(Math.log2(upperBound + 1));
if (numBits < expectedNumBits) {
return false;
}
// Verify each bit commitment proves a 0 or 1 value
for (let i = 0; i < numBits; i++) {
const commitment = bitCommitments[i];
const challenge = challenges[i];
const response = responses[i];
// Verify the first equation: g^0 * h^response = commitment
// or the second equation: g^1 * h^response = commitment * g^challenge
const zeroCase = numberTheory.modPow(h, response, p);
const oneCase = (numberTheory.modPow(g, 1, p)
* numberTheory.modPow(h, response, p)) % p;
const oneCaseAlt = (commitment * numberTheory.modPow(g, challenge, p)) % p;
if (zeroCase !== commitment && oneCase !== oneCaseAlt) {
return false;
}
}
// In a real implementation, we would also verify that the committed value
// equals the claimed sum of powers of 2 times the bit values
return true;
}
};
};
/**
* Implement Bulletproofs for confidential transactions
* (This is a simplified version for educational purposes)
* @param {Object} params - System parameters
* @returns {Object} - Bulletproofs implementation
*/
zeroKnowledgeProofs.bulletproofs = function (params = {}) {
const p = params.p || 2147483647; // Prime modulus
const g = params.g || 3; // Generator for value
const h = params.h || 5; // Generator for blinding
return {
/**
* Generate a bulletproof for a confidential value
* @param {number} value - Secret value to commit to
* @param {number} maxValue - Maximum possible value (for range verification)
* @returns {Object} - Proof and verification material
*/
prove(value, maxValue) {
if (value < 0 || value >= maxValue) {
throw new Error(`Value ${value} is outside valid range [0, ${maxValue})`);
}
// Create Pedersen commitment to the value
const blindingFactor = Math.floor(Math.random() * (p - 2)) + 1;
const commitment = (numberTheory.modPow(g, value, p)
* numberTheory.modPow(h, blindingFactor, p)) % p;
// In real Bulletproofs, we would now:
// 1. Convert value to binary representation
// 2. Create a vector commitment to these bits
// 3. Use inner product arguments to compress the proof
// 4. Apply Fiat-Shamir to make it non-interactive
// For this simplified version, we'll create a basic ZK range proof
const bits = [];
let temp = value;
while (temp > 0 || bits.length < Math.ceil(Math.log2(maxValue))) {
bits.push(temp % 2);
temp = Math.floor(temp / 2);
}
// Create commitments for each bit and aggregate them
const bitCommitments = [];
const bitBlindingFactors = [];
for (const bit of bits) {
const r = Math.floor(Math.random() * (p - 2)) + 1;
const bitCommitment = (numberTheory.modPow(g, bit, p)
* numberTheory.modPow(h, r, p)) % p;
bitCommitments.push(bitCommitment);
bitBlindingFactors.push(r);
}
// In a real bulletproof, this would be compressed logarithmically
// Here we're just sending all the bit commitments
return {
commitment,
bitCommitments,
numBits: bits.length,
// Typically not revealed, but included for verification in this simplified version
value,
blindingFactor,
bitBlindingFactors
};
},
/**
* Verify a bulletproof
* @param {Object} proof - The proof to verify
* @param {number} maxValue - Maximum possible value
* @returns {boolean} - True if the proof is valid
*/
verify(proof, maxValue) {
const {
commitment,
bitCommitments,
numBits,
value,
blindingFactor,
bitBlindingFactors
} = proof;
// Verify the main commitment matches the claimed value and blinding factor
const expectedCommitment = (numberTheory.modPow(g, value, p)
* numberTheory.modPow(h, blindingFactor, p)) % p;
if (commitment !== expectedCommitment) {
return false;
}
// Verify each bit commitment
const bits = [];
let temp = value;
for (let i = 0; i < numBits; i++) {
bits.push(temp % 2);
temp = Math.floor(temp / 2);
}
for (let i = 0; i < numBits; i++) {
const bitCommitment = bitCommitments[i];
const bit = bits[i];
const r = bitBlindingFactors[i];
const expectedBitCommitment = (numberTheory.modPow(g, bit, p)
* numberTheory.modPow(h, r, p)) % p;
if (bitCommitment !== expectedBitCommitment) {
return false;
}
}
// Verify the value is reconstructible from the bits
let reconstructedValue = 0;
for (let i = 0; i < bits.length; i++) {
reconstructedValue += bits[i] * 2 ** i;
}
return reconstructedValue === value && value < maxValue;
}
};
};
/**
* Implement a simplified zk-SNARK prover and verifier
* (Educational implementation - not cryptographically secure)
* @param {Object} params - System parameters
* @returns {Object} - zk-SNARK implementation
*/
zeroKnowledgeProofs.zkSnark = function (params = {}) {
// In a real implementation, we would use a trusted setup to generate
// proving key and verification key for a specific circuit
// For this simplified version, we'll focus on a specific statement:
// "I know x such that h = g^x mod p" (discrete logarithm)
const p = params.p || 2147483647; // Prime modulus
const g = params.g || 3; // Generator
return {
/**
* Generate a setup for the zk-SNARK system
* @returns {Object} - Proving and verification keys
*/
setup() {
// In a real trusted setup, these values would be generated
// in a secure multi-party computation
// Toxic waste (should be discarded in a real system)
const alpha = Math.floor(Math.random() * (p - 2)) + 1;
const beta = Math.floor(Math.random() * (p - 2)) + 1;
const gamma = Math.floor(Math.random() * (p - 2)) + 1;
// Generate verification key
const vk = {
alpha_g: numberTheory.modPow(g, alpha, p),
beta_g: numberTheory.modPow(g, beta, p),
gamma_g: numberTheory.modPow(g, gamma, p)
};
// Generate proving key
const pk = {
alpha_beta_g: numberTheory.modPow(g, (alpha * beta) % (p - 1), p)
// Additional values would be included for real circuits
};
return { vk, pk, toxicWaste: { alpha, beta, gamma } };
},
/**
* Generate a proof that the prover knows x such that h = g^x mod p
* @param {Object} pk - Proving key
* @param {number} x - Secret value
* @returns {Object} - Zero-knowledge proof
*/
prove(pk, x) {
// Compute public output h = g^x mod p
const h = numberTheory.modPow(g, x, p);
// In a real zk-SNARK, we would:
// 1. Represent the statement as a circuit
// 2. Convert to a QAP (Quadratic Arithmetic Program)
// 3. Use the proving key to create various commitments
// For this simplified implementation, we'll create a randomized proof
const r1 = Math.floor(Math.random() * (p - 2)) + 1;
const r2 = Math.floor(Math.random() * (p - 2)) + 1;
// Create proof elements (highly simplified)
const proof = {
A: numberTheory.modPow(g, r1, p),
B: numberTheory.modPow(pk.alpha_beta_g, r2, p),
C: numberTheory.modPow(g, (r1 * r2 * x) % (p - 1), p),
publicValue: h
};
return proof;
},
/**
* Verify a zk-SNARK proof
* @param {Object} vk - Verification key
* @param {Object} proof - Proof to verify
* @returns {boolean} - True if the proof is valid
*/
verify(vk, proof) {
// In a real zk-SNARK verification, we would check pairing equations
// Since we're using a simplified model, we'll verify some basic properties
const {
A, B, C, publicValue
} = proof;
// Simulate a bilinear pairing check (simplified)
// In a real system, this would use elliptic curve pairings
// Check if e(A, B) = e(vk.alpha_g, vk.beta_g) * e(vk.gamma_g^x, g)
// This is grossly simplified - just for educational purposes
// The verification would actually compute and compare pairings
// Here we're just checking that the proof elements are consistent
// Pseudocode for what would happen in a real system:
// return e(A, B) === e(vk.alpha_g, vk.beta_g) * e(publicValue, vk.gamma_g)
// Instead, we'll just verify that proofs are unique per public value
const checkValue = (A * B * C) % p;
const expectedCheck = (vk.alpha_g * vk.beta_g * numberTheory.modPow(publicValue, 3, p)) % p;
return checkValue === expectedCheck;
}
};
};
/**
* Implement a simplified zk-STARK prover and verifier
* (Educational implementation - not cryptographically secure)
* @returns {Object} - zk-STARK implementation
*/
zeroKnowledgeProofs.zkStark = function () {
// ZK-STARKs use Fast Reed-Solomon Interactive Oracle Proofs of Proximity
// They don't require a trusted setup, unlike zk-SNARKs
return {
/**
* Generate a proof for a computational statement
* @param {Function} computation - Function representing the computation
* @param {Array} inputs - Public and private inputs
* @param {Array} privateIndices - Indices of private inputs
* @returns {Object} - STARK proof
*/
prove(computation, inputs, privateIndices) {
// In a real ZK-STARK, we would:
// 1. Express the computation as a polynomial
// 2. Evaluate the polynomial at many points to create a trace
// 3. Use the FRI protocol to prove proximity to a low-degree polynomial
// 4. Apply Fiat-Shamir to make it non-interactive
// For this simplified version, we'll just compute the result
// and create a placeholder proof
const publicInputs = inputs.filter((_, i) => !privateIndices.includes(i));
const privateInputs = inputs.filter((_, i) => privateIndices.includes(i));
// Compute the actual result
const result = computation(...inputs);
// Create a commitment to the computation trace
// In a real STARK, this would be a Merkle tree of a low-degree extension
const traceCommitment = {
result,
// This would normally be cryptographic commitments to the execution trace
traceSize: privateInputs.length * 8 // Just a placeholder value
};
return {
publicInputs,
result,
traceCommitment
// Other proof elements would be included here
};
},
/**
* Verify a ZK-STARK proof
* @param {Function} computation - Function representing the computation
* @param {Object} proof - The proof to verify
* @returns {boolean} - True if the proof is valid
*/
verify(computation, proof) {
// In a real STARK verification, we would:
// 1. Verify the Merkle proof for the trace
// 2. Verify the FRI protocol for low-degree testing
// 3. Check that the computation is consistent with the trace
// For this simplified version, we'll just check if the
// computation can be verified with just the public inputs
const { publicInputs, result } = proof;
// Placeholder verification
// In a real system, this would verify cryptographic proofs
// Check that the claimed result is plausible
// Note: This is not an actual verification since we can't
// run the computation without the private inputs
return typeof result !== 'undefined';
}
};
};
/**
* Implement a simple zero-knowledge set membership proof
* @param {Array} set - The set to prove membership in
* @returns {Object} - Set membership proof functions
*/
zeroKnowledgeProofs.setMembership = function (set) {
// Create a commitment to the set using a Merkle tree
const merkleTree = cryptoeconomics.createMerkleTree(set);
return {
/**
* Prove that an element is in the set without revealing which one
* @param {*} element - The element to prove membership for
* @returns {Object} - Proof of set membership
*/
proveInSet(element) {
// Find the element in the set
const elementIndex = set.findIndex((item) => JSON.stringify(item) === JSON.stringify(element));
if (elementIndex === -1) {
throw new Error('Element not in set');
}
// Get the leaf node for this element
const leaf = merkleTree.leaves[elementIndex];
// Get Merkle proof
const merkleProof = merkleTree.getProof(leaf);
return {
rootHash: merkleTree.root.hash,
merkleProof,
leafHash: leaf.hash
};
},
/**
* Verify a set membership proof
* @param {*} proof - The membership proof
* @returns {boolean} - True if the proof is valid
*/
verifyInSet(proof) {
const { rootHash, merkleProof, leafHash } = proof;
// Verify the Merkle proof
return cryptoeconomics.verifyMerkleProof(
rootHash,
leafHash,
merkleProof
);
},
/**
* Get the Merkle root hash (public commitment to the set)
* @returns {string} - Merkle root hash
*/
getRootHash() {
return merkleTree.root.hash;
},
/**
* Get the size of the set (public information)
* @returns {number} - Set size
*/
getSetSize() {
return set.length;
}
};
};
/**
* Implement a simple zero-knowledge proof of a valid shuffle (e.g., for mixing in voting)
* @param {Array} input - Input array
* @returns {Object} - Shuffle proof functions
*/
zeroKnowledgeProofs.proveShuffle = function (input) {
return {
/**
* Prove that output is a valid permutation of input
* @param {Array} output - Shuffled array
* @param {Array} permutation - The permutation used (private)
* @returns {Object} - Proof of shuffle
*/
prove(output, permutation) {
if (input.length !== output.length || input.length !== permutation.length) {
throw new Error('Input, output, and permutation must have the same length');
}
// Check that permutation is valid (contains each index exactly once)
const sorted = [...permutation].sort((a, b) => a - b);
for (let i = 0; i < sorted.length; i++) {
if (sorted[i] !== i) {
throw new Error('Invalid permutation');
}
}
// Check that output is actually the result of applying the permutation
for (let i = 0; i < input.length; i++) {
if (input[i] !== output[permutation.indexOf(i)]) {
throw new Error('Output does not match the claimed permutation of input');
}
}
// In a real ZKP for shuffling, we would:
// 1. Commit to the permutation matrix
// 2. Prove that it's a valid permutation matrix
// 3. Prove that applying it to input gives output
// For this simplified version, we'll use a commitment to intermediate values
// Commitment to the mapping (simplified)
const commits = [];
for (let i = 0; i < input.length; i++) {
// A real implementation would use cryptographic commitments
const value = input[i] * permutation[i];
commits.push({
inputIndex: i,
outputIndex: permutation[i],
commitment: value
});
}
// Sort commitments by output index to hide the permutation
commits.sort((a, b) => a.outputIndex - b.outputIndex);
return {
inputHash: JSON.stringify(input).split('').reduce((a, b) => a + b.charCodeAt(0), 0),
outputHash: JSON.stringify(output).split('').reduce((a, b) => a + b.charCodeAt(0), 0),
commitments: commits.map((c) => ({ commitment: c.commitment }))
};
},
/**
* Verify a shuffle proof
* @param {Array} output - Claimed shuffled array
* @param {Object} proof - The proof to verify
* @returns {boolean} - True if the proof is valid
*/
verify(output, proof) {
const { inputHash, outputHash, commitments } = proof;
// Verify input and output hashes
const expectedInputHash = JSON.stringify(input).split('').reduce((a, b) => a + b.charCodeAt(0), 0);
const expectedOutputHash = JSON.stringify(output).split('').reduce((a, b) => a + b.charCodeAt(0), 0);
if (inputHash !== expectedInputHash || outputHash !== expectedOutputHash) {
return false;
}
// In a real verification, we would check the cryptographic proofs
// Here we just verify that the number of commitments matches
return commitments.length === input.length;
}
};
};
/**
* Create a zkSNARK for quadratic equations
* @returns {Object} - zkSNARK for proving knowledge of roots
*/
zeroKnowledgeProofs.quadraticRootProof = function () {
return {
/**
* Prove knowledge of x such that a*x^2 + b*x + c = 0
* @param {number} a - Coefficient a
* @param {number} b - Coefficient b
* @param {number} c - Coefficient c
* @param {number} x - Root of the equation (private)
* @returns {Object} - Proof of knowledge
*/
prove(a, b, c, x) {
// Verify that x is actually a root
const evaluation = a * x * x + b * x + c;
if (Math.abs(evaluation) > 1e-10) { // Allow for floating point imprecision
throw new Error(`${x} is not a root of the equation ${a}x^2 + ${b}x + ${c} = 0`);
}
// In a real zkSNARK, we would:
// 1. Represent the statement as a circuit
// 2. Use the zkSNARK proving system to create a proof
// For this simplified version, we'll create a commitment to x
// without revealing x itself
// Choose a random blinding factor
const r = Math.floor(Math.random() * 1000000);
// Create a commitment to x that hides its value
// In a real system, this would be a cryptographic commitment
const commitment = (x * x + r * r) % 1000000007;
// Create a proof that the commitment contains a root
// without revealing the root
// Calculate intermediate values that will be verified
const v1 = (a * x * x) % 1000000007;
const v2 = (b * x) % 1000000007;
const v3 = (v1 + v2 + c) % 1000000007;
return {
publicCoefficients: { a, b, c },
commitment,
proof: {
// In a real zkSNARK, these would be cryptographic proof elements
// Here they are just placeholders to simulate the structure
v1Commit: (v1 + r) % 1000000007,
v2Commit: (v2 + r * 2) % 1000000007,
v3Commit: v3 // Should be 0 if x is a root
}
};
},
/**
* Verify a proof of knowledge of a root
* @param {Object} proof - The proof to verify
* @returns {boolean} - True if the proof is valid
*/
verify(proof) {
const { publicCoefficients, commitment, proof: proofElements } = proof;
// In a real zkSNARK verification, we would:
// 1. Verify the cryptographic proof elements
// 2. Check that they satisfy the required relations
// For this simplified version, we just check that v3 is claimed to be 0
// which would be true if and only if x is a root
return proofElements.v3Commit === 0;
}
};
};
module.exports = zeroKnowledgeProofs;