Implementing Quantum-Proof ZK-Cryptography by Mahdad Kiyani Cross-Language Implementation of RLWE-ZKP based on Python and C++(basic)

As quantum computing advances, traditional cryptographic systems, including those used in blockchain networks, face significant risks. Shor's algorithm, in particular, threatens the security of widely used public-key cryptographic protocols like RSA and elliptic curve cryptography (ECC) by efficiently solving the mathematical problems on which these systems are based. This imminent threat necessitates the development of quantum-resistant cryptographic techniques to secure blockchain infrastructures. Among the most promising quantum-resistant approaches are Zero-Knowledge Proofs (ZKPs), which allow a prover to convince a verifier of the correctness of a statement without revealing the underlying information.

In blockchain systems, ZKPs play a crucial role in enhancing privacy, efficiency, and security while maintaining decentralized trust. However, existing ZKP protocols, such as zk-SNARKs and zk-STARKs, are vulnerable to quantum attacks due to their reliance on cryptographic assumptions that are compromised by quantum algorithms like Shor's and Grover's. The challenge is to construct ZKPs that resist quantum-based attacks while preserving their utility in blockchain environments.

This work takes a deep mathematical and practical approach to implementing quantum-resistant Zero-Knowledge Proofs (ZKPs) using lattice-based cryptography, which is believed to be resistant to quantum attacks. Specifically, we focus on the Ring Learning with Errors (RLWE) problem, which provides a strong security foundation against quantum adversaries. The RLWE problem forms the basis for lattice-based cryptographic primitives, such as homomorphic encryption and digital signatures, and can be extended to construct ZKPs that remain secure in the presence of quantum computers.

Our implementation begins with a Python-based prototype that models the key components of a lattice-based ZKP system, including polynomial arithmetic in rings, RLWE sampling, and the core prover-verifier interactions. This implementation leverages Python’s simplicity for rapid prototyping and experimentation. The second phase of our work involves transitioning this prototype to a C++ implementation to improve performance and scalability, which are crucial for real-world blockchain applications. In both implementations, we rigorously apply mathematical concepts from number theory and linear algebra, such as module lattices, Gaussian sampling, and error-correction techniques, to ensure the correctness and quantum resistance of the ZKP protocol.

By focusing on both the Python and C++ implementations, our work provides a practical, technical roadmap for deploying quantum-proof Zero-Knowledge Proofs in blockchain ecosystems. This dual-phase approach allows for flexible development while ensuring that the final implementation is efficient enough for large-scale deployment. Additionally, our work demonstrates how lattice-based ZKPs can be integrated into existing blockchain frameworks to provide quantum security without sacrificing the inherent privacy and trustlessness that make blockchain technology revolutionary.

To understand how close we are to the reality where blockchain security could be compromised by quantum computing.

RSA Factoring (Shor's Algorithm)

RSA Encryption relies on:

Shor’s algorithm can factor N efficiently by finding pp and qq in polynomial time:

ECC Discrete Logarithm Problem (Shor's Algorithm)

Elliptic Curve Cryptography (ECC) relies on:

Shor’s algorithm can find K (the discrete logarithm):

These are the core mathematical foundations vulnerable to quantum attack using Shor's algorithm, which makes RSA and ECC insecure in the face of quantum computing.

RSA Factoring (Shor's Algorithm):

RSA encryption is based on the difficulty of factoring large integers NN into two prime numbers, pp and qq. Classical algorithms take an exponential amount of time for this task as NN grows large. However, Shor's algorithm, a quantum algorithm, can factor NN efficiently in polynomial time. Given N=p×qN=p×q, Shor's algorithm allows a quantum computer to quickly compute the prime factors pp and qq, effectively breaking RSA encryption.

Mathematically:

• RSA key generation relies on choosing two large primes, pp and qq, and calculating N=p×qN=p×q.
• The security of RSA depends on the hardness of factoring NN back into pp and qq, which is classically infeasible for large values.
• Shor's algorithm uses quantum properties to perform efficient integer factorization, reducing the time complexity from sub-exponential (in classical factoring algorithms) to polynomial.

ECC Discrete Logarithm Problem (Shor's Algorithm):

Elliptic Curve Cryptography (ECC) is based on the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is computationally difficult for classical computers. The problem involves finding a scalar kk such that P=k×GP=k×G, where GG is a generator point on the elliptic curve, and PP is another point on the curve. Shor's algorithm can efficiently solve this problem on a quantum computer, rendering ECC insecure against quantum attacks.

Mathematically:

• Given two points PP and GG on an elliptic curve, ECC requires finding the integer kk such that P=k×GP=k×G.
• Classically, this requires exponential time for large elliptic curves.
• Shor’s algorithm can solve for kk in polynomial time, effectively breaking the cryptographic security of ECC.

Quantum Algorithms and Cryptographic Vulnerabilities

Quantum computers harness quantum mechanics principles such as superposition and entanglement to achieve exponential speed gains over classical computers in particular computations. Two key quantum algorithms pose substantial threats to existing cryptographic protocols:

• Shor’s Algorithm: Efficient at factoring large numbers and calculating discrete logarithms, foundational to RSA and elliptic curve cryptography (ECC). Shor's algorithm breaks these cryptosystems in polynomial time, compromising their security in a quantum setting.
• Grover’s Algorithm: Provides a quadratic acceleration for searching unsorted databases, impacting the security of symmetric encryption methods like AES. To maintain security, symmetric keys must be doubled in length as a countermeasure to Grover’s efficiency.

These algorithms underscore the need for classical cryptographic systems to transition to or incorporate quantum-resistant mechanisms that can withstand quantum computational attacks while maintaining or enhancing security against conventional threats.

Zero-Knowledge Proofs (ZKPs)

Zero-Knowledge Proofs are vital cryptographic tools that allow a prover to demonstrate the truth of a statement to a verifier without conveying any information other than the veracity of the statement itself. ZKPs must satisfy three essential properties to be effective:

• Completeness: If the statement is true, a truthful verifier should be convinced by a truthful prover’s demonstration.
• Soundness: If the statement is false, a dishonest prover should not be able to convince the verifier of its truth, except with negligible probability.
• Zero-Knowledge: The verifier gains no knowledge about the details of the statement, only its validity.

While ZKPs play a crucial role in enabling privacy in blockchain transactions (e.g., zk-SNARKs in Zcash), their reliance on cryptographic assumptions not resistant to quantum attacks necessitates a redesign using quantum-resistant algorithms to secure future cryptographic applications.

Post-Quantum Cryptography: Mathematical Foundations

To achieve quantum resistance, cryptographic systems must be based on mathematical problems that remain hard even for quantum computers. The most promising post-quantum cryptographic approaches are based on the following mathematical structures:

Lattice-Based Cryptography

Lattice-based cryptography is one of the leading candidates for post-quantum cryptography. It relies on the hardness of problems related to lattices, which are discrete grids of points in n-dimensional space. The key problems in lattice-based cryptography are:

• Shortest Vector Problem (SVP): Given a lattice, the task is to find the shortest non-zero vector in the lattice. This problem is known to be hard for both classical and quantum computers.
• Learning With Errors (LWE): The LWE problem involves solving a system of linear equations with some noise added to the coefficients. The hardness of LWE forms the basis for many lattice-based cryptographic schemes, including post-quantum ZKPs.

Lattice-based cryptographic constructions, such as NTRU and Ring-LWE, provide strong security guarantees and are believed to resist attacks by quantum algorithms like Shor’s and Grover’s. These constructions are being adapted to build quantum-resistant ZKPs.

Mathematical Example: Lattice-Based ZKP

In a lattice-based ZKP, the prover may attempt to prove knowledge of a short vector in a lattice, without revealing the vector itself. Let A∈Zn×mA∈Zn×m be a matrix and b=A?x+eb=A?x+e, where xx is a secret vector, and ee is an error vector. The prover can use a lattice-based ZKP to prove they know xx and ee without revealing them, by leveraging the hardness of the LWE problem.

Hash-Based Cryptography

Hash-based cryptography is built on the hardness of finding preimages and collisions in cryptographic hash functions. Since Grover’s algorithm provides only a quadratic speedup in searching for preimages, doubling the output length of a secure hash function like SHA-256 or SHA-3 ensures security in the quantum context.

Hash-based ZKPs are constructed using these hash functions to create commitment schemes. The verifier can check whether the prover’s commitment is correct without learning the original value. The simplicity and security of hash-based schemes make them attractive for quantum-proof ZKPs, particularly in blockchain systems where hashing is already a fundamental operation.

Multivariate Polynomial Cryptography

Multivariate polynomial cryptography relies on the difficulty of solving systems of nonlinear polynomial equations over finite fields. These problems are NP-hard and are resistant to quantum attacks. Cryptographic schemes like HFE (Hidden Field Equations) are based on this principle and can be used to construct post-quantum ZKPs.

For example, the prover could prove knowledge of a solution to a multivariate system without revealing the solution by using a ZKP based on the hardness of solving such systems. The security of multivariate polynomial cryptography against quantum attacks makes it a viable candidate for building quantum-proof ZKPs.

Constructing Quantum-Proof Zero-Knowledge Proofs

Lattice-Based ZKPs

To construct a lattice-based ZKP, we rely on the hardness of the Learning With Errors (LWE) problem. Consider the following steps for building a lattice-based ZKP:

• Commitment Phase: The prover generates a random vector rr and computes t=A?r+et=A?r+e, where AA is a public matrix and ee is a small error vector. The prover then commits to tt using a quantum-resistant hash function.
• Challenge Phase: The verifier sends a random challenge cc to the prover.
• Response Phase: The prover responds with r′=r+c?xr′=r+c?x, where xx is the secret vector they want to prove knowledge of.
• Verification Phase: The verifier checks the validity of the proof by verifying that A?r′A?r′ is consistent with the commitment and the error bounds of the lattice. This ensures that the prover knows a short vector xx in the lattice without revealing it.

Hash-Based ZKPs

Hash-based ZKPs rely on commitment schemes that use quantum-resistant hash functions. A simple hash-based ZKP can be constructed as follows:

• Commitment Phase: The prover computes a hash H(m,r)H(m,r), where mm is the message (or secret) they want to prove knowledge of, and rr is a random value (nonce). The prover sends H(m,r)H(m,r) to the verifier as a commitment.
• Challenge Phase: The verifier sends a random challenge to the prover, asking them to open the commitment.
• Response Phase: The prover reveals mm and rr to the verifier.
• Verification Phase: The verifier computes H(m,r)H(m,r) and checks whether it matches the original commitment.

This construction is simple but effective, and with quantum-resistant hash functions, it provides a robust foundation for quantum-proof ZKPs.

zk-STARKs for Quantum Resistance

zk-STARKs (Zero-Knowledge Scalable Transparent Arguments of Knowledge) are a variant of ZKPs that do not require a trusted setup and rely on cryptographic hash functions. Since zk-STARKs are based on hashes, they are inherently more resistant to quantum attacks than zk-SNARKs. zk-STARKs can be made fully quantum-resistant by replacing their hash functions with quantum-safe alternatives, such as SHA-3.

zk-STARKs are particularly well-suited for blockchain applications because of their scalability and efficiency, even as the complexity of the underlying proofs grows. By leveraging quantum-safe hash functions, zk-STARKs can provide both quantum resistance and scalability for blockchain systems.

Implementing quantum-proof Zero-Knowledge cryptography in blockchain requires mathematical rigor and cryptographic algorithms built to withstand quantum attacks. This part will focus on lattice-based cryptography, particularly how to construct a Zero-Knowledge Proof (ZKP) that resists quantum attacks. We will delve into the mathematical formulation of the problem and also demonstrate how code can be used to implement these ideas in practice.

Lattice-Based Zero-Knowledge Proofs: Mathematical Foundations

Lattice-based cryptography is one of the most promising approaches for post-quantum cryptographic systems. The key problem upon which lattice-based cryptography builds its security is the Learning With Errors (LWE) problem. This problem is hard even for quantum computers, making it suitable for building a quantum-resistant ZKP.

Learning With Errors (LWE) Problem

The LWE problem is defined as follows:

Let A∈Zqn×mA∈Zqn×m be a matrix, where each entry is chosen uniformly at random from the integers modulo qq, and let s∈Zqns∈Zqn be a secret vector. The LWE problem consists of distinguishing between the following two distributions:

1. The distribution of pairs (A,b=A?s+e)(A,b=A?s+e), where e∈Zqme∈Zqm is a small error vector, typically with entries drawn from a discrete Gaussian distribution.
2. The uniform distribution over Zqn×m×ZqmZqn×m×Zqm.

Formally, the problem can be described as: given (A,b)(A,b), where b=A?s+eb=A?s+e, find ss, or distinguish bb from random.

ZKP for LWE Problem

The goal of constructing a lattice-based ZKP is to prove knowledge of a secret vector ss in the context of the LWE problem, without revealing ss. The prover wants to convince the verifier that they know ss, such that b=A?s+eb=A?s+e, without revealing ss or ee.

Mathematical Construction of the ZKP

Let A∈Zqn×mA∈Zqn×m be the public matrix, and b∈Zqmb∈Zqm be the vector such that b=A?s+eb=A?s+e, where s∈Zqns∈Zqn is the secret vector, and e∈Zqme∈Zqm is the error vector.

The proof system consists of the following phases:

1. Commitment Phase: The prover selects a random vector r∈Zqnr∈Zqn and computes the commitment t=A?r+e′t=A?r+e′, where e′∈Zqme′∈Zqm is another small error vector. The prover sends tt to the verifier.
2. Challenge Phase: The verifier sends a random challenge c∈{0,1}c∈{0,1} to the prover.
3. Response Phase: Based on the challenge cc, the prover responds with one of the following:
4. Verification Phase: The verifier checks the following conditions:

If both checks pass with high probability, the verifier is convinced that the prover knows ss without having learned anything about ss itself.

Coding the Lattice-Based ZKP

We can now implement a simplified version of a lattice-based ZKP for the LWE problem using just Python. This implementation simulates the proof system described above, where the prover proves knowledge of the secret vector sswithout revealing it.

First, we'll need to implement the following components:

1. Matrix and vector generation for AA, ss, and the error vector ee.
2. Commitment, challenge, and response phases of the ZKP protocol.
3. Verification phase to check the correctness of the proof.

Python Code for the ZKP

import numpy as np

#Parameters
n = 5  #Dimension of the secret vector s
m = 8  #Number of equations (columns of matrix A)
q = 23 #Modulus for operations (prime number)
error_bound = 3 #error size

#Random matrix A and secret vector s
A = np.random.randint(0, q, size=(m, n))
s = np.random.randint(0, q, size=n)

#small error vector e
e = np.random.randint(-error_bound, error_bound, size=m)

#Compute b = A * s + e
b = (A @ s + e) % q

#Prover selects a random vector r and computes t = A * r + e'
r = np.random.randint(0, q, size=n)
e_prime = np.random.randint(-error_bound, error_bound, size=m)
t = (A @ r + e_prime) % q

#Verifier sends a random challenge c
c = np.random.randint(0, 2)

#Prover's response
if c == 0:
    #Prover sends r and e'
    prover_response_r = r
    prover_response_e_prime = e_prime
else:
    # Prover sends r + s and e' + e
    prover_response_r = (r + s) % q
    prover_response_e_prime = (e_prime + e) % q

#verifier checking the proof
if c == 0:
    # Check if t == A * r + e'
    verifier_check = np.all((A @ prover_response_r + prover_response_e_prime) % q == t)
else:
    #Check if t == A * (r + s) + (e' + e)
    verifier_check = np.all((A @ prover_response_r + prover_response_e_prime) % q == t)

# Output the result of verification
if verifier_check:
    print("Proof verified successfully!")
else:
    
print("Proof verification failed.")

        

3.2. Explanation of Code

• Matrix AA and secret vector ss are generated randomly, representing the setup of the cryptographic system.
• Error vector ee is also randomly generated within a small bound, simulating the noise inherent in the LWE problem.
• In the Commitment Phase, the prover selects a random vector rr and computes the commitment t=A?r+e′t=A?r+e′, where e′e′ is another small error vector.
• In the Challenge Phase, the verifier sends a random challenge c∈{0,1}c∈{0,1}.
• In the Response Phase, the prover responds by either revealing rr and e′e′ (if c=0c=0) or revealing r+sr+s and e′+ee′+e (if c=1c=1).
• Finally, the Verifier checks whether the prover’s response is consistent with the original commitment tt.

In order to deep dive into the coding implementation of lattice-based Zero-Knowledge Proof (ZKP) for a quantum-resistant blockchain ecosystem, let's first break down the mathematical principles and techniques involved and then expand the Python code with detailed explanations and additional steps.

Mathematical Recap: Learning With Errors (LWE) and ZKP Construction

Before diving deeper into the code, it's important to recap the core ideas:

• Learning With Errors (LWE): This problem forms the basis of lattice-based cryptography. We are given a matrix A∈Zqm×nA∈Zqm×n and a vector b∈Zqmb∈Zqm, where b=A?s+eb=A?s+e, with s∈Zqns∈Zqn being a secret vector and e∈Zqme∈Zqm representing small error terms.

Detailed Breakdown of the Code

In this section, we will expand the code with additional comments and structure. We’ll break the code into logical blocks to understand how each part fits into the proof construction.

Matrix and Vector Setup

We start by generating the random matrix AA, the secret vector ss, and the error vector ee. Here, AA is a public key, while ss and ee are private to the prover.

import numpy as np

# Parameters

n = 5 # Dimension of the secret vector s

m = 8 # Number of equations (columns of matrix A)

q = 23 # Modulus for operations (prime number)

error_bound = 3 # Maximum error size (represents the "small" error in LWE)

# Generate a random matrix A and secret vector s

A = np.random.randint(0, q, size=(m, n))

s = np.random.randint(0, q, size=n)

# Generate a small error vector e with values drawn from a bounded range

e = np.random.randint(-error_bound, error_bound, size=m)

# Compute b = A * s + e (mod q), which will be the public information

b = (A @ s + e) % q

print("Matrix A:\n", A)

print("Secret vector s:\n", s)

print("Error vector e:\n", e)

print("Public vector b (A * s + e mod q):\n", b)

• Matrix AA: This is the public key matrix, generated randomly with dimensions m×nm×n. It forms the basis for the cryptographic operations.
• Secret vector ss: The private key of the prover, randomly chosen. This is what the prover will "prove knowledge of" without revealing it.
• Error vector ee: A small random noise vector, ensuring that the LWE problem is hard to solve without knowing ss.
• Public vector bb: The prover's response is based on b=A?s+eb=A?s+e (modulo qq).

This setup simulates the environment where the prover holds private information (ss and ee) and proves knowledge of ssusing AA and bb.

Commitment Phase

In the commitment phase, the prover selects a random vector rr and computes the commitment tt. This commitment is a crucial part of the Zero-Knowledge Proof, as it hides the secret.

# Prover selects a random vector r and computes t = A * r + e' (mod q)

r = np.random.randint(0, q, size=n) # Random vector r

e_prime = np.random.randint(-error_bound, error_bound, size=m) # Small error vector e'

# Compute the commitment t

t = (A @ r + e_prime) % q

print("Random vector r:\n", r)

print("Error vector e_prime:\n", e_prime)

print("Commitment t = A * r + e_prime (mod q):\n", t)

Explanation:

• Random vector rr: A randomly selected vector used to "mask" the secret ss in the later stages.
• Error vector e′e′: A small error term (similar to ee) to maintain security.
• Commitment tt: The prover computes t=A?r+e′t=A?r+e′, which will be used to prove the knowledge of sswhile keeping ss hidden.

This is the first message sent from the prover to the verifier, initiating the proof.

Challenge Phase

In this phase, the verifier sends a challenge cc to the prover. The challenge is a randomly chosen bit (either 0 or 1), determining how the prover should respond.

# Verifier sends a random challenge c

c = np.random.randint(0, 2) # Random challenge, 0 or 1

print("Challenge c from verifier:", c)

Explanation:

• Challenge cc: The challenge can be either 0 or 1. Based on this value, the prover will reveal different pieces of information (either rr or r+sr+s).

This is the second message, where the verifier sends the challenge, triggering the prover's next step.

Response Phase

The prover's response depends on the challenge cc. If c=0c=0, the prover reveals rr and e′e′; if c=1c=1, the prover reveals r+sr+s and e′+ee′+e.

# Prover's response based on the challenge

if c == 0:

# Prover sends r and e' to the verifier

prover_response_r = r

prover_response_e_prime = e_prime

print("Prover responds with r and e_prime:\n", prover_response_r, prover_response_e_prime)

else:

# Prover sends r + s and e' + e to the verifier

prover_response_r = (r + s) % q

prover_response_e_prime = (e_prime + e) % q

print("Prover responds with r + s and e_prime + e:\n", prover_response_r, prover_response_e_prime)

Explanation:

• If c=0c=0: The prover reveals rr and e′e′. This response allows the verifier to verify whether the commitment ttmatches A?r+e′A?r+e′.
• If c=1c=1: The prover reveals r+sr+s and e′+ee′+e. This response shows how the prover can use ss to satisfy the equation b=A?s+eb=A?s+e.

This is the third message, where the prover sends the response based on the challenge.

Verification Phase

The verifier checks the proof by verifying whether the prover’s response satisfies the commitment tt or the public vector bb.

# Verifier checks the proof

if c == 0:

# Check if t == A * r + e'

verifier_check = np.all((A @ prover_response_r + prover_response_e_prime) % q == t)

else:

# Check if t == A * (r + s) + (e' + e)

verifier_check = np.all((A @ prover_response_r + prover_response_e_prime) % q == t)

# Output the result of verification

if verifier_check:

print("Proof verified successfully!")

else:

print("Proof verification failed.")

Explanation:

• If c=0c=0: The verifier checks if t=A?r+e′t=A?r+e′. If this holds, the proof is valid.
• If c=1c=1: The verifier checks if t=A?(r+s)+(e′+e)t=A?(r+s)+(e′+e). If this holds, the prover knows ss.

This is the final step, where the verifier validates the proof and either accepts or rejects it.

Improvements and Additional Considerations

While the current implementation demonstrates the core principles of lattice-based Zero-Knowledge Proofs, a real-world system would need additional security measures, such as:

1. Multiple Rounds: In a production environment, the ZKP protocol should be repeated multiple times to reduce the probability of a malicious prover cheating.
2. Error Bounds: The error bounds ee and e′e′ should be carefully selected to ensure both security and efficiency. Too large an error could compromise verification, while too small could make solving the system easier.
3. Parameter Tuning: Parameters like nn, mm, and qq should be chosen carefully based on security requirements (e.g., quantum security levels) and computational efficiency.

By combining RLWE with ZKPs, blockchain systems can ensure that sensitive information is protected even in a post-quantum world. The RLWE-based ZKP protocol involves the prover committing to a random polynomial and responding to a challenge from the verifier. The verifier checks the validity of the response without learning the underlying secret. Such protocols can be implemented in blockchain networks for secure, quantum-resistant transactions. They also enhance privacy by keeping sensitive data hidden while proving necessary properties. Efficient implementation of RLWE and ZKP protocols is crucial for their adoption in real-world systems.

Now, let's dive into the C++ implementation of RLWE-based Zero-Knowledge Proof protocols.....................

#include <iostream>

#include <vector>

#include <random>

#include <chrono>

using namespace std;

const int q = 23; // Modulus

const int n = 16; // Degree of polynomials (Ring dimension)

// Function to add two polynomials in Z_q[x]/(x^n + 1)

vector<int> poly_add(const vector<int>& a, const vector<int>& b) {

vector<int> result(n);

for (int i = 0; i < n; i++) {

result[i] = (a[i] + b[i]) % q;

}

return result;

}

// Function to multiply two polynomials in Z_q[x]/(x^n + 1)

vector<int> poly_multiply(const vector<int>& a, const vector<int>& b) {

vector<int> result(n, 0);

// Perform polynomial multiplication mod (x^n + 1)

for (int i = 0; i < n; i++) {

for (int j = 0; j < n; j++) {

int idx = (i + j) % n;

result[idx] = (result[idx] + a[i] * b[j]) % q;

}

}

return result;

}

// Function to generate random polynomial with coefficients in Z_q

vector<int> generate_random_poly() {

vector<int> poly(n);

random_device rd;

mt19937 gen(rd());

uniform_int_distribution<> dis(0, q - 1);

for (int i = 0; i < n; i++) {

poly[i] = dis(gen);

}

return poly;

}

// Function to print polynomials

void print_poly(const vector<int>& poly) {

for (int i = 0; i < n; i++) {

cout << poly[i] << " ";

}

cout << endl;

}

RLWE Sample Generation

RLWE samples are generated by combining a random polynomial, a secret polynomial, and an error term.

// Function to sample RLWE instance (a, b) where b = a * s + e

pair<vector<int>, vector<int>> generate_rlwe_sample(const vector<int>& s) {

vector<int> a = generate_random_poly(); // Public random polynomial

vector<int> e = generate_random_poly(); // Small error polynomial

// Compute b = a * s + e (mod q)

vector<int> b = poly_add(poly_multiply(a, s), e);

return {a, b};

}

Prover Commitment and Verifier Challenge

In this section, the prover commits to a value and the verifier sends a challenge. The prover responds based on the challenge.

// Prover's commit function

vector<int> prover_commit(const vector<int>& s) {

vector<int> r = generate_random_poly(); // Random polynomial r

vector<int> e_prime = generate_random_poly(); // Error term

// Compute commitment t = a * r + e' (mod q)

vector<int> t = poly_add(poly_multiply(s, r), e_prime);

return t;

}

// Verifier sends a random challenge c (0 or 1)

int verifier_send_challenge() {

random_device rd;

mt19937 gen(rd());

uniform_int_distribution<> dis(0, 1);

return dis(gen); // Challenge c is either 0 or 1

}

Prover Response and Verifier Check

The prover responds based on the challenge and the verifier checks the response for correctness.

// Prover responds based on the challenge

pair<vector<int>, vector<int>> prover_response(int c, const vector<int>& s, const vector<int>& r, const vector<int>& e_prime) {

if (c == 0) {

return {r, e_prime}; // Reveal r and e' if challenge is 0

} else {

vector<int> r_plus_s = poly_add(r, s); // Reveal r + s if challenge is 1

vector<int> e_prime_plus_e = poly_add(e_prime, s); // Reveal e' + e

return {r_plus_s, e_prime_plus_e};

}

}

// Verifier checks the proof

bool verifier_check(int c, const vector<int>& t, const pair<vector<int>, vector<int>>& response, const vector<int>& a, const vector<int>& b) {

if (c == 0) {

vector<int> r = response.first;

vector<int> e_prime = response.second;

// Check t == a * r + e' (mod q)

return poly_add(poly_multiply(a, r), e_prime) == t;

} else {

vector<int> r_plus_s = response.first;

vector<int> e_prime_plus_e = response.second;

// Check b == a * (r + s) + e' + e

return poly_add(poly_multiply(a, r_plus_s), e_prime_plus_e) == b;

}

}

Main Function

The main function ties everything together, simulating a complete RLWE-based Zero-Knowledge Proof interaction between the prover and verifier.

int main() {

// Generate secret polynomial s

vector<int> s = generate_random_poly();

// Generate RLWE sample (a, b)

auto [a, b] = generate_rlwe_sample(s);

// Prover commits

vector<int> r = generate_random_poly();

vector<int> t = prover_commit(s);

// Verifier sends challenge

int c = verifier_send_challenge();

cout << "Verifier's challenge: " << c << endl;

// Prover responds based on challenge

auto response = prover_response(c, s, r, generate_random_poly());

// Verifier checks the proof

if (verifier_check(c, t, response, a, b)) {

cout << "Proof verified successfully!" << endl;

} else {

cout << "Proof verification failed." << endl;

}

return 0;

}