Zero-Knowledge Proofs: Mathematical Magic for Privacy-Preserving Verification

import random
import networkx as nx
from typing import Dict, List, Tuple

class GraphIsomorphismZKP:
    def __init__(self, G1: nx.Graph, G2: nx.Graph, isomorphism: Dict[int, int]):
        self.G1 = G1
        self.G2 = G2
        self.iso = isomorphism  # Secret witness
        
    def prove_round(self) -> Tuple[nx.Graph, Dict[int, int], Dict[int, int]]:
        """Single round of the zero-knowledge proof protocol"""
        # Step 1: Prover creates random permutation of G1
        nodes = list(self.G1.nodes())
        random.shuffle(nodes)
        random_perm = {old: new for new, old in enumerate(nodes)}
        
        # Create H = π(G1) for random permutation π
        H = nx.Graph()
        H.add_nodes_from(range(len(nodes)))
        for u, v in self.G1.edges():
            H.add_edge(random_perm[u], random_perm[v])
            
        return H, random_perm, self.iso
    
    def respond_to_challenge(self, H: nx.Graph, random_perm: Dict[int, int], 
                           challenge: int) -> Dict[int, int]:
        """Respond to verifier's challenge"""
        if challenge == 0:
            # Reveal isomorphism between G1 and H
            return random_perm
        else:
            # Reveal isomorphism between G2 and H
            # Compose random_perm with secret isomorphism
            composed = {}
            for node in self.G1.nodes():
                composed[self.iso[node]] = random_perm[node]
            return composed

def verify_response(G_ref: nx.Graph, H: nx.Graph, response: Dict[int, int]) -> bool:
    """Verify that response represents valid isomorphism"""
    if len(G_ref.nodes()) != len(H.nodes()):
        return False
    
    # Check if response preserves edge structure
    for u, v in G_ref.edges():
        if not H.has_edge(response[u], response[v]):
            return False
    
    for u, v in H.edges():
        # Find pre-images
        pre_u = next(k for k, val in response.items() if val == u)
        pre_v = next(k for k, val in response.items() if val == v)
        if not G_ref.has_edge(pre_u, pre_v):
            return False
    
    return True

import random
from typing import Tuple

class DiscreteLogZKP:
    def __init__(self, p: int, g: int, h: int, x: int):
        self.p = p  # Prime modulus
        self.g = g  # Generator
        self.h = h  # Public value h = g^x mod p
        self.x = x  # Secret witness
        
    def commit(self) -> Tuple[int, int]:
        """Prover commits to random value"""
        self.r = random.randint(1, self.p - 2)
        commitment = pow(self.g, self.r, self.p)
        return commitment, self.r
    
    def respond(self, challenge: int, r: int) -> int:
        """Prover responds to challenge"""
        response = (r + challenge * self.x) % (self.p - 1)
        return response
    
    def verify(self, commitment: int, challenge: int, response: int) -> bool:
        """Verifier checks the proof"""
        left_side = pow(self.g, response, self.p)
        right_side = (commitment * pow(self.h, challenge, self.p)) % self.p
        return left_side == right_side

# Example usage
if __name__ == "__main__":
    # Safe prime p = 2q + 1 where q is also prime
    p = 23  # Small example for illustration
    g = 5
    x = 7   # Secret
    h = pow(g, x, p)  # Public value
    
    zkp = DiscreteLogZKP(p, g, h, x)
    
    # Simulate the protocol
    commitment, r = zkp.commit()
    challenge = random.randint(1, p-2)  # In practice, would be generated by verifier
    response = zkp.respond(challenge, r)
    
    is_valid = zkp.verify(commitment, challenge, response)
    print(f"Proof valid: {is_valid}")
    print(f"Public values: g={g}, h={h}, p={p}")
    print(f"Commitment: {commitment}, Challenge: {challenge}, Response: {response}")

import hashlib
import struct
from typing import Tuple

class NonInteractiveDiscreteLogZKP:
    def __init__(self, p: int, g: int, h: int, x: int):
        self.p = p
        self.g = g
        self.h = h
        self.x = x
        
    def generate_challenge(self, statement: Tuple[int, int, int], commitment: int) -> int:
        """Generate Fiat-Shamir challenge using hash function"""
        # Create hash input from statement and commitment
        data = struct.pack(">IIII", statement[0], statement[1], statement[2], commitment)
        hash_output = hashlib.sha256(data).digest()
        
        # Convert hash to integer in valid range
        challenge = int.from_bytes(hash_output[:8], 'big') % (self.p - 1)
        return challenge
    
    def prove(self) -> Tuple[int, int]:
        """Generate non-interactive zero-knowledge proof"""
        # Commitment phase
        r = random.randint(1, self.p - 2)
        commitment = pow(self.g, r, self.p)
        
        # Challenge generation (Fiat-Shamir)
        statement = (self.g, self.h, self.p)
        challenge = self.generate_challenge(statement, commitment)
        
        # Response phase
        response = (r + challenge * self.x) % (self.p - 1)
        
        return challenge, response
    
    def verify_proof(self, statement: Tuple[int, int, int], 
                    challenge: int, response: int) -> bool:
        """Verify non-interactive proof"""
        g, h, p = statement
        
        # Recompute commitment from challenge and response
        left_side = pow(g, response, p)
        right_side = pow(h, challenge, p)
        commitment = (left_side * pow(right_side, -1, p)) % p
        
        # Verify challenge was computed correctly
        expected_challenge = self.generate_challenge(statement, commitment)
        
        return challenge == expected_challenge

# Demonstrate the non-interactive protocol
if __name__ == "__main__":
    p = 1009  # Larger prime for better security
    g = 11
    x = 42
    h = pow(g, x, p)
    
    prover = NonInteractiveDiscreteLogZKP(p, g, h, x)
    challenge, response = prover.prove()
    
    # Verifier can check proof without interaction
    verifier = NonInteractiveDiscreteLogZKP(p, g, h, 0)  # x unknown to verifier
    statement = (g, h, p)
    is_valid = verifier.verify_proof(statement, challenge, response)
    
    print(f"Non-interactive proof valid: {is_valid}")
    print(f"Proof size: challenge ({challenge.bit_length()} bits) + response ({response.bit_length()} bits)")

from typing import List, Dict, Tuple

class ArithmeticCircuit:
    def __init__(self, field_size: int):
        self.field_size = field_size
        self.gates = []  # List of (operation, left_input, right_input, output)
        self.wire_count = 0
        self.inputs = []
        self.outputs = []
        
    def add_input(self) -> int:
        """Add an input wire and return its index"""
        wire_id = self.wire_count
        self.wire_count += 1
        self.inputs.append(wire_id)
        return wire_id
    
    def add_gate(self, operation: str, left: int, right: int) -> int:
        """Add a gate and return the output wire index"""
        output_wire = self.wire_count
        self.wire_count += 1
        self.gates.append((operation, left, right, output_wire))
        return output_wire
    
    def set_output(self, wire_id: int):
        """Mark a wire as an output"""
        self.outputs.append(wire_id)
    
    def evaluate(self, inputs: List[int]) -> Dict[int, int]:
        """Evaluate circuit with given inputs"""
        if len(inputs) != len(self.inputs):
            raise ValueError("Input count mismatch")
            
        wire_values = {}
        # Set input values
        for i, input_wire in enumerate(self.inputs):
            wire_values[input_wire] = inputs[i] % self.field_size
        
        # Evaluate gates in order
        for operation, left, right, output in self.gates:
            left_val = wire_values[left]
            right_val = wire_values[right]
            
            if operation == 'add':
                result = (left_val + right_val) % self.field_size
            elif operation == 'mul':
                result = (left_val * right_val) % self.field_size
            elif operation == 'const':
                result = right_val  # right input is the constant value
            else:
                raise ValueError(f"Unknown operation: {operation}")
                
            wire_values[output] = result
        
        return wire_values

def compile_hash_preimage_circuit(field_size: int) -> ArithmeticCircuit:
    """Compile a simple hash preimage circuit (simplified example)"""
    circuit = ArithmeticCircuit(field_size)
    
    # Input: secret preimage
    secret = circuit.add_input()
    
    # Simple "hash" function: h(x) = x^2 + 3x + 7 (mod field_size)
    # This is obviously not a real cryptographic hash!
    
    # x^2
    x_squared = circuit.add_gate('mul', secret, secret)
    
    # 3x (multiply by constant 3)
    const_3 = circuit.wire_count
    circuit.wire_count += 1
    three_x = circuit.add_gate('mul', secret, const_3)
    
    # x^2 + 3x
    partial_sum = circuit.add_gate('add', x_squared, three_x)
    
    # x^2 + 3x + 7
    const_7 = circuit.wire_count
    circuit.wire_count += 1
    hash_output = circuit.add_gate('add', partial_sum, const_7)
    
    circuit.set_output(hash_output)
    
    return circuit, {const_3: 3, const_7: 7}  # Return circuit and constants

# Example: proving knowledge of hash preimage
if __name__ == "__main__":
    field_size = 101  # Small prime field for example
    circuit, constants = compile_hash_preimage_circuit(field_size)
    
    # Secret preimage
    preimage = 13
    
    # Evaluate circuit to get hash
    wire_values = circuit.evaluate([preimage])
    
    # Set constant values
    for wire_id, value in constants.items():
        wire_values[wire_id] = value
    
    hash_value = wire_values[circuit.outputs[0]]
    print(f"Secret preimage: {preimage}")
    print(f"Hash output: {hash_value}")
    print(f"Circuit has {len(circuit.gates)} gates and {circuit.wire_count} wires")
    
    # In a real zk-SNARK, we would now generate a proof that we know
    # a preimage that produces this hash, without revealing the preimage

// Example of optimized finite field arithmetic for zk-SNARK implementation
use rayon::prelude::*;

#[derive(Clone, Copy, Debug, PartialEq)]
pub struct FieldElement {
    value: u64,
    modulus: u64,
}

impl FieldElement {
    pub fn new(value: u64, modulus: u64) -> Self {
        Self {
            value: value % modulus,
            modulus,
        }
    }
    
    // Optimized modular multiplication using Montgomery reduction
    pub fn mul(&self, other: &Self) -> Self {
        assert_eq!(self.modulus, other.modulus);
        
        // Use 128-bit intermediate to avoid overflow
        let product = (self.value as u128) * (other.value as u128);
        let result = (product % (self.modulus as u128)) as u64;
        
        Self {
            value: result,
            modulus: self.modulus,
        }
    }
    
    pub fn add(&self, other: &Self) -> Self {
        assert_eq!(self.modulus, other.modulus);
        
        let sum = self.value + other.value;
        let result = if sum >= self.modulus {
            sum - self.modulus
        } else {
            sum
        };
        
        Self {
            value: result,
            modulus: self.modulus,
        }
    }
}

// Parallel polynomial evaluation using Rayon
pub fn evaluate_polynomial_parallel(
    coefficients: &[FieldElement],
    points: &[FieldElement],
) -> Vec {
    points
        .par_iter()
        .map(|&x| {
            let mut result = coefficients[0];
            let mut x_power = x;
            
            for &coeff in &coefficients[1..] {
                result = result.add(&coeff.mul(&x_power));
                x_power = x_power.mul(&x);
            }
            
            result
        })
        .collect()
}

// Fast Number Theoretic Transform (NTT) for polynomial multiplication
pub fn ntt(
    input: &mut [FieldElement],
    inverse: bool,
    primitive_root: FieldElement,
) {
    let n = input.len();
    assert!(n.is_power_of_two());
    
    // Bit-reverse permutation
    let mut j = 0;
    for i in 1..n {
        let mut bit = n >> 1;
        while j & bit != 0 {
            j ^= bit;
            bit >>= 1;
        }
        j ^= bit;
        if i < j {
            input.swap(i, j);
        }
    }
    
    // Cooley-Tukey NTT
    let mut length = 2;
    while length <= n {
        let mut root = primitive_root;
        for _ in 0..(n / length).trailing_zeros() {
            root = root.mul(&root);  // Compute appropriate root of unity
        }
        
        if inverse {
            // For inverse NTT, use conjugate of root
            // This is a simplified version - real implementation needs modular inverse
        }
        
        for i in (0..n).step_by(length) {
            let mut w = FieldElement::new(1, root.modulus);
            for j in 0..(length / 2) {
                let u = input[i + j];
                let v = input[i + j + length / 2].mul(&w);
                input[i + j] = u.add(&v);
                input[i + j + length / 2] = u.add(&FieldElement::new(
                    root.modulus - v.value,
                    root.modulus,
                ));
                w = w.mul(&root);
            }
        }
        
        length *= 2;
    }
    
    if inverse {
        let inv_n = FieldElement::new(
            mod_inverse(n as u64, input[0].modulus),
            input[0].modulus,
        );
        for element in input.iter_mut() {
            *element = element.mul(&inv_n);
        }
    }
}

// Placeholder for modular inverse computation
fn mod_inverse(a: u64, m: u64) -> u64 {
    // Extended Euclidean Algorithm implementation
    // This is a simplified placeholder
    1
}

// Conceptual implementation of age verification without revealing exact age
class AgeVerificationZKP {
    constructor(minimumAge = 18) {
        this.minimumAge = minimumAge;
    }
    
    // Generate proof that age >= minimum without revealing exact age
    generateAgeProof(birthYear, currentYear, salt) {
        const age = currentYear - birthYear;
        
        // Create a constraint: age - minimumAge >= 0
        // This would be compiled to an arithmetic circuit in practice
        const constraint = age - this.minimumAge;
        
        if (constraint < 0) {
            throw new Error('Age verification failed');
        }
        
        // In a real implementation, this would generate a zk-SNARK proof
        // that the constraint is satisfied without revealing age or birthYear
        return {
            proof: this.simulateProofGeneration(constraint, salt),
            publicInputs: {
                minimumAge: this.minimumAge,
                currentYear: currentYear
                // Note: birthYear and exact age are NOT revealed
            }
        };
    }
    
    // Verify the age proof
    verifyAgeProof(proof, publicInputs) {
        // In practice, this would use a zk-SNARK verifier
        // that checks the proof cryptographically
        return this.simulateProofVerification(proof, publicInputs);
    }
    
    // Simplified simulation of proof generation
    simulateProofGeneration(constraint, salt) {
        // Real implementation would use arithmetic circuits and polynomial commitments
        const hash = this.hash(`${constraint}${salt}${this.minimumAge}`);
        return {
            pi_a: hash.substring(0, 16),
            pi_b: hash.substring(16, 32),
            pi_c: hash.substring(32, 48)
        };
    }
    
    // Simplified simulation of proof verification
    simulateProofVerification(proof, publicInputs) {
        // Real verification would involve elliptic curve pairings
        // and polynomial evaluations
        return proof && proof.pi_a && proof.pi_b && proof.pi_c;
    }
    
    // Simple hash function for simulation
    hash(input) {
        let hash = 0;
        for (let i = 0; i < input.length; i++) {
            const char = input.charCodeAt(i);
            hash = ((hash << 5) - hash) + char;
            hash = hash & hash; // Convert to 32-bit integer
        }
        return Math.abs(hash).toString(16).padStart(48, '0');
    }
}

// Usage example
const verifier = new AgeVerificationZKP(21); // Minimum age 21

try {
    // User proves they are at least 21 without revealing exact age
    const proof = verifier.generateAgeProof(1985, 2023, 'random-salt-12345');
    const isValid = verifier.verifyAgeProof(proof.proof, proof.publicInputs);
    
    console.log('Age verification proof generated successfully');
    console.log('Verifier confirmed age >= 21:', isValid);
    console.log('Exact age and birth year remain private');
    
    // What the verifier sees:
    console.log('Public information:', proof.publicInputs);
    
} catch (error) {
    console.error('Age verification failed:', error.message);
}