Zero-Knowledge Proofs: Privacy-Preserving Verification 2026

from dataclasses import dataclass
from typing import Tuple

@dataclass
class Point:
    x: int
    y: int
    
    def __add__(self, other: 'Point') -> 'Point':
        """Elliptic curve point addition."""
        if self == Point(0, 0):
            return other
        if other == Point(0, 0):
            return self
        
        if self.x == other.x:
            # Point doubling
            slope = (3 * self.x * self.x) * modinv(2 * self.y, MOD)
        else:
            # Point addition
            slope = (other.y - self.y) * modinv(other.x - self.x, MOD)
        
        x = (slope * slope - self.x - other.x) % MOD
        y = (slope * (self.x - x) - self.y) % MOD
        
        return Point(x, y)

# Generator point G for BN128 curve
G = Point(x=1, y=2)

class KZGCommitment:
    """KZG polynomial commitment scheme."""
    
    def __init__(self, G, H, setup_powers):
        self.G = G  # Generator point
        self.H = H  # Second generator for hiding
        self.setup_powers = setup_powers  # G^τ^i for i in range(n)
    
    def commit(self, polynomial):
        """
        Commit to a polynomial.
        Returns C = Σ polynomial[i] * G^τ^i
        """
        commitment = Point(0, 0)
        for i, coeff in enumerate(polynomial.coefficients):
            commitment = commitment + self.setup_powers[i] * coeff
        
        return commitment
    
    def open(self, polynomial, point, evaluation):
        """
        Create proof that polynomial(point) = evaluation.
        Uses quotient polynomial technique.
        """
        # q(x) = (p(x) - evaluation) / (x - point)
        quotient = polynomial.subtract_constant(evaluation)
        quotient = quotient.divide_by_monomial(point)
        
        # Commit to quotient
        proof = self.commit(quotient)
        
        return OpeningProof(commitment, evaluation, proof)
    
    def verify(self, commitment, point, evaluation, proof):
        """
        Verify the opening proof.
        """
        # e(C - evaluation*G, G) = e(proof, (point*G) - H)
        pass  # Pairing check

class Circuit:
    """Define a ZK circuit for a simple computation."""
    
    def __init__(self):
        self.inputs = []
        self.witness = []
        self.constraints = []
    
    def add_input(self, name, value=None):
        """Add a public or private input."""
        signal = Signal(name, len(self.inputs), value)
        self.inputs.append(signal)
        return signal
    
    def add_constraint(self, a, b, c):
        """
        Add a constraint: a * b = c
        R1CS constraint format.
        """
        self.constraints.append((a, b, c))
    
    def prove_knowledge_of_preimage(self, secret):
        """
        Prove knowledge of x such that hash(x) = public_hash.
        Circuit: x * x = y
        """
        x = self.add_input("secret", secret)
        y = self.add_input("public_output")
        
        # Constraint: x * x = y
        self.add_constraint(x, x, y)
        
        return self.generate_proof()

// circuit.circom
pragma circom 2.0.0;

include "circomlib/poseidon.circom";

template HashLeftRight() {
    signal input left;
    signal input right;
    signal output hash;
    
    component hasher = Poseidon(2);
    hasher.inputs[0] <== left;
    hasher.inputs[1] <== right;
    hash <== hasher.out;
}

template MerkleTree(levels) {
    signal input leaf;
    signal input siblings[levels];
    signal input positions[levels];
    signal output root;
    
    component hashers[levels];
    signal hashes[levels + 1];
    
    hashes[0] <== leaf;
    
    for (var i = 0; i < levels; i++) {
        hashers[i] = HashLeftRight();
        
        // Select left/right based on position
        signal left <== positions[i] == 0 ? siblings[i] : hashes[i];
        signal right <== positions[i] == 0 ? hashes[i] : siblings[i];
        
        hashers[i].left <== left;
        hashers[i].right <== right;
        
        hashes[i + 1] <== hashers[i].hash;
    }
    
    root <== hashes[levels];
}

component main {public [leaf]} = MerkleTree(20);

# Compile and generate proving key
circom circuit.circom --r1cs --wasm --sym
snarkjs powersoftau new bn128 15 pot15_0000.ptau -v
snarkjs powersoftau contribute pot15_0000.ptau pot15_final.ptau --name="First contribution" -v -e="random entropy"
snarkjs groth16 setup circuit.r1cs pot15_final.ptau circuit_0000.zkey
snarkjs zkey contribute circuit_0000.zkey circuit_0001.zkey --name="Contributor 1" -v -e="more entropy"
snarkjs zkey export verificationkey circuit_0001.zkey verification_key.json

// generate_proof.js
const { groth16 } = require("snarkjs");

async function generateProof(input) {
    const { proof, publicSignals } = await groth16.fullProve(
        input,
        "circuit.wasm",
        "circuit_0001.zkey"
    );
    
    return { proof, publicSignals };
}

async function verifyProof(verificationKey, proof, publicSignals) {
    const vKey = verificationKey;
    
    const res = await groth16.verify(
        vKey,
        publicSignals,
        proof
    );
    
    return res;
}

// Usage
const input = {
    leaf: 12345,
    siblings: [111, 222, 333, /* ... */],
    positions: [0, 1, 0, /* ... */]
};

const { proof, publicSignals } = await generateProof(input);
console.log("Proof generated:", proof);

// Verification happens off-chain or on-chain
const isValid = await verifyProof(vKey, proof, publicSignals);
console.log("Valid:", isValid);

class ShieldedTransaction:
    """
    A Zcash-style shielded transaction.
    """
    def __init__(self):
        self.spend_proofs = []
        self.output_cmnts = []
        self.binding_sig = None
    
    def create_spend_proof(self, note, tree_position, merkle_proof):
        """
        Prove:
        1. Note exists in merkle tree
        2. Prover knows the spending key
        3. Nullifier is computed correctly
        """
        # Create circuit inputs
        inputs = {
            'note_commitment': note.commitment,
            'merkle_root': merkle_proof.root,
            'nullifier': note.nullifier,
            'spending_key': note.spending_key,
        }
        
        # Generate zkSNARK proof
        proof = generate_zk_proof('spend_circuit', inputs)
        
        return SpendProof(note, proof)
    
    def create_output_commitment(self, value, address):
        """
        Create new note commitment (encrypted).
        """
        # Pedersen commitment: C = value * G + r * H
        commitment = value * G + note.randomness * H
        
        # Encrypt note data for recipient
        encrypted_note = encrypt_note(
            {'value': value, 'address': address},
            address.viewing_key
        )
        
        return OutputCommitment(commitment, encrypted_note)

class CredentialProof:
    """
    Prove claims without revealing underlying data.
    Example: Prove age > 18 without revealing exact age
    """
    
    def create_age_proof(birth_date, issuer_signature):
        """
        Create proof that age > 18.
        """
        # Age in years (calculated from birth_date)
        age = calculate_age(birth_date)
        
        # Constraint: age >= 18
        if age < 18:
            raise ValueError("Must be 18 or older")
        
        circuit_inputs = {
            'birth_year': birth_date.year,
            'current_year': 2026,
            'signature': issuer_signature,
        }
        
        # Generate proof showing age >= 18
        proof = generate_zk_proof('age_verification', circuit_inputs)
        
        return {
            'proof': proof,
            'metadata': {
                'issuer': 'Government',
                'schema': 'AgeCredential',
                'claimed_attributes': 'age >= 18'
            }
        }
    
    def verify_age_proof(proof, service_provider_public_key):
        """
        Verify age proof without learning exact age.
        """
        return verify_zk_proof(proof, 'age_verification')

class ZKRollup:
    """
    Scalable L2 using ZK proofs.
    """
    
    def __init__(self, account_tree, zk_circuit):
        self.account_tree = account_tree  # Merkle tree of accounts
        self.zk_circuit = zk_circuit
        self.pending_txs = []
    
    def add_transaction(self, tx):
        """Add transaction to pending batch."""
        self.pending_txs.append(tx)
    
    def process_batch(self):
        """
        Process a batch of transactions and generate proof.
        """
        # Prepare state transitions
        old_state = self.account_tree.root
        new_accounts = apply_transactions(
            self.account_tree, 
            self.pending_txs
        )
        new_state = merklize(new_accounts)
        
        # Generate ZK proof
        proof = self.zk_circuit.prove(
            old_state=old_state,
            new_state=new_state,
            transactions=self.pending_txs,
            signatures=[tx.signature for tx in self.pending_txs]
        )
        
        # Submit to L1
        return RollupBatch(
            prev_state=old_state,
            new_state=new_state,
            proof=proof,
            transactions=len(self.pending_txs)
        )

class PrivateQuery:
    """
    Query a database without revealing what you're looking for.
    """
    
    def create_equality_proof(database_commitment, query_value, secret_key):
        """
        Prove: I know x such that commit(x) = C AND x = query_value
        Without revealing x.
        """
        # Generate random blinding factor
        r = random_scalar()
        
        # Create commitment: C = x * G + r * H
        commitment = query_value * G + r * H
        
        # Generate proof
        proof = generate_zk_proof('equality', {
            'commitment': commitment,
            'query_hash': hash(query_value)
        })
        
        return PrivateQueryProof(commitment, proof)
    
    def verify_proof(proof, database_commitment):
        """Verify query without learning the value."""
        return verify_zk_proof(proof)

package main

import (
    "github.com/consensys/gnark-crypto/ecc"
    "github.com/consensys/gnark/backend/groth16"
    "github.com/consensys/gnark/frontend"
    "github.com/consensys/gnark/frontend/cs/r1cs"
)

// Define circuit
type Circuit struct {
    // Public input
    Hash frontend.Variable `gnark:"hash,public"`
    // Private input (witness)
    Preimage frontend.Variable `gnark:"preimage"`
}

func (c *Circuit) Define(api frontend.API) error {
    // Constraint: preimage * preimage = hash
    api.AssertIsEqual(
        api.Mul(c.Preimage, c.Preimage),
        c.Hash,
    )
    return nil
}

func main() {
    // Compile circuit
    ccs, err := groth16.NewCCS(ecc.BN254, r1cs.NewBuilder)
    if err != nil {
        panic(err)
    }
    
    // Generate proving key / verification key
    pk, vk, err := groth16.Setup(ccs)
    if err != nil {
        panic(err)
    }
    
    // Create witness
    witness := &Circuit{
        Hash:      25,
        Preimage:  5,
    }
    
    // Generate proof
    proof, err := groth16.Prove(ccs, pk, witness)
    if err != nil {
        panic(err)
    }
    
    // Verify
    publicWitness := &Circuit{Hash: 25}
    err = groth16.Verify(proof, vk, publicWitness)
    if err != nil {
        panic("Verification failed")
    }
    println("Proof verified!")
}

# Optimize circuit design
class OptimizedCircuit:
    """
    Techniques for reducing proof size and time.
    """
    
    # 1. Reduce constraint count
    def efficient_range_proof(self, value, bits):
        """
        Efficiently prove value is in range [0, 2^bits)
        Using bit decomposition instead of individual constraints
        """
        # Decompose into bits
        bits_array = decompose(value, bits)
        
        # Single constraint for power-of-two decomposition
        reconstructed = sum(bits_array[i] * (1 << i) for i in range(bits))
        self.add_constraint(1, reconstructed, value)
        
        # Bit constraints
        for bit in bits_array:
            self.add_constraint(bit, bit, bit)  # bit^2 = bit
    
    # 2. Use custom gates
    def poseidon_hash(self, inputs):
        """Use Poseidon hash for efficient hashing in circuits."""
        return poseidon(inputs)  # More efficient than SHA256
    
    # 3. Batch verification
    def batch_verify(self, proofs):
        """Verify multiple proofs more efficiently."""
        pass  # Use random linear combination