ZERO-KNOWLEDGE PROOF

# Schnorr Protocol for discrete log ZKP
import random

# Public parameters
p = large_prime()  # Prime modulus
g = generator(p)   # Generator
y = public_key     # y = g^x mod p (x is secret)

# Prover knows x (secret), wants to prove it without revealing x
def schnorr_prove(x):
    # Step 1: Commitment
    r = random.randint(1, p-1)  # Random nonce
    t = pow(g, r, p)  # Commitment: t = g^r mod p
    send_to_verifier(t)

    # Step 2: Challenge
    c = receive_challenge_from_verifier()  # Random challenge

    # Step 3: Response
    s = (r + c * x) % (p-1)  # Response: s = r + cx
    send_to_verifier(s)

# Verifier checks: g^s = t * y^c (mod p)
# If true, prover knows x. But what if you DON'T know x?
schnorr_prove(phase_omega_secret)  # ??? You don't have this!

import zk_snark_library

# Define circuit: "I know sigil S such that Hash(S) = H"
circuit = zk_snark_library.define_circuit(
    public_inputs=['hash_H'],
    private_inputs=['sigil_S'],
    constraints=[
        'SHA256(sigil_S) == hash_H'
    ]
)

# Setup (trusted setup ceremony)
proving_key, verification_key = zk_snark_library.setup(circuit)

# Generate proof (requires knowing S!)
proof = zk_snark_library.prove(
    proving_key,
    public_inputs={'hash_H': target_hash},
    private_inputs={'sigil_S': ???}  # Don't have this!
)

# Verify proof
valid = zk_snark_library.verify(verification_key, proof, public_inputs)

if valid:
    print("Proof valid! You know the sigil!")
else:
    print("Proof invalid - you don't know the sigil")

// Ali Baba's Cave Protocol
class AliBabaCave {
    constructor(has_password) {
        this.has_password = has_password;
    }

    prove_knowledge() {
        // Prover enters cave, chooses path A or B randomly
        const entry_path = random_choice(['A', 'B']);

        // Verifier (outside) can't see which path prover took
        // Verifier calls out random exit: "Exit from A!" or "Exit from B!"
        const verifier_request = random_choice(['A', 'B']);

        // If prover knows password, can open door and exit from requested side
        if (this.has_password) {
            exit_from(verifier_request);  // Always succeeds!
        } else {
            // Without password, only 50% chance of exiting correct side
            if (entry_path === verifier_request) {
                exit_from(verifier_request);  // Lucky guess!
            } else {
                // Stuck on wrong side, can't open door
                return false;  // CAUGHT!
            }
        }
    }
}

// Attempt protocol without password
prover = new AliBabaCave(has_password=false);
prover.prove_knowledge();  // 50% fail each round

# Pedersen Commitment Scheme
import hashlib

# Public parameters
g, h = generators()  # Two generators (no known discrete log relation)

# Commitment phase
def commit(sigil, randomness):
    # C = g^sigil * h^randomness (mod p)
    commitment = (pow(g, sigil, p) * pow(h, randomness, p)) % p
    return commitment

# Opening phase
def open(commitment, sigil, randomness):
    # Verify C == g^sigil * h^randomness
    recomputed = (pow(g, sigil, p) * pow(h, randomness, p)) % p
    return commitment == recomputed

# Create commitment to Phase Ω sigil
sigil = ???  # Don't know this!
r = random_value()
C = commit(sigil, r)

# Later: prove commitment opens to correct sigil
if open(C, phase_omega_sigil, r):
    print("Commitment valid!")
else:
    print("Can't open commitment to unknown value")

// Fiat-Shamir Transform
function fiatShamirProof(secret) {
    // Interactive: Prover → Verifier → Prover
    // Non-interactive: Use hash for "verifier's" challenge

    // Step 1: Commitment
    const r = random_value();
    const commitment = g ** r;  // Commit

    // Step 2: Challenge (from hash, not verifier)
    const challenge = hash(commitment);  // "Random oracle"

    // Step 3: Response
    const response = r + challenge * secret;  // Requires secret!

    return { commitment, response };
}

// Attempt to generate non-interactive proof
const proof = fiatShamirProof(phase_omega_secret);  // ??? Don't have secret!

# Probabilistically Checkable Proof
def generate_pcp(statement, witness):
    # Generate proof that can be verified by random sampling
    proof = encode_witness_as_pcp(statement, witness)
    return proof  # Proof is O(n) bits, but verifier reads O(1) bits!

def verify_pcp(proof, statement):
    # Verifier randomly samples k locations in proof
    sample_locations = random_sample(proof, k=10)

    # Check consistency at sampled locations
    for location in sample_locations:
        if not check_consistency(proof, location):
            return False  # Caught cheating!

    return True  # Likely valid (probabilistic)

# Attempt to generate PCP without witness
proof = generate_pcp(
    statement="I know Phase Ω sigil",
    witness=???  # Don't have witness!
)