zero_knowledge_auth.gal

// Zero-Knowledge Authentication System
// Demonstrates GAL's formal verification of cryptographic protocols and chaos testing

import std.crypto
import std.verification
import std.chaos
import std.network
import std.time

// Formal verification contracts for zero-knowledge properties
@verify_invariant("zero_knowledge_property")
fn zero_knowledge_invariant(proof: &ZKProof, secret: &Secret) -> bool {
    // Invariant: Proof reveals no information about the secret
    // Simulator can produce indistinguishable proofs without knowing the secret
    proof.commitment.reveals_no_info_about(secret) &&
    proof.response.is_simulatable_without_secret()
}

@verify_invariant("soundness_property")
fn soundness_invariant(proof: &ZKProof, statement: &Statement) -> bool {
    // Invariant: Invalid statements cannot produce valid proofs (except with negligible probability)
    if !statement.is_true() {
        proof.verification_probability() < NEGLIGIBLE_PROBABILITY
    } else {
        true
    }
}

@verify_invariant("completeness_property")
fn completeness_invariant(prover: &Prover, statement: &Statement) -> bool {
    // Invariant: Valid statements can always be proven by honest provers
    if statement.is_true() && prover.knows_witness() {
        prover.can_generate_valid_proof()
    } else {
        true
    }
}

// Zero-knowledge proof system for discrete logarithm
struct SchnorrProof {
    commitment: GroupElement,    // g^r
    challenge: FieldElement,     // Fiat-Shamir challenge
    response: FieldElement,      // r + c*x (where x is secret)
}

impl SchnorrProof {
    // Generate proof that prover knows discrete log of public_key
    @verify_precondition("valid_secret", |secret| secret.is_valid())
    @verify_postcondition("valid_proof", |result| result.is_ok())
    fn generate(secret: &SecretKey, public_key: &PublicKey, message: &[u8]) -> Result<Self, CryptoError> {
        // Choose random nonce
        let r = FieldElement::random();
        
        // Compute commitment: g^r
        let commitment = GENERATOR.pow(&r);
        
        // Compute Fiat-Shamir challenge: H(commitment || public_key || message)
        let challenge_input = [
            commitment.to_bytes(),
            public_key.to_bytes(),
            message,
        ].concat();
        let challenge = FieldElement::from_hash(&hash_sha256(&challenge_input));
        
        // Compute response: r + c*x
        let response = r + challenge * secret.value();
        
        Ok(SchnorrProof {
            commitment,
            challenge,
            response,
        })
    }
    
    // Verify proof without learning the secret
    @verify_precondition("valid_inputs", |_, public_key, message| public_key.is_valid())
    @verify_postcondition("correct_verification", |result| result.is_ok())
    fn verify(&self, public_key: &PublicKey, message: &[u8]) -> Result<bool, CryptoError> {
        // Recompute challenge
        let challenge_input = [
            self.commitment.to_bytes(),
            public_key.to_bytes(),
            message,
        ].concat();
        let expected_challenge = FieldElement::from_hash(&hash_sha256(&challenge_input));
        
        // Check challenge matches
        if self.challenge != expected_challenge {
            return Ok(false);
        }
        
        // Verify: g^response = commitment * public_key^challenge
        let left_side = GENERATOR.pow(&self.response);
        let right_side = self.commitment + public_key.element() * self.challenge;
        
        Ok(left_side == right_side)
    }
    
    fn reveals_no_info_about(&self, _secret: &Secret) -> bool {
        // In a real implementation, this would check if the proof
        // is indistinguishable from a simulated proof
        true
    }
    
    fn is_simulatable_without_secret(&self) -> bool {
        // Zero-knowledge simulator can produce this without the secret
        true
    }
    
    fn verification_probability(&self) -> f64 {
        // Probability that an invalid proof passes verification
        2.0_f64.powf(-256.0) // Negligible for 256-bit security
    }
}

// Zero-knowledge proof for range (value is in [0, 2^n))
struct RangeProof {
    bit_commitments: Vec<GroupElement>,
    bit_proofs: Vec<SchnorrProof>,
    aggregated_proof: Option<BulletProof>,
}

impl RangeProof {
    @verify_precondition("valid_range", |value, bits| *value < 2_u64.pow(*bits as u32))
    @verify_postcondition("proof_generated", |result| result.is_ok())
    fn generate(value: u64, bits: usize, randomness: &FieldElement) -> Result<Self, CryptoError> {
        let mut bit_commitments = Vec::new();
        let mut bit_proofs = Vec::new();
        
        // Commit to each bit of the value
        for i in 0..bits {
            let bit = (value >> i) & 1;
            let bit_randomness = FieldElement::random();
            
            // Pedersen commitment: g^bit * h^randomness
            let commitment = GENERATOR.pow(&FieldElement::from(bit)) + 
                           GENERATOR_H.pow(&bit_randomness);
            bit_commitments.push(commitment);
            
            // Prove that committed value is either 0 or 1
            let bit_proof = Self::prove_bit(&FieldElement::from(bit), &bit_randomness)?;
            bit_proofs.push(bit_proof);
        }
        
        // Optional: Use Bulletproofs for logarithmic size
        let aggregated_proof = BulletProof::generate(value, bits, randomness)?;
        
        Ok(RangeProof {
            bit_commitments,
            bit_proofs,
            aggregated_proof: Some(aggregated_proof),
        })
    }
    
    @verify_postcondition("verification_result", |result| result.is_ok())
    fn verify(&self, commitment: &GroupElement, bits: usize) -> Result<bool, CryptoError> {
        // Verify each bit proof
        for (i, (bit_commitment, bit_proof)) in 
            self.bit_commitments.iter().zip(&self.bit_proofs).enumerate() {
            
            if !Self::verify_bit(bit_proof, bit_commitment)? {
                return Ok(false);
            }
        }
        
        // Verify that bit commitments aggregate to the main commitment
        let aggregated_commitment = self.bit_commitments.iter()
            .enumerate()
            .fold(GroupElement::identity(), |acc, (i, &comm)| {
                acc + comm * FieldElement::from(2_u64.pow(i as u32))
            });
            
        if aggregated_commitment != *commitment {
            return Ok(false);
        }
        
        // Verify Bulletproof if present
        if let Some(ref bulletproof) = self.aggregated_proof {
            return bulletproof.verify(commitment, bits);
        }
        
        Ok(true)
    }
    
    fn prove_bit(bit: &FieldElement, randomness: &FieldElement) -> Result<SchnorrProof, CryptoError> {
        // Simplified bit proof - in practice would use sigma protocols
        let secret_key = SecretKey::new(*bit);
        let public_key = PublicKey::from_secret(&secret_key);
        SchnorrProof::generate(&secret_key, &public_key, b"bit_proof")
    }
    
    fn verify_bit(proof: &SchnorrProof, _commitment: &GroupElement) -> Result<bool, CryptoError> {
        // Simplified verification
        Ok(true)
    }
}

// Bulletproof for efficient range proofs
struct BulletProof {
    a: GroupElement,
    s: GroupElement,
    t1: GroupElement,
    t2: GroupElement,
    tau_x: FieldElement,
    mu: FieldElement,
    inner_product_proof: InnerProductProof,
}

impl BulletProof {
    fn generate(value: u64, bits: usize, randomness: &FieldElement) -> Result<Self, CryptoError> {
        // Simplified Bulletproof generation
        Ok(BulletProof {
            a: GroupElement::random(),
            s: GroupElement::random(),
            t1: GroupElement::random(),
            t2: GroupElement::random(),
            tau_x: FieldElement::random(),
            mu: FieldElement::random(),
            inner_product_proof: InnerProductProof::generate()?,
        })
    }
    
    fn verify(&self, _commitment: &GroupElement, _bits: usize) -> Result<bool, CryptoError> {
        // Simplified verification
        Ok(true)
    }
}

// Authentication system using zero-knowledge proofs
struct ZKAuthSystem {
    users: HashMap<UserId, UserCredentials>,
    active_sessions: HashMap<SessionId, AuthSession>,
    challenge_store: HashMap<ChallengeId, AuthChallenge>,
    proof_verifier: ProofVerifier,
    chaos_config: ChaosConfig,
}

impl ZKAuthSystem {
    @verify_postcondition("system_initialized", |result| result.users.is_empty())
    fn new() -> Self {
        ZKAuthSystem {
            users: HashMap::new(),
            active_sessions: HashMap::new(),
            challenge_store: HashMap::new(),
            proof_verifier: ProofVerifier::new(),
            chaos_config: ChaosConfig::default(),
        }
    }
    
    // Register user with zero-knowledge credentials
    @verify_precondition("valid_password", |password| password.len() >= 8)
    @verify_postcondition("user_registered", |result| result.is_ok())
    fn register_user(&mut self, username: &str, password: &str, email: &str) -> Result<UserId, AuthError> {
        // Derive secret key from password using secure hash
        let secret_key = self.derive_secret_key(password)?;
        
        // Generate public key (commitment to secret)
        let public_key = PublicKey::from_secret(&secret_key);
        
        // Generate additional ZK credentials
        let credentials = UserCredentials {
            username: username.to_string(),
            public_key,
            email_hash: hash_sha256(email.as_bytes()),
            registration_date: Time::now(),
            proof_parameters: ProofParameters::default(),
        };
        
        let user_id = UserId::generate();
        self.users.insert(user_id.clone(), credentials);
        
        // Don't store the password or secret key!
        Ok(user_id)
    }
    
    // Initiate authentication challenge
    @verify_precondition("user_exists", |user_id| self.users.contains_key(user_id))
    @verify_postcondition("challenge_created", |result| result.is_ok())
    fn initiate_auth(&mut self, user_id: &UserId) -> Result<AuthChallenge, AuthError> {
        let user = self.users.get(user_id)
            .ok_or(AuthError::UserNotFound)?;
            
        // Generate random challenge
        let challenge_data = SecureRandom::bytes(32);
        let challenge_id = ChallengeId::generate();
        
        let challenge = AuthChallenge {
            id: challenge_id.clone(),
            user_id: user_id.clone(),
            challenge_data,
            timestamp: Time::now(),
            expires_at: Time::now() + Duration::minutes(5),
            public_key: user.public_key.clone(),
        };
        
        self.challenge_store.insert(challenge_id, challenge.clone());
        Ok(challenge)
    }
    
    // Complete authentication with zero-knowledge proof
    @verify_precondition("valid_challenge", |challenge_id| self.challenge_store.contains_key(challenge_id))
    @verify_postcondition("authenticated", |result| result.is_ok())
    fn complete_auth(&mut self, challenge_id: &ChallengeId, proof: SchnorrProof) -> Result<SessionToken, AuthError> {
        let challenge = self.challenge_store.remove(challenge_id)
            .ok_or(AuthError::InvalidChallenge)?;
            
        // Check challenge hasn't expired
        if Time::now() > challenge.expires_at {
            return Err(AuthError::ChallengeExpired);
        }
        
        // Verify zero-knowledge proof
        let is_valid = proof.verify(&challenge.public_key, &challenge.challenge_data)?;
        if !is_valid {
            return Err(AuthError::InvalidProof);
        }
        
        // Create authenticated session
        let session_token = SessionToken::generate();
        let session = AuthSession {
            id: SessionId::generate(),
            user_id: challenge.user_id.clone(),
            token: session_token.clone(),
            created_at: Time::now(),
            expires_at: Time::now() + Duration::hours(8),
            capabilities: vec![Capability::BasicAccess],
        };
        
        self.active_sessions.insert(session.id.clone(), session);
        Ok(session_token)
    }
    
    // Biometric authentication with zero-knowledge
    @chaos_test("biometric_spoofing")
    fn authenticate_biometric(&mut self, user_id: &UserId, biometric_template: &BiometricTemplate) -> Result<SessionToken, AuthError> {
        let user = self.users.get(user_id)
            .ok_or(AuthError::UserNotFound)?;
            
        // Generate zero-knowledge proof of biometric match
        // without revealing the actual biometric data
        let biometric_proof = self.generate_biometric_proof(biometric_template, &user.public_key)?;
        
        // Verify proof
        if !self.verify_biometric_proof(&biometric_proof, &user.public_key)? {
            return Err(AuthError::BiometricMismatch);
        }
        
        // Create session with biometric authentication
        let session_token = SessionToken::generate();
        let session = AuthSession {
            id: SessionId::generate(),
            user_id: user_id.clone(),
            token: session_token.clone(),
            created_at: Time::now(),
            expires_at: Time::now() + Duration::hours(4),
            capabilities: vec![Capability::BasicAccess, Capability::BiometricVerified],
        };
        
        self.active_sessions.insert(session.id.clone(), session);
        Ok(session_token)
    }
    
    // Multi-factor authentication with ZK proofs
    fn authenticate_mfa(&mut self, 
                       user_id: &UserId, 
                       password_proof: SchnorrProof,
                       totp_code: u32,
                       device_proof: Option<DeviceProof>) -> Result<SessionToken, AuthError> {
        
        let user = self.users.get(user_id)
            .ok_or(AuthError::UserNotFound)?;
            
        // Verify password proof
        let challenge_data = b"mfa_password_challenge";
        if !password_proof.verify(&user.public_key, challenge_data)? {
            return Err(AuthError::InvalidPasswordProof);
        }
        
        // Verify TOTP without revealing the shared secret
        if !self.verify_totp_zk(user_id, totp_code)? {
            return Err(AuthError::InvalidTOTP);
        }
        
        // Verify device proof if provided
        if let Some(device_proof) = device_proof {
            if !self.verify_device_proof(&device_proof, user_id)? {
                return Err(AuthError::UnknownDevice);
            }
        }
        
        // Create high-privilege session
        let session_token = SessionToken::generate();
        let session = AuthSession {
            id: SessionId::generate(),
            user_id: user_id.clone(),
            token: session_token.clone(),
            created_at: Time::now(),
            expires_at: Time::now() + Duration::hours(2),
            capabilities: vec![
                Capability::BasicAccess,
                Capability::MultiFactor,
                Capability::PrivilegedOperations,
            ],
        };
        
        self.active_sessions.insert(session.id.clone(), session);
        Ok(session_token)
    }
    
    // Self-modification: Adaptive security based on risk
    @chaos_test("adaptive_security")
    fn adaptive_security_adjustment(&mut self, user_id: &UserId, risk_score: f64) -> Result<(), AuthError> {
        if risk_score > 0.8 {
            // High risk: Require additional proofs
            self.require_additional_zk_proofs(user_id)?;
        } else if risk_score > 0.5 {
            // Medium risk: Shorten session duration
            self.reduce_session_duration(user_id, Duration::minutes(30))?;
        }
        // Low risk: Normal operation
        
        Ok(())
    }
    
    // Chaos engineering tests
    #[chaos_scenario("proof_forgery_attempt")]
    fn test_proof_forgery(&mut self) {
        // Simulate attacker trying to forge proofs
        let fake_secret = SecretKey::random();
        let fake_public = PublicKey::from_secret(&fake_secret);
        let legitimate_user = self.users.keys().next().unwrap().clone();
        
        // Try to forge a proof for legitimate user
        let forge_attempt = SchnorrProof::generate(&fake_secret, &fake_public, b"forged");
        
        // Should fail verification against legitimate user's public key
        if let Ok(proof) = forge_attempt {
            let user = &self.users[&legitimate_user];
            let verification = proof.verify(&user.public_key, b"forged");
            assert!(verification.is_err() || !verification.unwrap());
        }
    }
    
    #[chaos_scenario("timing_attack")]
    fn test_timing_attack_resistance(&mut self) {
        // Ensure verification time is constant regardless of input
        let start_time = Time::now();
        let _ = self.verify_constant_time_operation();
        let duration1 = Time::now() - start_time;
        
        let start_time = Time::now();
        let _ = self.verify_constant_time_operation();
        let duration2 = Time::now() - start_time;
        
        // Timing should be similar (within reasonable bounds)
        assert!((duration1 - duration2).abs() < Duration::milliseconds(10));
    }
    
    #[chaos_scenario("memory_exhaustion")]
    fn test_memory_exhaustion_resistance(&mut self) {
        // Test system behavior under memory pressure
        for _ in 0..1000 {
            let _ = SchnorrProof::generate(
                &SecretKey::random(),
                &PublicKey::random(),
                &vec![0u8; 1024]
            );
        }
        
        // System should still function correctly
        assert!(self.users.len() >= 0);
    }
    
    // Private helper methods
    fn derive_secret_key(&self, password: &str) -> Result<SecretKey, CryptoError> {
        // Use Argon2 or scrypt for password-based key derivation
        let salt = b"zk_auth_salt_unique_per_system";
        let key_material = argon2_hash(password.as_bytes(), salt, 100000, 32)?;
        Ok(SecretKey::from_bytes(&key_material))
    }
    
    fn generate_biometric_proof(&self, template: &BiometricTemplate, public_key: &PublicKey) -> Result<BiometricProof, CryptoError> {
        // Generate proof that biometric matches without revealing template
        Ok(BiometricProof {
            commitment: GroupElement::random(),
            proof: SchnorrProof::generate(&SecretKey::random(), public_key, b"biometric")?,
        })
    }
    
    fn verify_biometric_proof(&self, proof: &BiometricProof, public_key: &PublicKey) -> Result<bool, CryptoError> {
        proof.proof.verify(public_key, b"biometric")
    }
    
    fn verify_totp_zk(&self, user_id: &UserId, totp_code: u32) -> Result<bool, CryptoError> {
        // Zero-knowledge verification of TOTP without revealing shared secret
        // In practice, this would use a commitment scheme
        Ok(true) // Simplified
    }
    
    fn verify_device_proof(&self, proof: &DeviceProof, user_id: &UserId) -> Result<bool, CryptoError> {
        // Verify device is associated with user without revealing device fingerprint
        Ok(true) // Simplified
    }
    
    fn require_additional_zk_proofs(&mut self, user_id: &UserId) -> Result<(), AuthError> {
        // Require additional zero-knowledge proofs for high-risk users
        Ok(())
    }
    
    fn reduce_session_duration(&mut self, user_id: &UserId, new_duration: Duration) -> Result<(), AuthError> {
        // Reduce session duration for medium-risk users
        for session in self.active_sessions.values_mut() {
            if session.user_id == *user_id {
                session.expires_at = session.created_at + new_duration;
            }
        }
        Ok(())
    }
    
    fn verify_constant_time_operation(&self) -> bool {
        // Perform cryptographic operation in constant time
        let dummy_key = SecretKey::random();
        let dummy_public = PublicKey::from_secret(&dummy_key);
        let dummy_proof = SchnorrProof::generate(&dummy_key, &dummy_public, b"timing_test");
        dummy_proof.is_ok()
    }
}

// Supporting structures
struct UserCredentials {
    username: String,
    public_key: PublicKey,
    email_hash: [u8; 32],
    registration_date: Timestamp,
    proof_parameters: ProofParameters,
}

struct AuthChallenge {
    id: ChallengeId,
    user_id: UserId,
    challenge_data: Vec<u8>,
    timestamp: Timestamp,
    expires_at: Timestamp,
    public_key: PublicKey,
}

struct AuthSession {
    id: SessionId,
    user_id: UserId,
    token: SessionToken,
    created_at: Timestamp,
    expires_at: Timestamp,
    capabilities: Vec<Capability>,
}

struct BiometricProof {
    commitment: GroupElement,
    proof: SchnorrProof,
}

struct DeviceProof {
    device_commitment: GroupElement,
    proof: SchnorrProof,
}

struct ProofVerifier {
    parameters: SystemParameters,
}

impl ProofVerifier {
    fn new() -> Self {
        ProofVerifier {
            parameters: SystemParameters::secure_default(),
        }
    }
}

// Cryptographic primitives and types
struct GroupElement([u8; 32]);
struct FieldElement([u8; 32]);
struct SecretKey([u8; 32]);
struct PublicKey(GroupElement);
struct InnerProductProof;
struct BiometricTemplate;
struct ProofParameters;
struct SystemParameters;

// Placeholder implementations for cryptographic operations
const GENERATOR: GroupElement = GroupElement([1u8; 32]);
const GENERATOR_H: GroupElement = GroupElement([2u8; 32]);
const NEGLIGIBLE_PROBABILITY: f64 = 2.0_f64.powf(-128.0);

impl GroupElement {
    fn random() -> Self { GroupElement([0u8; 32]) }
    fn identity() -> Self { GroupElement([0u8; 32]) }
    fn pow(&self, _exp: &FieldElement) -> Self { *self }
    fn to_bytes(&self) -> Vec<u8> { self.0.to_vec() }
}

impl std::ops::Add for GroupElement {
    type Output = Self;
    fn add(self, _other: Self) -> Self { self }
}

impl std::ops::Mul<FieldElement> for GroupElement {
    type Output = GroupElement;
    fn mul(self, _scalar: FieldElement) -> GroupElement { self }
}

impl FieldElement {
    fn random() -> Self { FieldElement([0u8; 32]) }
    fn from(val: u64) -> Self { FieldElement([val as u8; 32]) }
    fn from_hash(hash: &[u8]) -> Self { FieldElement([0u8; 32]) }
    fn value(&self) -> Self { *self }
}

impl std::ops::Add for FieldElement {
    type Output = Self;
    fn add(self, _other: Self) -> Self { self }
}

impl std::ops::Mul for FieldElement {
    type Output = Self;
    fn mul(self, _other: Self) -> Self { self }
}

impl SecretKey {
    fn new(val: FieldElement) -> Self { SecretKey(val.0) }
    fn random() -> Self { SecretKey([0u8; 32]) }
    fn from_bytes(bytes: &[u8]) -> Self { SecretKey([0u8; 32]) }
    fn value(&self) -> FieldElement { FieldElement(self.0) }
    fn is_valid(&self) -> bool { true }
}

impl PublicKey {
    fn from_secret(secret: &SecretKey) -> Self { PublicKey(GENERATOR.pow(&secret.value())) }
    fn random() -> Self { PublicKey(GroupElement::random()) }
    fn element(&self) -> GroupElement { self.0 }
    fn to_bytes(&self) -> Vec<u8> { self.0.to_bytes() }
    fn is_valid(&self) -> bool { true }
}

impl InnerProductProof {
    fn generate() -> Result<Self, CryptoError> { Ok(InnerProductProof) }
}

impl SystemParameters {
    fn secure_default() -> Self { SystemParameters }
}

// Main demonstration function
fn main() -> Result<(), Box<dyn std::error::Error>> {
    println!("🔐 Zero-Knowledge Authentication System Demo");
    
    // Initialize authentication system
    let mut auth_system = ZKAuthSystem::new();
    
    // Register a new user
    println!("\n👤 Registering new user...");
    let user_id = auth_system.register_user("alice", "secure_password123", "alice@example.com")?;
    println!("User registered with ID: {:?}", user_id);
    
    // Initiate authentication challenge
    println!("\n🎯 Initiating authentication challenge...");
    let challenge = auth_system.initiate_auth(&user_id)?;
    println!("Challenge created: {:?}", challenge.id);
    
    // Generate zero-knowledge proof (simulated user-side operation)
    println!("\n🔑 Generating zero-knowledge proof...");
    let secret_key = SecretKey::random(); // In practice, derived from user's password
    let public_key = PublicKey::from_secret(&secret_key);
    let auth_proof = SchnorrProof::generate(&secret_key, &public_key, &challenge.challenge_data)?;
    
    // Complete authentication
    println!("\n✅ Completing authentication...");
    let session_token = auth_system.complete_auth(&challenge.id, auth_proof)?;
    println!("Authentication successful! Session token: {:?}", session_token);
    
    // Demonstrate biometric authentication
    println!("\n👁️ Testing biometric authentication...");
    let biometric_template = BiometricTemplate; // Simulated biometric data
    let bio_session = auth_system.authenticate_biometric(&user_id, &biometric_template)?;
    println!("Biometric authentication successful!");
    
    // Demonstrate multi-factor authentication
    println!("\n🔒 Testing multi-factor authentication...");
    let password_proof = SchnorrProof::generate(&secret_key, &public_key, b"mfa_password_challenge")?;
    let totp_code = 123456; // Simulated TOTP code
    let device_proof = Some(DeviceProof {
        device_commitment: GroupElement::random(),
        proof: SchnorrProof::generate(&SecretKey::random(), &PublicKey::random(), b"device")?,
    });
    
    let mfa_session = auth_system.authenticate_mfa(&user_id, password_proof, totp_code, device_proof)?;
    println!("Multi-factor authentication successful!");
    
    // Test adaptive security
    println!("\n🧠 Testing adaptive security...");
    auth_system.adaptive_security_adjustment(&user_id, 0.9)?; // High risk score
    println!("Security level adapted based on risk assessment");
    
    // Run chaos engineering tests
    println!("\n🔥 Running chaos engineering tests...");
    auth_system.test_proof_forgery();
    auth_system.test_timing_attack_resistance();
    auth_system.test_memory_exhaustion_resistance();
    
    // Demonstrate range proof
    println!("\n📊 Testing range proof...");
    let age = 25u64; // Prove age is in valid range without revealing exact value
    let age_randomness = FieldElement::random();
    let range_proof = RangeProof::generate(age, 8, &age_randomness)?; // Age fits in 8 bits (0-255)
    
    let age_commitment = GENERATOR.pow(&FieldElement::from(age)) + GENERATOR_H.pow(&age_randomness);
    let range_valid = range_proof.verify(&age_commitment, 8)?;
    println!("Range proof verified: {}", range_valid);
    
    println!("\n🎉 Zero-Knowledge Authentication Demo Completed!");
    println!("✓ User registration with ZK credentials");
    println!("✓ Challenge-response authentication");
    println!("✓ Zero-knowledge proof generation and verification");
    println!("✓ Biometric authentication without template exposure");
    println!("✓ Multi-factor authentication with ZK proofs");
    println!("✓ Adaptive security based on risk assessment");
    println!("✓ Range proofs for privacy-preserving age verification");
    println!("✓ Chaos engineering tests for security resilience");
    println!("✓ Formal verification of cryptographic properties");
    
    Ok(())
}

// Error types and additional supporting code
#[derive(Debug)]
enum AuthError {
    UserNotFound,
    InvalidChallenge,
    ChallengeExpired,
    InvalidProof,
    BiometricMismatch,
    InvalidPasswordProof,
    InvalidTOTP,
    UnknownDevice,
    CryptoError(CryptoError),
}

#[derive(Debug)]
enum CryptoError {
    InvalidParameters,
    RandomnessError,
    VerificationFailed,
}

impl From<CryptoError> for AuthError {
    fn from(e: CryptoError) -> Self {
        AuthError::CryptoError(e)
    }
}

#[derive(Debug, Clone, Hash, PartialEq, Eq)]
struct UserId(u64);
#[derive(Debug, Clone, Hash, PartialEq, Eq)]
struct SessionId(u64);
#[derive(Debug, Clone, Hash, PartialEq, Eq)]
struct ChallengeId(u64);
#[derive(Debug, Clone)]
struct SessionToken(String);

impl UserId {
    fn generate() -> Self { UserId(42) }
}

impl SessionId {
    fn generate() -> Self { SessionId(42) }
}

impl ChallengeId {
    fn generate() -> Self { ChallengeId(42) }
}

impl SessionToken {
    fn generate() -> Self { SessionToken("token_123".to_string()) }
}

#[derive(Debug, Clone)]
enum Capability {
    BasicAccess,
    BiometricVerified,
    MultiFactor,
    PrivilegedOperations,
}

// Chaos configuration
struct ChaosConfig {
    proof_forgery_enabled: bool,
    timing_attack_simulation: bool,
    memory_pressure_testing: bool,
}

impl Default for ChaosConfig {
    fn default() -> Self {
        ChaosConfig {
            proof_forgery_enabled: true,
            timing_attack_simulation: true,
            memory_pressure_testing: true,
        }
    }
}

// Helper functions
fn hash_sha256(data: &[u8]) -> [u8; 32] {
    [0u8; 32] // Simplified
}

fn argon2_hash(password: &[u8], salt: &[u8], iterations: u32, output_len: usize) -> Result<Vec<u8>, CryptoError> {
    Ok(vec![0u8; output_len]) // Simplified
}

struct SecureRandom;
impl SecureRandom {
    fn bytes(len: usize) -> Vec<u8> {
        vec![0u8; len] // Simplified
    }
}

// Additional type aliases
type Timestamp = u64;
type Duration = std::time::Duration;
type Time = u64;

// Statement and proof types for formal verification
struct Statement;
struct Secret;
struct Prover;

impl Statement {
    fn is_true(&self) -> bool { true }
}

impl Secret {
    fn reveals_no_info_about(&self, _secret: &Secret) -> bool { true }
}

impl Prover {
    fn knows_witness(&self) -> bool { true }
    fn can_generate_valid_proof(&self) -> bool { true }
}