Cryptography-privacy
Cryptography and Privacy Engineering: From Zero-Knowledge to Post-Quantum Security
BugB Security Team
Security Researchers
2025-06-12
0 min read
BugB Security Team
Security Researchers
The intersection of cryptography and privacy engineering has become critical as organizations navigate increasing regulatory requirements, sophisticated threat actors, and the looming quantum computing revolution. Modern privacy-preserving systems require advanced cryptographic primitives that go far beyond traditional encryption to enable computation on encrypted data, verifiable privacy, and quantum-resistant security.
This comprehensive analysis explores cutting-edge cryptographic techniques and their practical implementation in privacy-preserving systems, demonstrating how theoretical advances translate into real-world security and privacy solutions.
Contemporary cryptographic systems must address multiple complex requirements simultaneously:
| Traditional Cryptography | Modern Privacy Engineering | Post-Quantum Era |
|---|---|---|
| Confidentiality | Data minimization | Quantum-resistant confidentiality |
| Integrity | Verifiable computation | Quantum-resistant authentication |
| Authentication | Zero-knowledge identity | Post-quantum digital signatures |
| Key Management | Forward secrecy | Quantum key distribution |
| Performance | Homomorphic operations | Efficient post-quantum algorithms |
| Compliance | Privacy-by-design | Crypto-agility frameworks |
┌─────────────────────────────────────────────────────────────┐
│ Modern Cryptographic Threats │
├─────────────────────────────────────────────────────────────┤
│ Classical Attacks │ → Brute force, cryptanalysis │
│ │ Side-channel attacks │
│ │ Implementation vulnerabilities │
├─────────────────────────────────────────────────────────────┤
│ Quantum Attacks │ → Shor's algorithm (RSA/ECC) │
│ │ Grover's algorithm (AES) │
│ │ Quantum cryptanalysis │
├─────────────────────────────────────────────────────────────┤
│ Privacy Attacks │ → Linkability analysis │
│ │ Inference attacks │
│ │ De-anonymization techniques │
├─────────────────────────────────────────────────────────────┤
│ AI-Enhanced Attacks │ → ML-based cryptanalysis │
│ │ Automated vulnerability discovery│
│ │ Adversarial AI techniques │
└─────────────────────────────────────────────────────────────┘
#!/usr/bin/env python3
"""
Zero-Knowledge Proof Implementation Framework
Advanced ZKP systems for privacy-preserving applications
"""
import hashlib
import random
import json
from typing import Tuple, List, Dict, Any
from dataclasses import dataclass
from abc import ABC, abstractmethod
@dataclass
class ZKProof:
"""Zero-knowledge proof structure"""
commitment: bytes
challenge: bytes
response: bytes
public_parameters: Dict[str, Any]
class ZKProofSystem(ABC):
"""Abstract base class for ZK proof systems"""
@abstractmethod
def setup(self, security_parameter: int) -> Dict[str, Any]:
"""Generate public parameters"""
pass
@abstractmethod
def prove(self, statement: Any, witness: Any, public_params: Dict[str, Any]) -> ZKProof:
"""Generate zero-knowledge proof"""
pass
@abstractmethod
def verify(self, statement: Any, proof: ZKProof) -> bool:
"""Verify zero-knowledge proof"""
pass
class SchnorrZKP(ZKProofSystem):
"""Schnorr Zero-Knowledge Proof implementation"""
def __init__(self):
# Use safe prime for demonstration (in practice, use standardized groups)
self.p = 0x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
self.g = 2
self.q = (self.p - 1) // 2 # Order of subgroup
def setup(self, security_parameter: int = 256) -> Dict[str, Any]:
"""Setup Schnorr proof system parameters"""
return {
"p": self.p,
"g": self.g,
"q": self.q,
"security_parameter": security_parameter
}
def prove(self, statement: int, witness: int, public_params: Dict[str, Any]) -> ZKProof:
"""
Generate Schnorr proof for discrete logarithm knowledge
Statement: y = g^x mod p
Witness: x (secret exponent)
"""
p, g, q = public_params["p"], public_params["g"], public_params["q"]
# Commitment phase: choose random r, compute A = g^r mod p
r = random.randint(1, q - 1)
commitment = pow(g, r, p)
# Challenge phase: compute challenge as hash of public info
challenge_input = f"{statement}_{commitment}_{g}_{p}".encode()
challenge = int(hashlib.sha256(challenge_input).hexdigest(), 16) % q
# Response phase: compute s = r + c*x mod q
response = (r + challenge * witness) % q
return ZKProof(
commitment=commitment.to_bytes((p.bit_length() + 7) // 8, 'big'),
challenge=challenge.to_bytes(32, 'big'),
response=response.to_bytes((q.bit_length() + 7) // 8, 'big'),
public_parameters=public_params
)
def verify(self, statement: int, proof: ZKProof) -> bool:
"""Verify Schnorr proof"""
try:
p = proof.public_parameters["p"]
g = proof.public_parameters["g"]
q = proof.public_parameters["q"]
# Extract proof components
commitment = int.from_bytes(proof.commitment, 'big')
challenge = int.from_bytes(proof.challenge, 'big')
response = int.from_bytes(proof.response, 'big')
# Verify challenge is correctly computed
challenge_input = f"{statement}_{commitment}_{g}_{p}".encode()
expected_challenge = int(hashlib.sha256(challenge_input).hexdigest(), 16) % q
if challenge != expected_challenge:
return False
# Verify proof equation: g^s = A * y^c mod p
left_side = pow(g, response, p)
right_side = (commitment * pow(statement, challenge, p)) % p
return left_side == right_side
except Exception as e:
print(f"Verification error: {e}")
return False
class BulletproofRangeProof:
"""Bulletproof range proof implementation (simplified)"""
def __init__(self, bit_length: int = 64):
self.bit_length = bit_length
self.setup_parameters()
def setup_parameters(self):
"""Setup Bulletproof parameters"""
# Simplified elliptic curve parameters (use proper curve in production)
self.p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
self.n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
# Generator points (simplified - use proper curve points)
self.G = (0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8)
self.H = (0xC6047F9441ED7D6D3045406E95C07CD85C778E4B8CEF3CA7ABAC09B95C709EE5,
0x1AE168FEA63DC339A3C58419466CEAEEF7F632653266D0E1236431A950CFE52A)
def prove_range(self, value: int, randomness: int) -> Dict[str, Any]:
"""
Generate range proof for value in [0, 2^bit_length)
Returns simplified proof structure
"""
if value < 0 or value >= 2**self.bit_length:
raise ValueError(f"Value must be in range [0, {2**self.bit_length})")
# Convert value to binary representation
binary_repr = [(value >> i) & 1 for i in range(self.bit_length)]
# Generate commitments for each bit
bit_commitments = []
bit_randomness = []
for bit in binary_repr:
r = random.randint(1, self.n - 1)
bit_randomness.append(r)
# Simplified commitment: C = bit * G + r * H (using scalar representation)
commitment = (bit * self.G[0] + r * self.H[0]) % self.p
bit_commitments.append(commitment)
# Generate proof of bit validity (simplified)
bit_proofs = []
for i, (bit, r) in enumerate(zip(binary_repr, bit_randomness)):
# Prove bit is 0 or 1 using Schnorr-like proof
schnorr_zkp = SchnorrZKP()
params = schnorr_zkp.setup()
# Simplified bit proof
bit_proof = {
"commitment": bit_commitments[i],
"is_valid_bit": True, # Simplified - real implementation would have ZKP
"randomness_commitment": r
}
bit_proofs.append(bit_proof)
return {
"type": "bulletproof_range",
"bit_length": self.bit_length,
"value_commitment": sum(bit_commitments) % self.p,
"bit_commitments": bit_commitments,
"bit_proofs": bit_proofs,
"aggregated_proof": self.aggregate_bit_proofs(bit_proofs)
}
def verify_range_proof(self, proof: Dict[str, Any], public_commitment: int) -> bool:
"""Verify range proof"""
try:
# Verify bit commitments sum to public commitment
computed_commitment = sum(proof["bit_commitments"]) % self.p
if computed_commitment != public_commitment:
return False
# Verify each bit proof (simplified)
for bit_proof in proof["bit_proofs"]:
if not bit_proof["is_valid_bit"]:
return False
# Verify aggregated proof
return self.verify_aggregated_proof(proof["aggregated_proof"])
except Exception as e:
print(f"Range proof verification error: {e}")
return False
def aggregate_bit_proofs(self, bit_proofs: List[Dict[str, Any]]) -> Dict[str, Any]:
"""Aggregate individual bit proofs (simplified)"""
return {
"aggregated": True,
"proof_size_reduction": len(bit_proofs) * 0.7, # Simplified metric
"verification_time": len(bit_proofs) * 0.1 # Simplified metric
}
def verify_aggregated_proof(self, aggregated_proof: Dict[str, Any]) -> bool:
"""Verify aggregated proof (simplified)"""
return aggregated_proof.get("aggregated", False)
class ZKSNARKSystem:
"""zk-SNARK implementation framework (simplified)"""
def __init__(self):
self.setup_done = False
self.proving_key = None
self.verification_key = None
def trusted_setup(self, circuit_description: Dict[str, Any]) -> Tuple[Dict, Dict]:
"""
Perform trusted setup for zk-SNARK
WARNING: This is a simplified version - real implementation requires
secure multiparty computation for setup
"""
# Generate random toxic waste (should be deleted after setup)
tau = random.randint(1, 2**256)
alpha = random.randint(1, 2**256)
beta = random.randint(1, 2**256)
gamma = random.randint(1, 2**256)
# Setup parameters based on circuit
circuit_size = circuit_description.get("num_constraints", 100)
# Proving key (simplified structure)
proving_key = {
"alpha": alpha,
"beta": beta,
"circuit_size": circuit_size,
"constraint_system": circuit_description,
"toxic_waste_used": True # In real implementation, this would be deleted
}
# Verification key (public)
verification_key = {
"gamma": gamma,
"circuit_hash": hashlib.sha256(str(circuit_description).encode()).hexdigest(),
"public_inputs_count": circuit_description.get("public_inputs", 0)
}
self.proving_key = proving_key
self.verification_key = verification_key
self.setup_done = True
return proving_key, verification_key
def generate_proof(self, public_inputs: List[int], private_witness: List[int]) -> Dict[str, Any]:
"""Generate zk-SNARK proof"""
if not self.setup_done:
raise RuntimeError("Trusted setup must be performed first")
# Simulate constraint satisfaction
if not self.check_constraint_satisfaction(public_inputs, private_witness):
raise ValueError("Constraint system not satisfied")
# Generate proof components (simplified)
proof = {
"A": self.compute_proof_element_A(private_witness),
"B": self.compute_proof_element_B(private_witness),
"C": self.compute_proof_element_C(private_witness),
"public_inputs": public_inputs,
"proof_size": 3 * 32, # Simplified: 3 group elements of 32 bytes each
"generation_time": len(private_witness) * 0.01 # Simplified timing
}
return proof
def verify_proof(self, proof: Dict[str, Any]) -> bool:
"""Verify zk-SNARK proof"""
if not self.setup_done:
return False
try:
# Verify proof equation (simplified)
# Real implementation would use elliptic curve pairing
A = proof["A"]
B = proof["B"]
C = proof["C"]
# Simplified verification equation
verification_result = (A * B) % (2**256) == C % (2**256)
return verification_result
except Exception as e:
print(f"SNARK verification error: {e}")
return False
def check_constraint_satisfaction(self, public_inputs: List[int], private_witness: List[int]) -> bool:
"""Check if inputs satisfy circuit constraints (simplified)"""
# Simplified constraint checking
if len(public_inputs) + len(private_witness) < self.proving_key["circuit_size"]:
return False
# Example constraint: sum of inputs equals first public input
total_sum = sum(public_inputs[1:]) + sum(private_witness)
return total_sum == public_inputs[0]
def compute_proof_element_A(self, witness: List[int]) -> int:
"""Compute proof element A (simplified)"""
return sum(w * (i + 1) for i, w in enumerate(witness)) % (2**256)
def compute_proof_element_B(self, witness: List[int]) -> int:
"""Compute proof element B (simplified)"""
return sum(w * w for w in witness) % (2**256)
def compute_proof_element_C(self, witness: List[int]) -> int:
"""Compute proof element C (simplified)"""
A = self.compute_proof_element_A(witness)
B = self.compute_proof_element_B(witness)
return (A * B) % (2**256)
def demonstrate_zkp_applications():
"""Demonstrate practical ZKP applications"""
applications = {
"private_voting": {
"description": "Anonymous voting with verifiable tallies",
"zkp_type": "Range proofs + NIZKs",
"privacy_guarantee": "Vote secrecy",
"verifiability": "Public tally verification"
},
"credential_verification": {
"description": "Prove credential validity without revealing credential",
"zkp_type": "zk-SNARKs",
"privacy_guarantee": "Credential privacy",
"use_cases": ["Age verification", "Qualification proof", "Access control"]
},
"financial_privacy": {
"description": "Private transactions with public auditability",
"zkp_type": "Bulletproofs + zk-STARKs",
"privacy_guarantee": "Transaction amount privacy",
"compliance": "Regulatory reporting without data exposure"
},
"supply_chain_verification": {
"description": "Prove supply chain compliance without revealing trade secrets",
"zkp_type": "Merkle tree proofs + NIZKs",
"privacy_guarantee": "Process confidentiality",
"verification": "Compliance without data sharing"
},
"identity_systems": {
"description": "Self-sovereign identity with selective disclosure",
"zkp_type": "CL signatures + zk-SNARKs",
"privacy_guarantee": "Attribute privacy",
"features": ["Selective disclosure", "Unlinkable presentations", "Revocation"]
}
}
return applications
#!/usr/bin/env python3
"""
Homomorphic Encryption Implementation Framework
Privacy-preserving computation on encrypted data
"""
import random
import math
from typing import List, Tuple, Union
from dataclasses import dataclass
@dataclass
class HECiphertext:
"""Homomorphic encryption ciphertext"""
data: Union[int, List[int]]
noise_level: float
scheme: str
@dataclass
class HEPublicKey:
"""Homomorphic encryption public key"""
n: int # Modulus
g: int # Generator (for some schemes)
scheme: str
@dataclass
class HEPrivateKey:
"""Homomorphic encryption private key"""
lambda_val: int
mu: int
scheme: str
class PaillierHE:
"""Paillier homomorphic encryption (additively homomorphic)"""
def __init__(self, key_size: int = 2048):
self.key_size = key_size
self.public_key = None
self.private_key = None
def generate_keypair(self) -> Tuple[HEPublicKey, HEPrivateKey]:
"""Generate Paillier key pair"""
# Generate two large primes
p = self._generate_prime(self.key_size // 2)
q = self._generate_prime(self.key_size // 2)
n = p * q
lambda_val = self._lcm(p - 1, q - 1)
# Choose generator g
g = n + 1 # Common choice for efficiency
# Compute mu = (L(g^lambda mod n^2))^(-1) mod n
# where L(x) = (x - 1) / n
g_lambda = pow(g, lambda_val, n * n)
l_result = (g_lambda - 1) // n
mu = self._mod_inverse(l_result, n)
public_key = HEPublicKey(n=n, g=g, scheme="paillier")
private_key = HEPrivateKey(lambda_val=lambda_val, mu=mu, scheme="paillier")
self.public_key = public_key
self.private_key = private_key
return public_key, private_key
def encrypt(self, plaintext: int, public_key: HEPublicKey = None) -> HECiphertext:
"""Encrypt plaintext using Paillier encryption"""
if public_key is None:
public_key = self.public_key
if public_key is None:
raise ValueError("Public key not provided")
n = public_key.n
g = public_key.g
# Choose random r coprime to n
r = self._random_coprime(n)
# Compute ciphertext: c = g^m * r^n mod n^2
ciphertext = (pow(g, plaintext, n * n) * pow(r, n, n * n)) % (n * n)
return HECiphertext(
data=ciphertext,
noise_level=0.0, # Paillier is exact
scheme="paillier"
)
def decrypt(self, ciphertext: HECiphertext, private_key: HEPrivateKey = None) -> int:
"""Decrypt Paillier ciphertext"""
if private_key is None:
private_key = self.private_key
if private_key is None:
raise ValueError("Private key not provided")
n = self.public_key.n
lambda_val = private_key.lambda_val
mu = private_key.mu
# Compute c^lambda mod n^2
c_lambda = pow(ciphertext.data, lambda_val, n * n)
# Apply L function: L(x) = (x - 1) / n
l_result = (c_lambda - 1) // n
# Compute plaintext: m = L(c^lambda mod n^2) * mu mod n
plaintext = (l_result * mu) % n
return plaintext
def add_encrypted(self, ct1: HECiphertext, ct2: HECiphertext) -> HECiphertext:
"""Homomorphic addition of encrypted values"""
if ct1.scheme != "paillier" or ct2.scheme != "paillier":
raise ValueError("Ciphertexts must be Paillier encrypted")
n = self.public_key.n
# Homomorphic addition: E(m1) * E(m2) mod n^2 = E(m1 + m2)
result_data = (ct1.data * ct2.data) % (n * n)
return HECiphertext(
data=result_data,
noise_level=max(ct1.noise_level, ct2.noise_level),
scheme="paillier"
)
def multiply_by_constant(self, ciphertext: HECiphertext, constant: int) -> HECiphertext:
"""Homomorphic multiplication by plaintext constant"""
n = self.public_key.n
# E(m)^k mod n^2 = E(k * m)
result_data = pow(ciphertext.data, constant, n * n)
return HECiphertext(
data=result_data,
noise_level=ciphertext.noise_level,
scheme="paillier"
)
def _generate_prime(self, bits: int) -> int:
"""Generate random prime of specified bit length (simplified)"""
while True:
candidate = random.getrandbits(bits)
candidate |= (1 << bits - 1) | 1 # Ensure it's odd and has correct bit length
if self._is_prime(candidate):
return candidate
def _is_prime(self, n: int, k: int = 10) -> bool:
"""Miller-Rabin primality test"""
if n < 2:
return False
if n == 2 or n == 3:
return True
if n % 2 == 0:
return False
# Write n-1 as d * 2^r
r = 0
d = n - 1
while d % 2 == 0:
r += 1
d //= 2
# Perform k rounds of testing
for _ in range(k):
a = random.randrange(2, n - 1)
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
for _ in range(r - 1):
x = pow(x, 2, n)
if x == n - 1:
break
else:
return False
return True
def _lcm(self, a: int, b: int) -> int:
"""Compute least common multiple"""
return abs(a * b) // math.gcd(a, b)
def _mod_inverse(self, a: int, m: int) -> int:
"""Compute modular inverse using extended Euclidean algorithm"""
if math.gcd(a, m) != 1:
raise ValueError("Modular inverse does not exist")
def extended_gcd(a, b):
if a == 0:
return b, 0, 1
gcd, x1, y1 = extended_gcd(b % a, a)
x = y1 - (b // a) * x1
y = x1
return gcd, x, y
_, x, _ = extended_gcd(a % m, m)
return x % m
def _random_coprime(self, n: int) -> int:
"""Generate random number coprime to n"""
while True:
r = random.randrange(1, n)
if math.gcd(r, n) == 1:
return r
class CKKSHE:
"""CKKS homomorphic encryption (approximate, supports floating point)"""
def __init__(self, polynomial_degree: int = 8192, coefficient_modulus_bits: int = 438):
self.polynomial_degree = polynomial_degree
self.coefficient_modulus_bits = coefficient_modulus_bits
self.scaling_factor = 2**40 # For precision control
self.setup_parameters()
def setup_parameters(self):
"""Setup CKKS parameters (simplified)"""
# In real implementation, would use Number Theoretic Transform (NTT) friendly primes
self.modulus = 2**self.coefficient_modulus_bits - 1
self.primitive_root = self._find_primitive_root(self.polynomial_degree)
def generate_keypair(self) -> Tuple[dict, dict]:
"""Generate CKKS key pair (simplified)"""
# Generate secret key as polynomial with small coefficients
secret_key = [random.randint(-1, 1) for _ in range(self.polynomial_degree)]
# Generate public key (a, b) where b = a*s + e
a = [random.randint(0, self.modulus - 1) for _ in range(self.polynomial_degree)]
error = [random.gauss(0, 3.2) for _ in range(self.polynomial_degree)] # Gaussian error
b = [(a[i] * secret_key[i] + int(error[i])) % self.modulus
for i in range(self.polynomial_degree)]
public_key = {"a": a, "b": b}
private_key = {"s": secret_key}
return public_key, private_key
def encode(self, values: List[float]) -> List[int]:
"""Encode floating point values into polynomial coefficients"""
if len(values) > self.polynomial_degree // 2:
raise ValueError("Too many values to encode")
# Pad with zeros if necessary
while len(values) < self.polynomial_degree // 2:
values.append(0.0)
# Scale and round to integers
scaled_values = [int(v * self.scaling_factor) for v in values]
# Embed into polynomial using complex encoding (simplified)
polynomial = scaled_values + [0] * (self.polynomial_degree - len(scaled_values))
return polynomial
def decode(self, polynomial: List[int]) -> List[float]:
"""Decode polynomial coefficients back to floating point values"""
# Extract the first half (real parts in simplified encoding)
scaled_values = polynomial[:self.polynomial_degree // 2]
# Unscale to floating point
values = [v / self.scaling_factor for v in scaled_values]
return values
def encrypt(self, plaintext_poly: List[int], public_key: dict) -> dict:
"""Encrypt polynomial using CKKS (simplified)"""
a, b = public_key["a"], public_key["b"]
# Generate random polynomial u with small coefficients
u = [random.randint(-1, 1) for _ in range(self.polynomial_degree)]
# Generate error polynomials
e0 = [random.gauss(0, 3.2) for _ in range(self.polynomial_degree)]
e1 = [random.gauss(0, 3.2) for _ in range(self.polynomial_degree)]
# Compute ciphertext (c0, c1)
c0 = [(b[i] * u[i] + int(e0[i]) + plaintext_poly[i]) % self.modulus
for i in range(self.polynomial_degree)]
c1 = [(a[i] * u[i] + int(e1[i])) % self.modulus
for i in range(self.polynomial_degree)]
return {
"c0": c0,
"c1": c1,
"noise_budget": self.coefficient_modulus_bits - 20 # Simplified noise tracking
}
def decrypt(self, ciphertext: dict, private_key: dict) -> List[int]:
"""Decrypt CKKS ciphertext (simplified)"""
c0, c1 = ciphertext["c0"], ciphertext["c1"]
s = private_key["s"]
# Compute c0 + c1 * s
decrypted = [(c0[i] + c1[i] * s[i]) % self.modulus
for i in range(self.polynomial_degree)]
return decrypted
def add_encrypted(self, ct1: dict, ct2: dict) -> dict:
"""Homomorphic addition of CKKS ciphertexts"""
c0_result = [(ct1["c0"][i] + ct2["c0"][i]) % self.modulus
for i in range(self.polynomial_degree)]
c1_result = [(ct1["c1"][i] + ct2["c1"][i]) % self.modulus
for i in range(self.polynomial_degree)]
return {
"c0": c0_result,
"c1": c1_result,
"noise_budget": min(ct1["noise_budget"], ct2["noise_budget"]) - 1
}
def multiply_encrypted(self, ct1: dict, ct2: dict) -> dict:
"""Homomorphic multiplication of CKKS ciphertexts (simplified)"""
# Real implementation requires relinearization
# This is a simplified version
c0_result = [(ct1["c0"][i] * ct2["c0"][i]) % self.modulus
for i in range(self.polynomial_degree)]
c1_result = [(ct1["c1"][i] * ct2["c1"][i]) % self.modulus
for i in range(self.polynomial_degree)]
return {
"c0": c0_result,
"c1": c1_result,
"noise_budget": min(ct1["noise_budget"], ct2["noise_budget"]) - 5 # Noise grows faster
}
def _find_primitive_root(self, n: int) -> int:
"""Find primitive nth root of unity (simplified)"""
# In real implementation, would use NTT-friendly parameters
return 3 # Simplified placeholder
def demonstrate_privacy_preserving_ml():
"""Demonstrate privacy-preserving machine learning using HE"""
# Initialize CKKS for floating point computation
ckks = CKKSHE()
public_key, private_key = ckks.generate_keypair()
# Example: Private neural network inference
def private_neural_network_inference():
# Input data (encrypted)
input_data = [1.5, 2.3, -0.8, 1.2]
encoded_input = ckks.encode(input_data)
encrypted_input = ckks.encrypt(encoded_input, public_key)
# Model weights (can be public or encrypted)
weights = [0.5, -0.3, 0.8, 0.2]
encoded_weights = ckks.encode(weights)
encrypted_weights = ckks.encrypt(encoded_weights, public_key)
# Homomorphic computation: dot product
encrypted_result = ckks.multiply_encrypted(encrypted_input, encrypted_weights)
# Decrypt result
decrypted_poly = ckks.decrypt(encrypted_result, private_key)
result = ckks.decode(decrypted_poly)
return {
"input_privacy": "Preserved",
"computation": "Homomorphic",
"result": result[0], # First element of result
"noise_level": encrypted_result["noise_budget"]
}
# Example: Private statistical analysis
def private_statistics():
# Dataset (encrypted)
dataset = [10.5, 20.3, 15.8, 25.2, 18.7]
encoded_data = ckks.encode(dataset)
encrypted_data = ckks.encrypt(encoded_data, public_key)
# Compute encrypted sum (simplified)
encrypted_sum = encrypted_data
for _ in range(len(dataset) - 1):
encrypted_sum = ckks.add_encrypted(encrypted_sum, encrypted_data)
# Decrypt sum
decrypted_sum_poly = ckks.decrypt(encrypted_sum, private_key)
sum_result = ckks.decode(decrypted_sum_poly)
return {
"operation": "Sum computation on encrypted data",
"privacy_guarantee": "Individual values never revealed",
"result": sum_result[0] / len(dataset), # Average
"data_size": len(dataset)
}
return {
"neural_network": private_neural_network_inference(),
"statistics": private_statistics(),
"capabilities": [
"Private model inference",
"Encrypted statistical analysis",
"Secure multi-party computation",
"Privacy-preserving analytics"
]
}
#!/usr/bin/env python3
"""
Post-Quantum Cryptography Implementation Framework
Quantum-resistant cryptographic algorithms
"""
import hashlib
import random
import numpy as np
from typing import Tuple, List, Dict, Any
from dataclasses import dataclass
@dataclass
class LatticeKey:
"""Lattice-based cryptographic key"""
matrix: np.ndarray
error_distribution: str
security_parameter: int
@dataclass
class PostQuantumSignature:
"""Post-quantum digital signature"""
signature: bytes
algorithm: str
security_level: int
class NTRU:
"""NTRU lattice-based encryption (simplified implementation)"""
def __init__(self, n: int = 509, p: int = 3, q: int = 2048):
"""
Initialize NTRU parameters
n: polynomial degree
p, q: moduli (p < q, gcd(p,q) = 1)
"""
self.n = n
self.p = p
self.q = q
self.public_key = None
self.private_key = None
def generate_keypair(self) -> Tuple[Dict[str, Any], Dict[str, Any]]:
"""Generate NTRU key pair"""
# Generate private key polynomials f and g
f = self._generate_small_polynomial()
g = self._generate_small_polynomial()
# Ensure f is invertible modulo p and q
while not self._is_invertible(f, self.p) or not self._is_invertible(f, self.q):
f = self._generate_small_polynomial()
# Compute f^(-1) mod p and f^(-1) mod q
f_inv_p = self._polynomial_inverse(f, self.p)
f_inv_q = self._polynomial_inverse(f, self.q)
# Compute public key h = g * f^(-1) mod q
h = self._polynomial_multiply(g, f_inv_q, self.q)
public_key = {
"h": h,
"n": self.n,
"p": self.p,
"q": self.q
}
private_key = {
"f": f,
"f_inv_p": f_inv_p,
"g": g
}
self.public_key = public_key
self.private_key = private_key
return public_key, private_key
def encrypt(self, message: List[int], public_key: Dict[str, Any] = None) -> List[int]:
"""Encrypt message using NTRU"""
if public_key is None:
public_key = self.public_key
h = public_key["h"]
# Generate random polynomial r
r = self._generate_small_polynomial()
# Compute ciphertext e = r * h + m mod q
rh = self._polynomial_multiply(r, h, self.q)
ciphertext = [(rh[i] + message[i]) % self.q for i in range(self.n)]
return ciphertext
def decrypt(self, ciphertext: List[int], private_key: Dict[str, Any] = None) -> List[int]:
"""Decrypt NTRU ciphertext"""
if private_key is None:
private_key = self.private_key
f = private_key["f"]
f_inv_p = private_key["f_inv_p"]
# Compute a = f * e mod q
a = self._polynomial_multiply(f, ciphertext, self.q)
# Reduce a modulo p
a_mod_p = [a[i] % self.p for i in range(self.n)]
# Compute message m = a * f^(-1) mod p
message = self._polynomial_multiply(a_mod_p, f_inv_p, self.p)
return message
def _generate_small_polynomial(self) -> List[int]:
"""Generate polynomial with small coefficients"""
# Generate polynomial with coefficients in {-1, 0, 1}
return [random.choice([-1, 0, 1]) for _ in range(self.n)]
def _polynomial_multiply(self, a: List[int], b: List[int], modulus: int) -> List[int]:
"""Multiply polynomials in ring Z[x]/(x^n - 1, modulus)"""
result = [0] * self.n
for i in range(self.n):
for j in range(self.n):
# Convolution with wraparound (x^n = 1)
k = (i + j) % self.n
result[k] = (result[k] + a[i] * b[j]) % modulus
return result
def _is_invertible(self, poly: List[int], modulus: int) -> bool:
"""Check if polynomial is invertible modulo given modulus (simplified)"""
# Simplified check - in practice would use extended Euclidean algorithm
return True # Placeholder
def _polynomial_inverse(self, poly: List[int], modulus: int) -> List[int]:
"""Compute polynomial inverse (simplified implementation)"""
# Simplified inverse computation - real implementation uses extended Euclidean algorithm
# This is a placeholder that returns a valid-looking polynomial
return [(i + 1) % modulus for i in range(self.n)]
class DilithiumSignature:
"""Dilithium post-quantum signature scheme (simplified)"""
def __init__(self, security_level: int = 3):
"""
Initialize Dilithium parameters
security_level: 1-5 (higher = more secure but slower)
"""
self.security_level = security_level
self.setup_parameters()
def setup_parameters(self):
"""Setup Dilithium parameters based on security level"""
if self.security_level == 3:
self.n = 256
self.q = 8380417
self.d = 13
self.tau = 49
self.gamma1 = (1 << 19)
self.gamma2 = (self.q - 1) // 32
self.k = 6 # Number of rows in A
self.l = 5 # Number of columns in A
else:
# Default parameters
self.n = 256
self.q = 8380417
self.d = 13
self.tau = 49
self.gamma1 = (1 << 19)
self.gamma2 = (self.q - 1) // 32
self.k = 4
self.l = 4
def generate_keypair(self) -> Tuple[Dict[str, Any], Dict[str, Any]]:
"""Generate Dilithium key pair"""
# Generate random seed
seed = random.randbytes(32)
# Generate matrix A from seed (would use SHAKE-128 in real implementation)
A = self._generate_matrix_A(seed)
# Generate secret vectors s1, s2
s1 = self._generate_secret_vector(self.l)
s2 = self._generate_secret_vector(self.k)
# Compute t = A * s1 + s2
t = self._matrix_vector_multiply(A, s1)
t = self._vector_add(t, s2)
# Public key
public_key = {
"A_seed": seed,
"t": t,
"security_level": self.security_level
}
# Private key
private_key = {
"s1": s1,
"s2": s2,
"t": t
}
return public_key, private_key
def sign(self, message: bytes, private_key: Dict[str, Any]) -> PostQuantumSignature:
"""Generate Dilithium signature"""
s1 = private_key["s1"]
s2 = private_key["s2"]
# Hash message (simplified)
message_hash = hashlib.sha256(message).digest()
# Generate random y
y = self._generate_random_vector(self.l)
# Compute w = A * y (need to reconstruct A from public key)
# This is simplified - real implementation maintains A
A = np.random.randint(0, self.q, size=(self.k, self.l)) # Placeholder
w = self._matrix_vector_multiply(A, y)
# Compute challenge c from message hash and w
c = self._compute_challenge(message_hash, w)
# Compute z = y + c * s1
cs1 = self._scalar_vector_multiply(c, s1)
z = self._vector_add(y, cs1)
# Rejection sampling (simplified)
if self._norm(z) > self.gamma1 - 1:
# Restart with new y (simplified - just continue)
pass
# Compute hint h (simplified)
h = self._compute_hint(s2, c, w)
# Signature is (z, h, c)
signature_data = {
"z": z,
"h": h,
"c": c
}
# Serialize signature (simplified)
signature_bytes = str(signature_data).encode()
return PostQuantumSignature(
signature=signature_bytes,
algorithm="Dilithium",
security_level=self.security_level
)
def verify(self, message: bytes, signature: PostQuantumSignature, public_key: Dict[str, Any]) -> bool:
"""Verify Dilithium signature"""
try:
# Deserialize signature (simplified)
signature_data = eval(signature.signature.decode())
z = signature_data["z"]
h = signature_data["h"]
c = signature_data["c"]
# Reconstruct A from seed
A = self._generate_matrix_A(public_key["A_seed"])
t = public_key["t"]
# Hash message
message_hash = hashlib.sha256(message).digest()
# Verify signature equation (simplified)
# Real verification: Az - ct = w' and check h is correct
Az = self._matrix_vector_multiply(A, z)
ct = self._scalar_vector_multiply(c, t)
w_prime = self._vector_subtract(Az, ct)
# Verify challenge (simplified)
expected_c = self._compute_challenge(message_hash, w_prime)
return c == expected_c
except Exception as e:
print(f"Signature verification error: {e}")
return False
def _generate_matrix_A(self, seed: bytes) -> np.ndarray:
"""Generate matrix A from seed (simplified)"""
# Real implementation would use SHAKE-128
random.seed(int.from_bytes(seed[:4], 'big'))
return np.random.randint(0, self.q, size=(self.k, self.l))
def _generate_secret_vector(self, length: int) -> List[int]:
"""Generate secret vector with small coefficients"""
return [random.randint(-self.tau, self.tau) for _ in range(length)]
def _generate_random_vector(self, length: int) -> List[int]:
"""Generate random vector"""
return [random.randint(0, self.gamma1 - 1) for _ in range(length)]
def _matrix_vector_multiply(self, matrix: np.ndarray, vector: List[int]) -> List[int]:
"""Multiply matrix by vector modulo q"""
result = matrix.dot(vector) % self.q
return result.tolist()
def _vector_add(self, a: List[int], b: List[int]) -> List[int]:
"""Add two vectors modulo q"""
return [(a[i] + b[i]) % self.q for i in range(len(a))]
def _vector_subtract(self, a: List[int], b: List[int]) -> List[int]:
"""Subtract two vectors modulo q"""
return [(a[i] - b[i]) % self.q for i in range(len(a))]
def _scalar_vector_multiply(self, scalar: int, vector: List[int]) -> List[int]:
"""Multiply vector by scalar modulo q"""
return [(scalar * v) % self.q for v in vector]
def _compute_challenge(self, message_hash: bytes, w: List[int]) -> int:
"""Compute challenge from hash and commitment (simplified)"""
combined = message_hash + str(w).encode()
return int.from_bytes(hashlib.sha256(combined).digest()[:4], 'big') % self.q
def _compute_hint(self, s2: List[int], c: int, w: List[int]) -> List[int]:
"""Compute hint for signature (simplified)"""
return [1 if random.random() < 0.1 else 0 for _ in range(len(w))]
def _norm(self, vector: List[int]) -> int:
"""Compute infinity norm of vector"""
return max(abs(v) for v in vector)
class SPHINCS:
"""SPHINCS+ hash-based signature scheme (simplified)"""
def __init__(self, security_parameter: int = 128):
self.security_parameter = security_parameter
self.setup_parameters()
def setup_parameters(self):
"""Setup SPHINCS+ parameters"""
self.n = 32 # Hash output size (bytes)
self.h = 64 # Height of hypertree
self.d = 8 # Number of layers
self.tree_height = self.h // self.d
self.w = 16 # Winternitz parameter
def generate_keypair(self) -> Tuple[Dict[str, Any], Dict[str, Any]]:
"""Generate SPHINCS+ key pair"""
# Generate random seed
seed = random.randbytes(self.n)
# Generate secret key from seed
sk_seed = hashlib.sha256(seed + b"SK").digest()
sk_prf = hashlib.sha256(seed + b"PRF").digest()
pub_seed = hashlib.sha256(seed + b"PUB").digest()
# Compute public key root
pk_root = self._compute_root(sk_seed, pub_seed)
public_key = {
"pk_root": pk_root,
"pub_seed": pub_seed
}
private_key = {
"sk_seed": sk_seed,
"sk_prf": sk_prf,
"pub_seed": pub_seed
}
return public_key, private_key
def sign(self, message: bytes, private_key: Dict[str, Any]) -> PostQuantumSignature:
"""Generate SPHINCS+ signature"""
sk_seed = private_key["sk_seed"]
sk_prf = private_key["sk_prf"]
pub_seed = private_key["pub_seed"]
# Generate randomizer
opt_rand = hashlib.sha256(sk_prf + message).digest()
# Hash message with randomizer
msg_hash = hashlib.sha256(opt_rand + message).digest()
# Generate FORS signature
fors_sig = self._fors_sign(msg_hash, sk_seed, pub_seed)
# Generate hypertree signature
ht_sig = self._ht_sign(fors_sig["pk"], sk_seed, pub_seed)
# Combine signatures
signature_data = {
"randomizer": opt_rand,
"fors_signature": fors_sig,
"hypertree_signature": ht_sig
}
signature_bytes = json.dumps(signature_data, default=lambda x: x.hex() if isinstance(x, bytes) else x).encode()
return PostQuantumSignature(
signature=signature_bytes,
algorithm="SPHINCS+",
security_level=self.security_parameter
)
def verify(self, message: bytes, signature: PostQuantumSignature, public_key: Dict[str, Any]) -> bool:
"""Verify SPHINCS+ signature"""
try:
# Parse signature
sig_data = json.loads(signature.signature.decode())
# Recreate message hash
opt_rand = bytes.fromhex(sig_data["randomizer"])
msg_hash = hashlib.sha256(opt_rand + message).digest()
# Verify FORS signature
fors_pk = self._fors_verify(msg_hash, sig_data["fors_signature"], public_key["pub_seed"])
# Verify hypertree signature
verified_root = self._ht_verify(fors_pk, sig_data["hypertree_signature"], public_key["pub_seed"])
# Check against public key
return verified_root == public_key["pk_root"]
except Exception as e:
print(f"SPHINCS+ verification error: {e}")
return False
def _compute_root(self, sk_seed: bytes, pub_seed: bytes) -> bytes:
"""Compute root of hypertree (simplified)"""
return hashlib.sha256(sk_seed + pub_seed + b"ROOT").digest()
def _fors_sign(self, message: bytes, sk_seed: bytes, pub_seed: bytes) -> Dict[str, Any]:
"""Generate FORS signature (simplified)"""
signature = []
auth_paths = []
# Generate FORS signature elements
for i in range(self.w):
sig_element = hashlib.sha256(sk_seed + message + i.to_bytes(4, 'big')).digest()
signature.append(sig_element)
# Generate authentication path (simplified)
auth_path = [hashlib.sha256(sig_element + j.to_bytes(4, 'big')).digest()
for j in range(self.tree_height)]
auth_paths.append(auth_path)
# Compute FORS public key
fors_pk = hashlib.sha256(b"".join(signature)).digest()
return {
"signature": signature,
"auth_paths": auth_paths,
"pk": fors_pk
}
def _fors_verify(self, message: bytes, fors_sig: Dict[str, Any], pub_seed: bytes) -> bytes:
"""Verify FORS signature and return public key (simplified)"""
return fors_sig["pk"]
def _ht_sign(self, fors_pk: bytes, sk_seed: bytes, pub_seed: bytes) -> Dict[str, Any]:
"""Generate hypertree signature (simplified)"""
return {
"tree_signature": hashlib.sha256(fors_pk + sk_seed).digest(),
"auth_path": [hashlib.sha256(fors_pk + i.to_bytes(4, 'big')).digest()
for i in range(self.h)]
}
def _ht_verify(self, fors_pk: bytes, ht_sig: Dict[str, Any], pub_seed: bytes) -> bytes:
"""Verify hypertree signature (simplified)"""
return hashlib.sha256(fors_pk + pub_seed + b"VERIFIED").digest()
def demonstrate_post_quantum_migration():
"""Demonstrate post-quantum cryptography migration strategies"""
migration_strategies = {
"hybrid_classical_pq": {
"description": "Use both classical and post-quantum algorithms",
"advantages": ["Backward compatibility", "Security against both threats"],
"disadvantages": ["Increased overhead", "Complexity"],
"use_cases": ["Transition period", "High-security applications"]
},
"crypto_agility": {
"description": "Design systems to easily swap cryptographic algorithms",
"principles": [
"Algorithm abstraction layers",
"Standardized interfaces",
"Configuration-driven crypto selection",
"Automated algorithm negotiation"
],
"benefits": ["Future-proofing", "Quick response to threats"]
},
"pq_readiness_assessment": {
"areas_to_evaluate": [
"Current cryptographic inventory",
"Performance impact analysis",
"Compliance requirements",
"Integration complexity",
"Timeline constraints"
],
"tools": ["Crypto discovery scanners", "Performance benchmarks", "Migration planners"]
},
"implementation_timeline": {
"phase_1": "Assessment and planning (6-12 months)",
"phase_2": "Hybrid deployment (12-24 months)",
"phase_3": "Full PQ migration (24-36 months)",
"phase_4": "Classical algorithm deprecation (36+ months)"
}
}
return migration_strategies
This comprehensive cryptography and privacy engineering guide covers advanced topics from zero-knowledge proofs to post-quantum cryptography, providing practical implementations and frameworks for modern privacy-preserving systems.
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