Post-Quantum Cryptography

Post-Quantum Cryptography (PQC) refers to cryptographic algorithms that are secure against attacks by both classical and quantum computers. This document covers the major families of post-quantum cryptographic algorithms and their implementations.

Lattice-Based Cryptography

Learning With Errors (LWE)

Security: Based on the hardness of solving noisy linear equations
Parameters:
- Dimension (n): 512-1024
- Modulus (q): 2^23-2^32
- Error distribution: Discrete Gaussian

Python Implementation (LWE Encryption)

import numpy as np
from cryptography.hazmat.primitives import hashes
from cryptography.hazmat.primitives.kdf.hkdf import HKDF
import secrets

class LWE:
    def __init__(self, n=512, q=2**32-1):
        self.n = n  # Dimension
        self.q = q  # Modulus
        
    def keygen(self):
        # Generate secret key (small coefficients)
        s = np.random.randint(-1, 2, size=self.n)
        # Generate public key (A, b = A*s + e)
        A = np.random.randint(0, self.q, size=(self.n, self.n))
        e = np.random.normal(0, 3.2, size=self.n).astype(int)
        b = (A @ s + e) % self.q
        return (A, b), s
    
    def encrypt(self, pk, m):
        A, b = pk
        r = np.random.randint(0, 2, size=self.n)
        u = (A.T @ r) % self.q
        v = (b @ r + m * (self.q // 2)) % self.q
        return (u, v)
    
    def decrypt(self, s, ct):
        u, v = ct
        return 0 if (v - s @ u) % self.q < self.q // 2 else 1

NTRU (N-th Degree Truncated Polynomial Ring)

Security: Based on the shortest vector problem in a lattice
Parameters:
- N: 509-1277 (polynomial degree)
- q: 2048-12289 (modulus)

Hash-Based Cryptography

SPHINCS+

Type: Stateless hash-based signature scheme
Security: Based on hash function security
Parameters:
- Hash function: SHA-256, SHAKE-256, or Haraka
- Signature size: 8-41 KB

Go Implementation (SPHINCS+ Signature)

package main

import (
    "crypto/rand"
    "fmt"
    "github.com/cloudflare/circl/sign/sphincsplus"
)

func main() {
    // Select security level (128, 192, or 256 bits)
    mode := sphincsplus.SHAKE256s
    
    // Generate key pair
    publicKey, privateKey, err := sphincsplus.GenerateKey(rand.Reader, mode)
    if err != nil {
        panic(err)
    }
    
    // Sign message
    message := []byte("Sign this message with SPHINCS+")
    signature := sphincsplus.Sign(privateKey, message)
    
    // Verify signature
    valid := sphincsplus.Verify(publicKey, message, signature)
    fmt.Printf("Signature valid: %v\n", valid)
}

Code-Based Cryptography

Classic McEliece

Security: Based on decoding random linear codes
Parameters:
- n: 3488-8192 (code length)
- k: 2720-6528 (dimension)
- t: 64-128 (error correction capability)

Multivariate Cryptography

Rainbow Signature Scheme

Security: Based on solving systems of multivariate quadratic equations
Parameters:
- v1: Number of vinegar variables in first layer
- o1, o2: Number of oil variables in each layer
- q: Finite field size (typically 256)

Isogeny-Based Cryptography

SIKE (Supersingular Isogeny Key Encapsulation)

Security: Based on computing isogenies between supersingular elliptic curves
Parameters:
- p: Large prime of form 2^a * 3^b * f - 1
- l1, l2: Small primes (typically 2 and 3)

Implementation Challenges

1. LWE Parameter Selection

Task: Implement parameter selection for LWE-based schemes
Challenge: Balance security and performance
Target: 128-bit post-quantum security

2. NTRU Key Generation

Task: Implement NTRU key generation with protection against key recovery attacks
Challenge: Ensure proper parameter selection to prevent lattice reduction attacks

3. SPHINCS+ Optimization

Task: Optimize SPHINCS+ for embedded systems
Challenge: Reduce signature size while maintaining security

Migration to Post-Quantum Cryptography

Hybrid Schemes

Approach: Combine classical and post-quantum algorithms
Example: ECDH + Kyber for key exchange
Benefits: Maintains security against both classical and quantum attacks

Performance Considerations

Key Sizes: Post-quantum keys are typically larger
Computational Overhead: Some PQC algorithms require more computation
Bandwidth: Larger keys and signatures increase communication overhead