Python | Number Theoretic Transformation

Python | Number Theoretic Transformation

The Number Theoretic Transform (NTT) is an analogue of the Fast Fourier Transform (FFT) for the integers modulo a prime number. It's used in various applications like polynomial multiplication and in the implementation of certain cryptographic protocols. Here's a basic guide to implementing NTT in Python:

Understanding the Basics

  • The NTT works similarly to the FFT but in the context of modular arithmetic.
  • It requires a prime number p such that p=1mod(2n) where n is the length of the input array, and n is a power of 2.
  • It also requires a primitive n-th root of unity modulo p.

Python Implementation

A basic NTT implementation includes the NTT itself, its inverse, and auxiliary functions for modular arithmetic:

def modpow(base, exp, modulus): """Modular exponentiation.""" result = 1 while exp > 0: if exp % 2 == 1: result = (result * base) % modulus base = (base * base) % modulus exp = exp // 2 return result def find_primitive_root(modulus): """Find a primitive root modulo a given prime.""" # Implementation depends on the prime and its factorization # For simplicity, this could be precomputed or hardcoded for known primes pass def ntt(a, modulus, root): """Compute the Number Theoretic Transform.""" n = len(a) if n <= 1: return a even = ntt(a[0::2], modulus, root) odd = ntt(a[1::2], modulus, root) zeta = modpow(root, (modulus - 1) // n, modulus) w = 1 for i in range(n // 2): even_val = even[i] odd_val = (w * odd[i]) % modulus a[i] = (even_val + odd_val) % modulus a[i + n // 2] = (even_val - odd_val) % modulus w = (w * zeta) % modulus return a def intt(a, modulus, root): """Compute the Inverse Number Theoretic Transform.""" n = len(a) a = ntt(a, modulus, modpow(root, modulus - 2, modulus)) inv_n = modpow(n, modulus - 2, modulus) return [(val * inv_n) % modulus for val in a]

Using the NTT

  1. Choose a prime p and find a primitive root of unity modulo p.
  2. Ensure your input array length is a power of 2, and pad with zeros if necessary.
  3. Call ntt to transform, perform your operation (like convolution), and then use intt to invert.

Notes and Considerations

  • This is a basic, recursive implementation. There are more efficient, iterative approaches.
  • For practical use, especially in cryptography, you'll need a secure way to find a suitable prime and root of unity.
  • This code omits some optimizations and error checks for brevity.
  • NTT is a specialized tool; for general-purpose polynomial multiplication, FFT or other algorithms might be more suitable.

This implementation serves as a starting point and can be expanded and optimized based on specific requirements and constraints of your application.

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