Python Calculate Primitive Root Mod

What is a Primitive Root?

A primitive root modulo n is an integer g that is a generator of the multiplicative group of integers modulo n. This means that the powers of g modulo n produce all the integers from 1 to φ(n), where φ is Euler's totient function.

For a number n, a primitive root exists if and only if n is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime. The primitive root is unique modulo n when it exists.

Python Implementation

Here's a Python function to calculate primitive roots modulo n:

import math

def find_primitive_root(n):
    if n == 1:
        return 0
    if n == 2:
        return 1
    if n == 4:
        return 3

    # Check if n is a power of 2
    if (n & (n - 1)) == 0:
        return None

    # Check if n is twice a power of an odd prime
    m = n
    if m % 2 == 0:
        m //= 2
        if (m & (m - 1)) != 0:
            return None

    # Find the prime factors of φ(n)
    phi = euler_phi(n)
    factors = set()
    temp = phi
    for i in range(2, int(math.sqrt(temp)) + 1):
        while temp % i == 0:
            factors.add(i)
            temp //= i
    if temp > 1:
        factors.add(temp)

    # Test each number from 2 to n-1
    for g in range(2, n):
        ok = True
        for p in factors:
            if pow(g, phi // p, n) == 1:
                ok = False
                break
        if ok:
            return g
    return None

def euler_phi(n):
    result = n
    p = 2
    while p * p <= n:
        if n % p == 0:
            while n % p == 0:
                n //= p
            result -= result // p
        p += 1
    if n > 1:
        result -= result // n
    return result

The function first checks if a primitive root exists for the given n. If it does, it then finds the smallest primitive root by testing each number from 2 to n-1.

Example Calculation

Let's find a primitive root modulo 7:

1. First, calculate φ(7) = 6 (since 7 is prime).
2. The prime factors of 6 are 2 and 3.
3. Test numbers from 2 to 6:
   • 2³ ≡ 1 mod 7 (6/3=2, so 2 is not a primitive root)
   • 3¹ ≡ 3 mod 7 (3/1=3, 3/2=9≡2 mod 7, so 3 is a primitive root)

The smallest primitive root modulo 7 is 3.

Common Errors

When calculating primitive roots, several common mistakes can occur:

• Incorrect φ(n) calculation: Euler's totient function must be calculated correctly for the algorithm to work.
• Missing prime factorization: The prime factors of φ(n) must be found to properly test potential primitive roots.
• Inefficient testing: Testing all numbers up to n-1 can be slow for large n. Optimizations are needed for practical use.