Calculate P From N Rsa Python

What is RSA encryption?

RSA (Rivest-Shamir-Adleman) is a public-key cryptosystem widely used for secure data transmission. The algorithm relies on the mathematical difficulty of factoring large integers. The key components are:

• n - The modulus (product of two primes)
• e - The public exponent
• d - The private exponent
• p and q - Large prime numbers

The security of RSA depends on the difficulty of factoring large numbers. While modern RSA uses very large primes (typically 2048 bits or more), smaller examples are used for educational purposes.

How to calculate p from n

Given the modulus n = p × q, finding one of the prime factors is equivalent to factoring n. For small numbers, trial division is practical, but for large numbers, more sophisticated algorithms are needed.

Trial Division Method

The simplest approach is to test divisibility by all odd numbers up to √n:

1. Start with the smallest odd prime (3)
2. Check if n is divisible by the current number
3. If yes, that's one of the factors
4. Repeat until a factor is found

Python implementation

Here's a Python function to find a prime factor of n using trial division:

import math

def find_prime_factor(n):
    """Find a prime factor of n using trial division."""
    if n % 2 == 0:
        return 2

    max_divisor = math.isqrt(n) + 1
    for d in range(3, max_divisor, 2):
        if n % d == 0:
            return d
    return None  # n is prime

The function first checks for divisibility by 2, then tests odd numbers up to √n. For larger numbers, you would use more advanced algorithms like Pollard's Rho or Quadratic Sieve.

Worked example

Let's find a prime factor of n = 55:

1. 55 is odd, so we skip checking 2
2. Check 3: 55 ÷ 3 ≈ 18.333 → not divisible
3. Check 5: 55 ÷ 5 = 11 → divisible

The prime factor is 5. The other factor is 11 (since 5 × 11 = 55).