Implementing-cryptography-using-python

110 Chapter 4 ■ Cryptographic Math and Frequency Analysis
If the modulus is prime, Python provides an internal function called pow that
takes a, p − 2, and p to compute the inverse modulo. Consider the following
example:
>>> a = 14
>>> p = 101
>>> a_inv = pow(a, p-2, p)
>>> a_check = a * a_inv % p
>>> print("The modular inverse is ", a_inv)
The modular inverse is 65
>>> print("The check value should equal 1. It equals ", a_check)
The check value should equal 1. It equals 1
Fermat’s Little Theorem to Find the Inverse
As mentioned earlier in this chapter, we can use Fermat’s little theorem to calculate
the inverse if we know that m is prime:
a (m − 1) ≡ 1 (mod m)
a −1 ≡ a m − 2 (mod m)
Here’s the Python code to calculate the inverse if we know that m is prime:
# Fermat's little theorem.
# modular inverse of a under modulo m using
# Assumption: m is prime
def gcd(a,b):
if (b == 0):
return a
else:
return gcd(b, a % b)
# To compute x^y under modulo m
def power(x,y,m):
if (y == 0):
return 1
p = power(x, y // 2, m) % m
p = (p * p) % m
return p if(y % 2 == 0) else (x * p) % m
# Function to find modular inverse of a under modulo m
def modInverse(a,m):
if (gcd(a, m) != 1):
print("Inverse doesn't exist")
else:
# If a and m are relatively prime, then