Implementing-cryptography-using-python

Chapter 4 ■ Cryptographic Math and Frequency Analysis 111
# modulo inverse is a^(m-2) mode m
print("Modular multiplicative inverse is ",
power(a, m - 2, m))
a = 3
m = 11
modInverse(a, m)
NOTE In the preceding code listing, the code uses a gcd() function instead of
using the built-in math library to help you understand how to code the GCD. You will
find this helpful when you use the extended GCD.
Extending the GCD
To find the inverse when you have a nonprime modulus, you need to add a new
trick: the extended Euclidean algorithm. The extended Euclidean algorithm is
useful when two integers (a, b) are coprime; two numbers are said to be coprime
if the only positive number that divides both numbers is 1.
The next example returns the inverse using the extended Euclidean algorithm:
def egcd(a, b):
if a == 0:
return (b, 0, 1)
else:
g, y, x = egcd(b % a, a)
return (g, x - (b // a) * y, y)
def modinv(a, m):
g, x, y = egcd(a, m)
if g != 1:
raise Exception('modular inverse does not exist')
else:
return x % m
a = 102721840089015263978980446
p = 6768924473842051155435121
print modinv(a, p)
4751454584755824717584097
Euler’s Theorem
Earlier in this chapter, you were introduced to Fermat’s little theorem. We will
now examine a generalization of Fermat’s theorem known as Euler’s theorem.