Implementing-cryptography-using-python

236 Chapter 8 ■ Cryptographic Applications and PKI
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Bob publishes F, h = g a , q, and g as his public key and retains a as his
private key.
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Alice encrypts data using Bob’s public key:
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Alice selects an element k from cyclic group F such that gcd(k, q) = 1.
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Then she computes p = g k and s = h k = g ak .
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She multiplies s with M.
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Then she sends (p, M*s) = (g k , M*s).
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Bob decrypts the message:
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Bob calculates s′ = p a = g ak .
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He divides M*s by s′ to obtain M as s = s′.
The following Python code will help you gain an understanding of these
steps. You should notice many of the functions you have already learned about,
but if you need a refresher, see Chapter 4.
# Chapter 8 - ElGamal encryption
import random
from math import pow
a = random.randint(2, 10)
# Compute the GCD
def gcd(a, b):
if a < b:
return gcd(b, a)
elif a % b == 0:
return b;
else:
return gcd(b, a % b)
# Generating large random numbers
def gen_key(q):
key = random.randint(pow(10, 20), q)
while gcd(q, key) != 1:
key = random.randint(pow(10, 20), q)
return key
# Compute the power
def power(a, b, c):
x = 1
y = a
while b > 0:
if b % 2 == 0: