Implementing-cryptography-using-python

Chapter 4 ■ Cryptographic Math and Frequency Analysis 97
A = B * Q + R and B ≠ 0 then GCD(A, B) = GCD(B,R)
Therefore, write A using the quotient remainder form: A = B * Q + R
Find GCD(B, R)
Euclid’s algorithm works by continuously dividing one number into another
and calculating the quotient and remainder at each step. Each phase produces a
decreasing sequence of remainders, which terminate at zero, and the last nonzero
remainder in the sequence is the GCD. You will revisit Euclid’s algorithm
shortly when you examine the modular inverses; for now, you can use the
algorithm to write a GCD function in Python:
def gcd(a,b):
if b == 0:
return a
else:
return gcd(b, a % b)
print(gcd(12,18))
Now that you know how to create your own GCD function, note that it is
very inefficient due to its use of recursion. Prefer using Python’s built-in GCD
function, which is part of the standard Python math library. You will see an
example of this when you explore Euler’s totient function later in this chapter.
Prime Numbers
Prime numbers in cryptography are vital to the security of our encryption
schemes. Prime factorization, also known as integer factorization, is a mathematical
problem that is used to secure public-key encryption schemes. This is achieved
by using extremely large semiprime numbers that are a result of the multiplication
of two prime numbers. As you may remember, a prime number is
any number that is only divisible by 1 and itself. The first prime number is 2.
Additional prime numbers include 3, 5, 7, 11, 13, 17, 19, 23, and so on. An infinite
number of prime numbers exist, and all numbers have one prime factorization.
A semiprime number, also known as biprime, 2-almost prime, or a pq number,
is a natural number that is the product of two prime numbers. The semiprimes
less than 50 are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, and 49.
Prime numbers are significant in cryptography. Here is a simple Python script
that tests if an integer value is prime:
def isprime(x):
x = abs(int(x))
if x < 2:
return "Less 2", False