Implementing-cryptography-using-python

98 Chapter 4 ■ Cryptographic Math and Frequency Analysis
elif x == 2:
return True
elif x % 2 == 0:
return False
else:
for n in range(3, int(x**0.5)+2, 2):
if x % n == 0:
return n, False
return True
print (isprime(100000007))
Computationally speaking, it is easy to generate large prime numbers that
will require most humans to either perform math on paper or use a computer.
You can calculate a fairly large number and then check if there are any available
factors. Take, for example, 19 × 13 = 589. Both 19 and 13 are prime numbers as
their only factors are themselves and 1; the product of the two numbers results
in a semiprime. Multiplying 19 and 13 together is straightforward. However,
finding the factors of 589 is a bit more challenging. To find all factors, you will
need to examine all of the primes that are less than 589 until you find which
prime numbers are used. You can achieve this reasonably quickly for smaller
numbers, but once you start dealing with larger numbers, the amount of possible
numbers that are required to check becomes so large that even modern
computers are not able to calculate them within a reasonable time frame.
Prime Number Theorem
To estimate the generation of prime numbers, you can use the prime number
theorem. The prime number theorem returns an approximate value for the
number of primes less than or equal to a provided positive real number x. The
usual notation for this number is π(x), so that π(2) = 1, π(3.5) = 2, and π(10) = 4.
To generate large prime numbers, you will need to take an approximation and
then use a primality test to verify the number. You will see how this works at
the end of this section when you review the code listing for generating large
prime numbers.
You can write a number of primality tests in Python. Some of the examples
provided in this chapter include the school primality test, Fermat’s little theorem,
and the Miller-Rabin primality test.
School Primality Test
The school primality test solves the problem of whether an integer is prime or
not by iterating through all of the numbers starting from 2 to (n/2) using a loop
for every number check to see if it divides n. If the program finds any number