Implementing-cryptography-using-python

112 Chapter 4 ■ Cryptographic Math and Frequency Analysis
Both Fermat’s and Euler’s theorems play an important role in public-key cryptography,
which will be explored in greater detail in Chapter 8.
In number theory, Euler’s theorem, also known as Euler’s totient theorem or
the Fermat–Euler theorem, states that if n and a are coprime positive integers,
then aφ (n) ≡ 1 mod n where φ(n) is Euler’s totient function. In 1736, Leonhard
Euler published his proof of Fermat’s little theorem, which Fermat had presented
without proof. Subsequently, Euler presented other proofs of the theorem, culminating
with “Euler’s theorem” in his paper of 1763, in which he attempted to
find the smallest exponent for which Fermat’s little theorem was always true.
Euler investigated the properties of numbers; he specifically studied the
distribution of prime numbers. One crucial function he defined is named the
PHI function; the PHI function measures the breakability of a number. Assume
you have the number n; the function calculates the number of integers that are
less than or equal to n and do not share any common factor with n; you see it
in the following notation: ɸ [n]. For example, if you wanted to examine ɸ [8],
you would examine all values from 1 to 8 and count all integers with which 8
does not share a factor greater than 1; the numbers are 1, 3, 5, 7. The function
produces 4. As it turns out, calculating the PHI of a prime number P is simple.
ɸ [P] = P − 1. To calculate ɸ [7], you count all integers except 7 since none of the
integers share a factor with 7; therefore, ɸ [7] = 6. Assume a larger prime such
as 21,377. ɸ [21,377] = 21,376. The equation looks like the following:
ap − 1 ≡ 1 mod p
To take full advantage of this trick, which helps when you explore RSA, you
need to see the general version. If you know the prime factorization of a modulus,
then computing the Euler’s totient works as follows:
ɸ(p 1
K1
... p n Kn ) = ɸ(p 1 K1 ) ... ɸ(p n
Ki − 1 ) where ɸ(p i Ki ) – p i
Ki − 1
= p Ki – 1 ◦ (p i – 1)
We extend the ideas presented in this chapter into some classical number
theory that also happens to be practical for cryptographers. Euler totient functions
offer benefits to speed up modular inverse computations. You also learn why
raising numbers to substantial powers is not expensive when you are working
with modular arithmetic.
For instance:
ɸ(3 ◦ 5 ◦ 23) = ɸ(3) ◦ ɸ(5) ◦ ɸ(23) = 2 ◦ 4 ◦ 22 = 32
Euler’s totient function Φ(n) for an input n is the count of numbers in the
format of {1, 2, 3, 4, 5, n} that are relatively prime to n, i.e., the numbers whose
GCD with n is 1. Examine the following six examples, which calculate the Euler’s
totient function Φ (n) in respect to the inputs 1 through 6. The output will be
the number of positive integers that do not exceed n and also have no common
divisors with n other than the common divisor 1:
Φ (1) = 1
gcd (1, 1) is 1