Random math for cryptography

Greatest common divisor

Given two integers $a$ and $b$ , with $b \neq 0$ , we say that $b$ divides $a$ if $\exists$ and integer $c$ such that the following equality is satisfied:

a = bc

If that $c$ exists, then we write $b \mid a$ , otherwise $b \nmid a$ .

Despite we are used to consider the concept of division fairly simple, it plays a fundamental role in modern cryptography and it deserve additional attention.

[Proposition]

Let $a, b, c \in \mathbb{Z}$ be three integers:

If $a \mid b$ and $b \mid c$ $\Rightarrow a \mid c$
If $a \mid b$ and $b \mid a$ $\Rightarrow a \pm b$
If $a \mid b$ and $a \mid a$ $\Rightarrow a \mid (b + c)$ and $a \mid (b - c)$

Amongst all the divisors that are in common between two numbers, we are particularly interested in the greatest common division.

[Definition] Greatest common division

Let $a, b \in \mathbb{Z}$ be two integers, we define the greatest common divisor as the largest integer $d$ such that:

d \mid a,\quad d \mid b

The greatest common divisor between two numbers is commonly written as. $\gcd(a,b)$

When the $\gcd(a,b) = 1$ , then we say that the two numbers are relatively prime.

Is it possible to compute efficiently the $\gcd(a,b)$ using the method of long division, also called the Euclidean algorithm. To understand how the algorithm works, it is useful to consider the generic division form between two numbers:

a = bq + r \quad \text{with} \quad 0 \le r < b

In this case, $q$ is called the quotient, and $r$ is the reminder. With this equation, we can present an equality used in the next theorem:

\gcd(a,b) = \gcd(b,r)

This equality states that the greatest common divisor between two numbers, is equal to the greatest common divisor between the lowest of the two, and the reminder of their division.

[Theorem] The Euler Algorithm

Let $a$ and $b$ two positive numbers in $\mathbb{Z}$ such that $a \ge b$ . We can compute the $\gcd(a,b)$ in a finite number of steps with the following algorithm:

Let $r_0 = a$ , $r_1=b$ , and set $i=1$ .
Divide $r_{i-1}$ by $r_i$ to get the quotient $q$ and the reminder $r_{i+1}$ ;

r_{i-1} = r_i \cdot q + r_{i+1}

We have to situations here:
1. If the reminder is equal to zero, we terminate with:
  $\gcd(a,b) = r_i$
2. Otherwise, we set $i = i +1$ and go back to point 2.

From the application of the Euclidean algorithm, we can obtain another important theorem for cryptography, which allows us to write the $\gcd(a,b)$ as a linear combination of $a$ and $b$ .

[Theorem] Extended Euclidean Algorithm

Let $a$ and $b$ two positive numbers in $\mathbb{Z}$ . The following combination has always a solution with $u$ and $v$ in $\mathbb{Z}$ :

au + bv = \gcd(a,b)

An interesting alternative formulation of the same equation, is the one for relatively prime numbers:

\frac{A}{\gcd(A,B)}u + \frac{B}{\gcd(A,B)}v = 1

Before we mentioned that we have an algorithm to compute the $\gcd$ efficiently. But, what does it mean to compute something in an efficient way?

The Euclidean algorithm has $\mathcal{O}(\log_2n)$ complexity since it is possible to demonstrate that $r_{i+2} < r_i$ , so every two steps we have a reduction by half.

The fast powering algorithm

TODO

References

Hoffstein, Pipher, Silverman, An Introduction to Mathematical Cryptography, Chapter 2