Euclidean Algorithm

Introduction

Although we have functions that can compute our $\gcd$easily it's important enough that we need to give and study an algorithm for it: the euclidean algorithm.

It's extended version will help us calculate modular inverses which we will define a bit later.

Euclidean Algorithm

Important Remark

If $a = b \cdot q + r$and $d = \gcd(a, b)$ then $d | r$. Therefore $\gcd(a, b) = \gcd(b, r)$

Algorithm

We write the following:

$a = q_0 \cdot b + r_0 \\ b = q_1 \cdot r_0 + r_1 \\ r_0 = q_2 \cdot r_1 + r_2 \\ \vdots \\ r_{n-2} = r_ {n-1} \cdot q_{n - 1} + r_n \\ r_n = 0$

Or iteratively $r_{k-2} = q_k \cdot r_{k-1} + r_k$ until we find a $0$. Then we stop

Now here's the trick:

$$\gcd(a, b) = gcd(b, r_0) = gcd(r_0, r_1) = \dots = \gcd(r_{n-2}, r_{n-1}) = r_{n-1} = d$$

If $d = \gcd(a, b)$ then $d$ divides $r_0, r_1, ... r_{n-1}$

Pause and ponder. Make you you understand why that works.

Example:

Calculate $\gcd(24, 15)$

$24 = 1 \cdot 15 + 9 \\ 15 = 1 \cdot 9 + 6 \\ 9 = 1 \cdot 6 + 3 \\ 6 = 2 \cdot 3 + 0 \Rightarrow 3 = \gcd(24, 15)$

Code

def my_gcd(a, b):
    # If a < b swap them
    if a < b: 
        a, b = b, a
    # If we encounter 0 return a
    if b == 0: 
        return a
    else:
        r = a % b
        return my_gcd(b, r)

print(my_gcd(24, 15))
# 3

Exercises:

1. Pick 2 numbers and calculate their $\gcd$by hand.
2. Implement the algorithm in Python / Sage and play with it. Do not copy paste the code

Extended Euclidean Algorithm

This section needs to be expanded a bit.

Bezout's identity

Let $d = \gcd(a, b)$. Then there exists $u, v$ such that $au + bv = d$

The extended euclidean algorithm aims to find $d = \gcd(a, b), \text{ and }u, v$given $a, b$

# In sage we have the `xgcd` function
a = 24
b = 15
g, u, v = xgcd(a, b)
print(g, u, v)
# 3 2 -3 

print(u * a + v * b)
# 3 -> because 24 * 2 - 15 * 3 = 48 - 45 = 3