## Extended Euclidean Algorithm

### Euclidean Algorithm

The greatest common divisor `gcd(a, b)`
of two natural numbers `a` and `b` is the
greatest number that divides both `a` and `b`.
This number plays an important role in the number theory
and public cryptography.

There is conceptually a simple way to find the gcd.
Factor both numbers `a` and `b`
into a product of primes and multiply together all common factors.
The problem with this approach is that it is
computationally extremely difficult to factor large numbers into primes.

Already in the 3rd century BC,
the greek mathematician Euclid described an ingenious and very efficient
algorithm to compute the gcd.
The algorithm is based on the following fact
(assume a ≥ b > 0 ):

Here `q` is the quotient, i.e. the result of the integer division
`a // b`, and `r` is the remainder
`r = a % b = a mod b`.
Now we have two cases:

- Either
`r = 0`. Then`b`divides`a`and therefore we found`gcd(a, b) = b.` - Or
`r > 0`. Because r = a - q ⋅ b all divisors of`a, b`are also divisors of`b, r`and vice versa. So,gcd(a, b) = gcd(b, r) and b < a, r < b(2)a' = q' ⋅ b' + r' and r' = 0, b' = gcd(a, b)(3)

The described process can immediately be cast into a recursive algorithm in Python:

```
def gcd_rec(a, b):
if b == 0:
return a
return gcd_rec(b, a % b)
```

An iterative algorithm is also straightforward (Python):

```
def gcd(a, b):
while b:
q = a // b
a, b = b, a % b
return a
```

The last assignment in the above program is a parallel (vectorial) assignment that is possible in Python.

### Bézout's Identity

The gcd(a, b) has an interesting and very useful property:
it is a linear combination of `a` and `b` with integer coefficients:

This relation is called the Bézout's identity. We will prove it
constructively and at the same time present the
Extended Euclidean Algorithm to compute the coefficients `x` and `y`
in equation (4).

### Extended Euclidean Algorithm

The Euclidean Algorithm computes a sequence of residues and can be represented as follows:

_{0}= a

_{1}= b

_{2}= r

_{0}- q

_{1}⋅ r

_{1}

_{n+1}= r

_{n}- q

_{n-1}⋅ r

_{n-1}

We have prepended the equations (5a) and (5b) to make the presentation simpler.
The equation (5c) corresponds to equation (1).
q_{n} are the quotients.
We claim that all the residues are a linear combination of `a` and `b`
and prove it by induction.
Consequently also the last non-zero residue, which is
`gcd(a, b)`, is a linear combination of of `a` and `b`.
First we have:

_{0}= a = a ⋅ 1 + b ⋅ 0

_{1}= b = a ⋅ 0 + b ⋅ 1

Suppose that

_{n}= a ⋅ x

_{n}+ b ⋅ y

_{n}

for all `i = 0, 1, .. n` and compute

_{n+1}=

_{n}- q

_{n-1}⋅ r

_{n-1}

_{n}+ b ⋅ y

_{n}) - q

_{n-1}⋅ (a ⋅ x

_{n-1}+ b ⋅ y

_{n-1})

_{n}- q

_{n-1}⋅ x

_{n-1}) + b ⋅ (y

_{n}- q

_{n-1}⋅ y

_{n-1})

_{n+1}+ b ⋅ y

_{n+1}(6d)

with

_{n+1}= x

_{n}- q

_{n-1}⋅ x

_{n-1}

_{n+1}= y

_{n}- q

_{n-1}⋅ y

_{n-1}

The above algorithm can immediately be transformed into a program:

```
def gcd_ext(a, b):
x0, y0 = 1, 0
x1, y1 = 0, 1
while b:
q = a // b
x0, x1 = x1, x0 - q * x1
y0, y1 = y1, y0 - q * y1
a, b = b, a % b
return a, x0, y0
```

The program returns a triple `g, x, y`
such that:

### Multiplicative Inverse (mod p)

In the public cryptography we need to find efficiently a number
`y` that is the multiplicative inverse
of a number `b` `(mod p)`, i.e.

Let us mention that this equation does not always have a solution.
E.g. when `b` and `p` are both even numbers
then b ⋅ y (mod p) is even and can never produce 1.
However, when `b` and `p` are relatively prime
then we know from the Bézout's identity that

for some `x` and `y` and
and hence `y` is the solution of equation (8).
So, we can compute `x` using the Extended Euclidean Algorithm
applied to `p` and `b`:

```
def inv(b, p):
g, x, y = gcd_ext(p, b)
return y % p
```

The last statement in this algorithm guarantees that

### Links:

- P. Keef and D. Guichard : Introduction to Higher Mathematics
- The Euclidean Algorithm and the Extended Euclidean Algorithm (DI Management Services)

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