# Euclidean algorithms (Basic and Extended)

GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common factors.

Basic Euclidean Algorithm for GCD
The algorithm is based on below facts.

• If we subtract smaller number from larger (we reduce larger number), GCD doesn’t change. So if we keep subtracting repeatedly the larger of two, we end up with GCD.
• Now instead of subtraction, if we divide smaller number, the algorithm stops when we find remainder 0.

Below is a recursive C function to evaluate gcd using Euclid’s algorithm.

`// C program to demonstrate Basic Euclidean Algorithm`
`#include <stdio.h>`
`// Function to return gcd of a and b`
`int` `gcd(``int` `a, ``int` `b)`
`{`
`    ``if` `(a == 0)`
`        ``return` `b;`
`    ``return` `gcd(b%a, a);`
`}`
`// Driver program to test above function`
`int` `main()`
`{`
`    ``int` `a = 10, b = 15;`
`    ``printf``(``"GCD(%d, %d) = %dn"``, a, b, gcd(a, b));`
`    ``a = 35, b = 10;`
`    ``printf``(``"GCD(%d, %d) = %dn"``, a, b, gcd(a, b));`
`    ``a = 31, b = 2;`
`    ``printf``(``"GCD(%d, %d) = %dn"``, a, b, gcd(a, b));`
`    ``return` `0;`
`}`

Output:

```GCD(10, 15) = 5
GCD(35, 10) = 5
GCD(31, 2) = 1```

Time Complexity: O(Log min(a, b))
Extended Euclidean Algorithm:
Extended Euclidean algorithm also finds integer coefficients x and y such that:

`  ax + by = gcd(a, b)`

Examples:

```Input: a = 30, b = 20
Output: gcd = 10
x = 1, y = -1
(Note that 30*1 + 20*(-1) = 10)

Input: a = 35, b = 15
Output: gcd = 5
x = 1, y = -2
(Note that 10*0 + 5*1 = 5)```

The extended Euclidean algorithm updates results of gcd(a, b) using the results calculated by recursive call gcd(b%a, a). Let values of x and y calculated by the recursive call be x1and y1. x and y are updated using below expressions.

```x = y1 - ⌊b/a⌋ * x1
y = x1

```

Below is C implementation based on above formulas.

`// C program to demonstrate working of extended`
`// Euclidean Algorithm`
`#include <stdio.h>`
`// C function for extended Euclidean Algorithm`
`int` `gcdExtended(``int` `a, ``int` `b, ``int` `*x, ``int` `*y)`
`{`
`    ``// Base Case`
`    ``if` `(a == 0)`
`    ``{`
`        ``*x = 0;`
`        ``*y = 1;`
`        ``return` `b;`
`    ``}`
`    ``int` `x1, y1; ``// To store results of recursive call`
`    ``int` `gcd = gcdExtended(b%a, a, &x1, &y1);`
`    ``// Update x and y using results of recursive`
`    ``// call`
`    ``*x = y1 - (b/a) * x1;`
`    ``*y = x1;`
`    ``return` `gcd;`
`}`
`// Driver Program`
`int` `main()`
`{`
`    ``int` `x, y;`
`    ``int` `a = 35, b = 15;`
`    ``int` `g = gcdExtended(a, b, &x, &y);`
`    ``printf``(``"gcd(%d, %d) = %d"``, a, b, g);`
`    ``return` `0;`
`}`

Output:

`gcd(35, 15) = 5`

How does Extended Algorithm Work?

```As seen above, x and y are results for inputs a and b,
a.x + b.y = gcd                      ----(1)

And x1 and y1 are results for inputs b%a and a
(b%a).x1 + a.y1 = gcd

When we put b%a = (b - (⌊b/a⌋).a) in above,
we get following. Note that ⌊b/a⌋ is floor(a/b)

(b - (⌊b/a⌋).a).x1 + a.y1  = gcd

Above equation can also be written as below
b.x1 + a.(y1 - (⌊b/a⌋).x1) = gcd      ---(2)

After comparing coefficients of 'a' and 'b' in (1) and
(2), we get following
x = y1 - ⌊b/a⌋ * x1
y = x1
```

How is Extended Algorithm Useful?
The extended Euclidean algorithm is particularly useful when a and b are coprime (or gcd is 1). Since x is the modular multiplicative inverse of “a modulo b”, and y is the modular multiplicative inverse of “b modulo a”. In particular, the computation of the modular multiplicative inverse is an essential step in RSA public-key encryption method.

Disclaimer: This does not belong to TechCodeBit, its an article taken from the below
source and credits.
source and credits: http://www.geeksforgeeks.org
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