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.

GCD

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.

 

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source and credits.
source and credits: http://www.geeksforgeeks.org
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rakesh

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