Euclidean algorithms (Basic and Extended)

The Euclidean algorithm is a way to find the greatest common divisor of two positive integers. 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 prime factors.

Examples:

input: a = 12, b = 20
Output: 4
Explanation: The Common factors of (12, 20) are 1, 2, and 4 and greatest is 4.

input: a = 18, b = 33
Output: 3
Explanation: The Common factors of (18, 33) are 1 and 3 and greatest is 3.

👁 GCD

Basic Euclidean Algorithm for GCD

The algorithm is based on the below facts. 

• If we subtract a smaller number from a larger one (we reduce a 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 the larger number, the algorithm stops when we find the remainder 0.

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

Time Complexity: O(log min(a, b))
Auxiliary Space: 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
Explanation: 30*1 + 20*(-1) = 10

Input: a = 35, b = 15
Output: gcd = 5, x = 1, y = -2
Explanation: 35*1 + 15*(-2) = 5

The extended Euclidean algorithm updates the results of gcd(a, b) using the results calculated by the recursive call gcd(b%a, a). Let values of x and y calculated by the recursive call be x₁ and y₁. x and y are updated using the below expressions. 

ax + by = gcd(a, b)
gcd(a, b) = gcd(b%a, a)
gcd(b%a, a) = (b%a)x₁ + ay₁
ax + by = (b%a)x₁ + ay₁
ax + by = (b - [b/a] * a)x₁ + ay₁
ax + by = a(y₁ - [b/a] * x₁) + bx₁

Comparing LHS and RHS,
x = y₁ - * x₁
 y = x₁

5

Time Complexity: O(log min(a, b))
Auxiliary Space: O(log (min(a,b)))

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 = in above, 
we get following. Note that is floor(b/a)

Above equation can also be written as below

     ---(2)

After comparing coefficients of 'a' and 'b' in (1) and 
(2), we get following, 

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.