The extended Euclidean algorithm is an extension to the Euclidean algorithm, which computes, besides the greatest common divisor of integers a and b, the coefficients of Bézout’s identity, that is integers x and y such that
For example,
gcd(30, 50) = 10
Here x = 2 y = -1 as 30*2 + 50*-1 = 10
gcd(2740, 1760) = 20
Here, x = 9 y = -14 as 2740*9 + 1760*-14 = 20
The gcd is the only number that can simultaneously satisfy this equation and divide the inputs. The extended Euclid’s algorithm will allows us to simultaneously calculate the gcd and coefficients of the Bézout’s identity x and y at no extra cost.
Following is the implementation of Extended Euclidean algorithm in C, C++ and Python.
C
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#include<stdio.h> // Function for Extended Euclidean algorithm // It uses pointers to return multiple values int extended_gcd(int a, int b, int *x, int *y) { if (a == 0) { *x = 0; *y = 1; return b; } int _x, _y; int gcd = extended_gcd(b % a, a, &_x, &_y); *x = _y - (b/a) * _x; *y = _x; return gcd; } // main function int main() { int a = 30; int b = 50; int x, y; printf("GCD is %d\n", extended_gcd(a, b, &x, &y)); printf("x = %d, y = %d", x, y); return 0; } |
Output:
GCD is 10
x = 2, y = -1
C++
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#include <iostream> #include <tuple> // std::tuple, std::make_tuple, std::tie using namespace std; // Function for Extended Euclidean algorithm // It returns multiple values using tuple in C++ tuple<int, int, int> extended_gcd(int a, int b) { if (a == 0) return make_tuple(b, 0, 1); int gcd, x, y; // unpack tuple returned by function into variables tie(gcd, x, y) = extended_gcd(b % a, a); return make_tuple(gcd, (y - (b/a) * x), x); } // main function int main() { int a = 30; int b = 50; tuple<int, int, int> t = extended_gcd(a, b); int gcd = get<0>(t); int x = get<1>(t); int y = get<2>(t); cout << "GCD is " << gcd << endl; cout << "x = " << x << " y = " << y << endl; cout << a << "*" << x << " + " << b << "*" << y << " = " << gcd; return 0; } |
Output:
GCD is 10
x = 2 y = -1
30*2 + 50*-1 = 10
Python
The extended Euclidean algorithm is particularly useful when a and b are co-prime, since x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Both extended Euclidean algorithms are widely used in cryptography. The computation of the modular multiplicative inverse is an essential step in RSA public-key encryption algorithm.
References: https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
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