Extended Euclidean Algorithm Algorithm
In arithmetical and computer programming, the extended euclidean algorithm is an extension to the euclidean algorithm, and computes, in addition to the greatest common divisor of integers a and b, also the coefficients of Bézout's identity, which are integers X and Y such that with that provision, X is the modular multiplicative inverse of a modulo b, and Y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended euclidean algorithm lets one to calculate the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order.
/*
Petar 'PetarV' Velickovic
Algorithm: Extended Euclidean Algorithm
*/
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <assert.h>
#include <iostream>
#include <vector>
#include <list>
#include <string>
#include <algorithm>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <complex>
typedef long long lld;
typedef unsigned long long llu;
using namespace std;
/*
The Extended Euclidean Algorithm, similarly to its basic variant,
calculates the Greatest Common Divisor (GCD) of two integers, a and b.
However, it also computes two integers, x and y, such that a*x + b*y = gcd(a, b).
This is the preferred method for calculating the modular inverse of a number,
as well as partitioning an integer into subgroups (as per the Chinese Remainder Theorem).
Complexity: O(log N)
*/
inline pair<lld, pair<lld, lld> > egcd(lld a, lld b)
{
lld aa = 1, ab = 0, ba = 0, bb = 1;
while (true)
{
lld q = a / b;
if (a == b * q) return make_pair(b, make_pair(ba, bb));
lld tmp_a = a;
lld tmp_aa = aa;
lld tmp_ab = ab;
a = b;
b = tmp_a - b * q;
aa = ba;
ab = bb;
ba = tmp_aa - ba * q;
bb = tmp_ab - bb * q;
}
}
inline lld modinv(lld a, lld b)
{
lld b0 = b;
lld aa = 1, ba = 0;
while (true)
{
lld q = a / b;
if (a == b * q)
{
if (b != 1)
{
// Modular inverse does not exist!
return -1;
}
while (ba < 0) ba += b0;
return ba;
}
lld tmp_a = a;
lld tmp_aa = aa;
a = b;
b = tmp_a - b * q;
aa = ba;
ba = tmp_aa - ba * q;
}
}
int main()
{
lld a = 2250, b = 360;
pair<lld, pair<lld, lld> > ret = egcd(a, b);
printf("gcd(%lld, %lld) = %lld = %lld * %lld + %lld * %lld\n", a, b, ret.first, a, ret.second.first, b, ret.second.second);
a = 806515533049393LL, b = 1304969544928657LL;
lld inv = modinv(a, b);
printf("modinv(%lld, %lld) = %lld\n", a, b, inv);
a = 4505490, b = 7290036;
inv = modinv(a, b);
printf("modinv(%lld, %lld) = %lld\n", a, b, inv);
return 0;
}
