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documentation for extended euclid

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This commit is contained in:
Krishna Vedala
2020-05-27 15:06:46 -04:00
parent 691cbdb1f5
commit 757970dbae
1 changed files with 69 additions and 5 deletions
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74
math/extended_euclid_algorithm.cpp
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@@ -1,9 +1,70 @@
/**
* @file
* @brief GCD using [extended Euclid's algorithm]
* (https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm)
*
* Finding coefficients of a and b ie x and y in Bézout's identity
* \f[\text{gcd}(a, b) = a \times x + b \times y \f]
* This is also used in finding Modular
* multiplicative inverse of a number. (A * B)%M == 1 Here B is the MMI of A for
* given M, so extendedEuclid (A, M) gives B.
*/
#include <algorithm> // for swap function
#include <iostream>
// Finding coefficients of a and b ie x and y in gcd(a, b) = a * x + b * y
// d is gcd(a, b)
// This is also used in finding Modular multiplicative inverse of a number.
// (A * B)%M == 1 Here B is the MMI of A for given M,
// so extendedEuclid (A, M) gives B.
/**
* function to update the coefficients per iteration
* \f[r_0,\,r = r,\, r_0 - \text{quotient}\times r\f]
*
* @param[in,out] r signed
* @param[in,out] r0 signed
* @param[in] quotient unsigned input
*/
template <typename T, typename T2>
inline void update_step(T &r, T &r0, const T2 quotient) {
T temp = r;
r = r0 - (quotient * temp);
r0 = temp;
}
/**
* Implementation using iterative algorithm from
* [Wikipedia](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Pseudocode)
*
* @param[in] A unsigned
* @param[in] B unsigned
* @param[out] GCD unsigned
* @param[out] x signed
* @param[out] y signed
*/
template <typename T1, typename T2>
void extendedEuclid_1(T1 A, T1 B, T1 *GCD, T2 *x, T2 *y) {
if (B > A) std::swap(A, B); // Ensure that A >= B
T2 s = 0, s0 = 1;
T2 t = 1, t0 = 0;
T1 r = B, r0 = A;
while (r != 0) {
T1 quotient = r0 / r;
update_step(r, r0, quotient);
update_step(s, s0, quotient);
update_step(t, t0, quotient);
}
*GCD = r0;
*x = s0;
*y = t0;
}
/**
* Implementation using recursive algorithm
*
* @param[in] A unsigned
* @param[in] B unsigned
* @param[out] GCD unsigned
* @param[in,out] x signed
* @param[in,out] y signed
*/
template <typename T, typename T2>
void extendedEuclid(T A, T B, T *GCD, T2 *x, T2 *y) {
if (B > A) std::swap(A, B); // Ensure that A >= B
@@ -20,11 +81,14 @@ void extendedEuclid(T A, T B, T *GCD, T2 *x, T2 *y) {
}
}
/// Main function
int main() {
uint32_t a, b, gcd;
int32_t x, y;
std::cin >> a >> b;
extendedEuclid(a, b, &gcd, &x, &y);
std::cout << gcd << " " << x << " " << y << std::endl;
extendedEuclid_1(a, b, &gcd, &x, &y);
std::cout << gcd << " " << x << " " << y << std::endl;
return 0;
}