galois.egcd

galois.egcd(a, b)

Finds the integer multiplicands of \(a\) and \(b\) such that \(a x + b y = \mathrm{gcd}(a, b)\).

This function implements the Extended Euclidean Algorithm.

Parameters

• a (int) – Any integer.

• b (int) – Any integer.

Returns

• int – Greatest common divisor of \(a\) and \(b\).

• int – Integer \(x\), such that \(a x + b y = \mathrm{gcd}(a, b)\).

• int – Integer \(y\), such that \(a x + b y = \mathrm{gcd}(a, b)\).

References

• 20. Moon, “Error Correction Coding”, Section 5.2.2: The Euclidean Algorithm and Euclidean Domains, p. 181

• https://en.wikipedia.org/wiki/Euclidean_algorithm#Extended_Euclidean_algorithm

Examples

In [1]: a = 12

In [2]: b = 28

In [3]: gcd, x, y = galois.egcd(a, b)

In [4]: gcd, x, y
Out[4]: (4, -2, 1)

In [5]: a*x + b*y == gcd
Out[5]: True