galois.extended_euclidean_algorithm

galois.extended_euclidean_algorithm(a, b)

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

Parameters

• a (int) – Any integer.

• b (int) – Any integer.

Returns

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

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

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

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, b = 2, 13

In [2]: x, y, gcd = galois.extended_euclidean_algorithm(a, b)

In [3]: x, y, gcd
Out[3]: (-6, 1, 1)

In [4]: a*x + b*y == gcd
Out[4]: True