galois.egcd¶

galois.egcd(a: int, b: int) → tuple[int, int, int]¶

galois.egcd(a: Poly, b: Poly) → tuple[Poly, Poly, Poly]

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

Parameters

a: int¶

a: Poly

The first integer or polynomial argument.

b: int¶

b: Poly

The second integer or polynomial argument.

Returns

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

• The multiplicand \(s\) of \(a\).

• The multiplicand \(t\) of \(b\).

See also

gcd, lcm, prod

Notes

This function implements the Extended Euclidean Algorithm.

References

• Algorithm 2.107 from https://cacr.uwaterloo.ca/hac/about/chap2.pdf

• Algorithm 2.221 from https://cacr.uwaterloo.ca/hac/about/chap2.pdf

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

Examples

Integers

Compute the extended GCD of two integers.

In [1]: a, b = 12, 16

In [2]: gcd, s, t = galois.egcd(a, b)

In [3]: gcd, s, t
Out[3]: (4, -1, 1)

In [4]: a*s + b*t == gcd
Out[4]: True

Polynomials

Generate irreducible polynomials over \(\mathrm{GF}(7)\).

In [5]: GF = galois.GF(7)

In [6]: f1 = galois.irreducible_poly(7, 1); f1
Out[6]: Poly(x, GF(7))

In [7]: f2 = galois.irreducible_poly(7, 2); f2
Out[7]: Poly(x^2 + 1, GF(7))

In [8]: f3 = galois.irreducible_poly(7, 3); f3
Out[8]: Poly(x^3 + 2, GF(7))

Compute the extended GCD of \(f_1(x)^2 f_2(x)\) and \(f_1(x) f_3(x)\).

In [9]: a = f1**2 * f2

In [10]: b = f1 * f3

In [11]: gcd, s, t = galois.egcd(a, b)

In [12]: gcd, s, t
Out[12]: (Poly(x, GF(7)), Poly(2x^2 + 4x + 1, GF(7)), Poly(5x^2 + 3x + 4, GF(7)))

In [13]: a*s + b*t == gcd
Out[13]: True