Extended GCD Algorithm

The extended Euclidean algorithm is a modification of the classical GCD algorithm. From 2 natural inegers a and b, its steps allow to calculate their GCD and their Bézout coefficients (see the identity of Bezout).

Example: \(a=12\) and \(b=30\), thus \( \mbox{pgcd}(12, 30) = 6 \)

$$ 12 \times -10 + 30 \times 3 = 6 \\ 12 \times -3 + 30 \times 1 = 6 \\ 12 \times 4 + 30 \times -1 = 6 \\ 12 \times 11 + 30 \times -3 = 6 \\ 12 \times 18 + 30 \times -5 = 6 \ 12 \times −2+30 \times 1 = 6 $$

Here is an implementation of the pseudo-code algorithm:
function extended_gcd(a, b) {// a, b natural integers
r1 = a, r2 = b, u1 = 1, v1 = 0, u2 = 0, v2 = 1
while (r2! = 0) do
q = r1 ÷ r2 (integer division)
r3 = r1, u3 = u1, v3 = v1,
r1 = r2, u1 = u2, v1 = v2,
r2 = r3 - q * r2, u2 = u3 - q * u2, v2 = v3 - q * v2
end while
return (r1, u1, v1) (r1 natural integer and u1, v1 rational integers)

The values are such that r1 = pgcd(a, b) = a * u1 + b * v1

Using the absolute values for a and b, the rest of the calculation is identical thanks to the property: $$ a(\text{sign}(a)\cdot x)+b(\text{sign}(b)\cdot y)=1 $$