Tool to apply the extended GCD algorithm (Euclidean method) in order to find the values of the Bezout coefficients and the value of the GCD of 2 numbers.

Extended GCD Algorithm - dCode

Tag(s) : Arithmetics

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The extended Euclidean algorithm is a modification of the classical GCD algorithm allowing to find a linear combination. 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 $ gcd(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 eGCD implementation of the pseudo-code algorithm to find the linear combination gcd(a,b) = a.u+b.v:

function extended_gcd(a, b) {// a, b natural integers a < b

r1 = b, r2 = a, u1 = 0, v1 = 1, u2 = 1, v2 = 0

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: $$ a(\text{sign}(a)\cdot x)+b(\text{sign}(b)\cdot y)=1 $$

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Cite as source (bibliography):

*Extended GCD Algorithm* on dCode.fr [online website], retrieved on 2024-07-18,

extended,gcd,egcd,identity,bezout,coefficient

https://www.dcode.fr/extended-gcd

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