Extended Euclid

Since gcd(a,b) = ax + by, x, y $\in$ Z, finding x and y will be useful for computing modular multiplicative inverses

Extended-Euclid(a,b)
$\;\;\;\;\;$if b = 0
$\;\;\;\;\;$then return(a, 1, 0)
$\;\;\;\;\;$(d', x', y') = Extended-Euclid(b, a mod b)
$\;\;\;\;\;$(d, x, y) = (d', y', x' - $\lfloor$ $\rfloor$y')
$\;\;\;\;\;$return(d, x, y)

Running time same as Euclid algorithm.

Example

(d, x, y) = Extended-Euclid(6, 3)
   (d', x', y') = Extended-Euclid(3, 0)
                = (3, 1, 0)
   d = 3, x = 0, y = 1 - 6/3 * 0 = 1
   = (3, 0, 1)

d = 3, x = 0, y = 1

6x + 3y = d
6*0 + 3*1 = 3