Editing Extended Euclidean algorithm (section)

{{short description|Method for computing the relation of two integers with their greatest common divisor}}
In [[arithmetic]] and [[computer programming]], the '''extended Euclidean algorithm''' is an extension to the [[Euclidean algorithm]], and computes, in addition to the [[greatest common divisor]] (gcd) of integers ''a'' and ''b'', also  the coefficients of [[Bézout's identity]], which are integers ''x'' and ''y'' such that
: <math>ax + by = \gcd(a, b).</math>
This is a [[certifying algorithm]], because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs.<ref>{{cite web |last1=McConnell |first1=Ross |last2=Mehlhorn |first2=Kurt |last3=Näher |first3=Stefan |last4=Schweitzer |first4=Pascal |title=Certifying Algorithms |url=https://people.mpi-inf.mpg.de/~mehlhorn/ftp/CertifyingAlgorithms.pdf |access-date=29 September 2024}}</ref>
It allows one to compute also, with almost no extra cost, the quotients of ''a'' and ''b'' by their greatest common divisor.

{{em|Extended Euclidean algorithm}} also refers to a [[Polynomial greatest common divisor#Bézout's identity and extended GCD algorithm|very similar algorithm]] for computing the [[polynomial greatest common divisor]] and the coefficients of Bézout's identity of two [[univariate polynomial]]s.

The extended Euclidean algorithm is particularly useful when ''a'' and ''b'' are [[coprime]]. With that provision, ''x'' is the [[modular multiplicative inverse]] of ''a'' [[modular arithmetic|modulo]] ''b'', and ''y'' is the modular multiplicative inverse of ''b'' modulo ''a''. Similarly, the polynomial extended Euclidean algorithm allows one to compute the [[multiplicative inverse]] in [[algebraic field extension]]s and, in particular in [[finite field]]s of non prime order. It follows that both extended Euclidean algorithms are widely used in [[cryptography]]. In particular, the computation of the [[modular multiplicative inverse]] is an essential step in the derivation of key-pairs in the [[RSA (algorithm)|RSA]] public-key encryption method.