A One-Way Function Based On The Extended Euclidean Algorithm

A problem based on the Extended Euclidean Algorithm applied to a class of polynomials with many factors is presented and believed to be hard. If so, it is a one-way function well suited for applications in digital signatures. Let ) (x P and ) (x Q be polynomials defined over the Integers modulo a prime integer p with ) 1 ( ) ( ) ( 1    p x x Q x P p mod . Also let ) (x A be a polynomial such that 1 ) ( ) (  x P x A ) ( mod x Q mod p . In general, the pair   ) ( ), ( x Q x P is not uniquely determined (mod p ) from just ) (x A and p ; using Mathematica, we have found counterexamples. However, when  ) ( deg x P 2 / ) 1 ( ) ( deg   p x Q , we have not found counterexamples, but neither have we proved that the solution is unique. More pertinent for this correspondence is the question: is finding such a pair of polynomials a hard problem? That is, can one find such a pair from just ) (x A and p without testing all possible factorizations of 1 1   p x and checking for the congruence until a match is found? Because