Efficient Inversion of Rational Maps Over Finite Fields

Abstract

We study the problem of finding the inverse image of a point in the image of a rational map F : F-q(n) -> F-n(q) over a finite field F-q. Our interest mainly stems from the case where F encodes a permutation given by some public-key cryptographic scheme. Given an element y((o)) epsilon F(F-q(n)), we axe able to compute the set of values x((0)) epsilon F-q(n) for which F(x((o))) = y((0)) holds with O(Tn(4.38)D(2.38)delta log(2)q) bit operations, up to logarithmic terms. Here T is the cost of the evaluation of F1,...,F, D is the degree of F and delta is the degree of the graph of F.