Investigating rational perfect nonlinear functions

Abstract

Perfect nonlinear (PN) functions over a finite field, whose study is also motivated by practical applications to Cryptography, have been the subject of several recent papers where the main problems, such as effective constructions and non-existence results, are considered. So far, all contributions have focused on PN functions represented by polynomials, and their constructions. Unfortunately, for polynomial PN functions, the approach based on Hasse–Weil type bounds applied to function fields can only provide non-existence results in a small degree regime. In this paper, we investigate the non-existence problem of rational perfect nonlinear functions over a finite field. Our approach makes it possible to use deep results about the number of points of algebraic varieties over finite fields. Our main result is that no PN rational function f/g with \(f,g\in \mathbb {F}_q[X]\) exists when certain mild arithmetical conditions involving the degree of f(X) and g(X) are satisfied.