# Bounds on the degree of APN polynomials: the case of x^(-1) + g(x)

Gregor Leander, François Rodier

Designs, Codes and Cryptography April 2011, Volume 59, Issue 1-3, pp 207-222

## Abstract

In this paper we consider APN functions f:F2^m?F2^m of the form f(x) = x^(?1) + g(x) where g is any non F_2-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields F_(2^m). Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ? 3 where these functions are equivalent to x³.

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