One-to-One Highly Nonlinear Power Functions on GF(2n)

Abstract

It is an important problem to determine all one-to-one power functions on GF(2ⁿ) possessing a maximal nonlinearity. Unfortunately the maximal nonlinearity is not known explicitly for even n. But there is a very reasonable conjecture for it. Assuming that this conjecture is actually true we give an overview about all known cases and add two new results for even n. We expect that now all one-to-one power functions with maximal nonlinearity are found, with exception of conjectures of Welch and Niho for odd n.