Almost perfect nonlinear families which are not equivalent to permutations

Abstract An important problem on almost perfect nonlinear (APN) functions is the existence of APN permutations on even-degree extensions of F 2 larger than 6. Browning et al. (2010) gave the first known example of an APN permutation on the degree-6 extension of F 2 . The APN permutation is CCZ-equivalent to the previously known quadratic Kim κ-function (Browning et al. (2009)). Aside from the computer based CCZ-inequivalence results on known APN functions on even-degree extensions of F 2 with extension degrees less than 12, no theoretical CCZ-inequivalence result on infinite families is known. In this paper, we show that Gold and Kasami APN functions are not CCZ-equivalent to permutations on infinitely many even-degree extensions of F 2 . In the Gold case, we show that Gold APN functions are not equivalent to permutations on any even-degree extension of F 2 , whereas in the Kasami case we are able to prove inequivalence results for every doubly-even-degree extension of F 2 .