On The Inequivalence Of Ness-Helleseth APN Functions

In this paper, the Ness-Helleseth functions over Fpn defined by the form f(x) = ux pn−1 2 −1 + x n −2 are proven to be a new class of almost perfect nonlinear (APN) functions and they are CCZ-inequivalent with all other known APN functions when p ≥ 7. The original method of Ness and Helleseth showing the functions are APN for p = 3 and odd n ≥ 3 is also suitable for showing their APN property for any prime p ≥ 7 with p ≡ 3 (mod 4) and odd n.