Trims and extensions of quadratic APN functions

Abstract

In this work, we study functions that can be obtained by restricting a vectorial Boolean function F:Fn2→Fn2 to an affine hyperplane of dimension n−1 and then projecting the output to an n−1-dimensional space. We show that a multiset of 2⋅(2n−1)2 EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on Fn2. Further, for all of the known quadratic APN functions in dimension n<10, we determine the restrictions that are also APN. Moreover, we construct 6,368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function F:Fn2→Fn2 with linearity of 2n−1 by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity 27 up to EA-equivalence.