Trims and extensions of quadratic APN functions

In this work, we study functions that can be obtained by restricting a vectorial Boolean function F: F-2(n) -> F-2(n) 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.(2(n) - 1)(2) EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on F-2(n). Further, for all of the known quadratic APN functions in dimension n < 10, we determine the restrictions that are also APN. Moreover, we construct 6368 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: F-2(n) -> F-2(n) with linearity of 2(n-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 2(7) up to EA-equivalence.