Trims and extensions of quadratic APN functions

Abstract

AbstractIn this work, we study functions that can be obtained by restricting a vectorial Boolean function$$F :\mathbb {F}_{2}^n \rightarrow \mathbb {F}_{2}^n$$F:F2n→F2nto an affine hyperplane of dimension$$n-1$$n-1and then projecting the output to an$$n-1$$n-1-dimensional space. We show that a multiset of$$2 \cdot (2^n-1)^2$$2·(2n-1)2EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on$$\mathbb {F}_{2}^n$$F2n. Further, for all of the known quadratic APN functions in dimension$$n < 10$$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$$F :\mathbb {F}_{2}^n \rightarrow \mathbb {F}_{2}^n$$F:F2n→F2nwith linearity of$$2^{n-1}$$2n-1by 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$$2^7$$27up to EA-equivalence.