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Christof Beierle, Nils-Gregor Leander, Léo Perrin

• In this work, we study functions that can be obtained by restricting a vectorial Boolean function \(\it F\) : \(\mathbb{F}^{n}_{2}\) \(\rightarrow\) \(\mathbb{F}^{n}_{2}\) to an affine hyperplane of dimension \(\it n\)−1 and then projecting the output to an \(\it n\)−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 \(\mathbb{F}^{n}_{2}\). Further, for all of the known quadratic APN functions in dimension \(\it n\)<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 \(\it F\) : \(\mathbb{F}^{n}_{2}\) \(\rightarrow\) \(\mathbb{F}^{n}_{2}\) 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.