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Trims and Extensions of Quadratic APN Functions

by   Christof Beierle, et al.

In this work, we study functions that can be obtained by restricting a vectorial Boolean function F ๐”ฝ_2^n โ†’๐”ฝ_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 ยท (2^n-1)^2 EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on ๐”ฝ_2^n. Further, for all of the known quadratic APN functions in dimension n โ‰ค 10, we determine the restrictions that are also APN. Moreover, we construct 5,167 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 ๐”ฝ_2^n โ†’๐”ฝ_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.


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