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Article Dans Une Revue Designs, Codes and Cryptography Année : 2022

Trims and extensions of quadratic APN functions

Résumé

Abstract In this work, we study functions that can be obtained by restricting a vectorial Boolean function $F :\mathbb {F}_{2}^n \rightarrow \mathbb {F}_{2}^n$ 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 $2 \cdot (2^n-1)^2$ EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on $\mathbb {F}_{2}^n$. 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 $F :\mathbb{F}_{2}^n \rightarrow \mathbb {F}_{2}^n$ 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.

Dates et versions

hal-03901649 , version 1 (15-12-2022)

Identifiants

Citer

Christof Beierle, Gregor Leander, Léo Perrin. Trims and extensions of quadratic APN functions. Designs, Codes and Cryptography, 2022, 90 (4), pp.1009-1036. ⟨10.1007/s10623-022-01024-4⟩. ⟨hal-03901649⟩
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