Trims and extensions of quadratic APN functions
- 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.
Author: | Christof BeierleORCiDGND, Nils-Gregor LeanderORCiDGND, Léo PerrinORCiDGND |
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URN: | urn:nbn:de:hbz:294-88379 |
DOI: | https://doi.org/10.1007/s10623-022-01024-4 |
Parent Title (English): | Designs, Codes and Cryptography |
Publisher: | Springer Science+Business Media B.V. |
Place of publication: | Dordrecht |
Document Type: | Article |
Language: | English |
Date of Publication (online): | 2022/04/21 |
Date of first Publication: | 2022/03/11 |
Publishing Institution: | Ruhr-Universität Bochum, Universitätsbibliothek |
Tag: | Almost perfect nonlinear; EA-equivalence; EA-invariant; Extension; Linearity; Restriction |
Volume: | 90 |
Issue: | 4 |
First Page: | 1009 |
Last Page: | 1036 |
Institutes/Facilities: | Horst Görtz Institut für IT-Sicherheit |
open_access (DINI-Set): | open_access |
faculties: | Fakultät für Mathematik |
Licence (English): | Creative Commons - CC BY 4.0 - Attribution 4.0 International |