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The classification of quadratic APN functions in 7 variables and combinatorial approaches to search for APN functions

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Abstract

Almost perfect nonlinear functions possess optimal resistance to differential cryptanalysis and are widely studied. Most known APN functions are defined using their representation as a polynomial over a finite field and very little is known about combinatorial constructions of them on \(\mathbb {F}_{2}^{n}\). In this work we propose two approaches for obtaining quadratic APN functions on \(\mathbb {F}_{2}^{n}\). The first approach exploits a secondary construction idea, it considers how to obtain a quadratic APN function in n + 1 variables from a given quadratic APN function in n variables using special restrictions on the new terms. The second approach is searching for quadratic APN functions that have a matrix representation partially filled with the standard basis vectors in a cyclic manner. This approach allows us to find a new APN function in 7 variables. We prove that the updated list of quadratic APN functions in dimension 7 is complete up to CCZ-equivalence. Also, we observe that the quadratic parts of some APN functions have a low differential uniformity. This observation allows us to introduce a new subclass of APN functions, the so-called stacked APN functions. These are APN functions of algebraic degree d such that eliminating monomials of degrees k + 1,…, d for any k < d results in APN functions of algebraic degree k. We provide cubic examples of stacked APN functions for dimensions up to 6.

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Acknowledgements

We sincerely thank the anonymous reviewers for their careful reading of this manuscript and suggesting substantial improvements. We would like to cordially thank Natalia Tokareva for her valuable remarks. We are deeply thankful to Christof Beierle for pointing out some inaccuracies. We are much indebted to the reviewers of the SETA-2020 conference for their helpful reviews. We are grateful to Anastasia Gorodilova and Nikolay Kolomeec for their useful observations and fruitful discussions. The work is supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. We are grateful to the Supercomputing Center of the Novosibirsk State University for the provided computational resources.

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  1. Sobolev Institute of Mathematics, Institute of Computational Mathematics and Mathematical Geophysics of the Siberian Branch of the RAS, Novosibirsk State University, Novosibirsk, Russia

    Konstantin Kalgin

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Valeriya Idrisova

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  1. Konstantin Kalgin
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Correspondence to Konstantin Kalgin.

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This article belongs to the Topical Collection: Sequences and Their Applications III

Guest Editors: Chunlei Li, Tor Helleseth and Zhengchun Zhou

The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. We are grateful to the Supercomputing Center of the Novosibirsk State University for provided computational resources.

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Kalgin, K., Idrisova, V. The classification of quadratic APN functions in 7 variables and combinatorial approaches to search for APN functions. Cryptogr. Commun. 15, 239–256 (2023). https://doi.org/10.1007/s12095-022-00588-1

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