On an algorithm generating 2-to-1 APN functions and its applications to “the big APN problem”

Abstract

Almost perfect nonlinear (APN) functions are of great interest to many researchers since they have the optimal resistance to the differential attack. The existence of bijective APN functions in even number of variables is an important open problem, and there is only one known example of such a function at present. In this paper we consider a special subclass of 2-to-1 vectorial Boolean functions that can allow us to search and construct APN permutations. We proved that each 2-to-1 function is potentially EA-equivalent to a permutation and proposed an algorithm that generates special symbol sequences for constructing 2-to-1 APN functions. Also, we described two methods for searching APN permutations, that are based on sequences generated by this algorithm.

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Acknowledgements

We would like to cordially thank Anastasiya Gorodilova and Natalia Tokareva for useful observations and fruitful discussions all along this work. We sincerely thank Nikolay Kolomeec for his helpful comments and careful reading. We are much indebted to the reviewers for their valuable remarks and for providing the proof of Proposition 5.

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Correspondence to Valeriya Idrisova.

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This article is part of the Topical Collection on Special Issue on Boolean Functions and Their Applications

The author was supported by the Russian Foundation for Basic Research (projects no. 17-41-543364 and no. 18-31-00374), by the program of fundamental scientific researches of the SB RAS no. I.5.1.(project no. 0314-2016-0017) and by Russian Ministry of Education and Science (the 5-100 Excellence Program and Project No. 1.12875.2018/12.1).

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Idrisova, V. On an algorithm generating 2-to-1 APN functions and its applications to “the big APN problem”. Cryptogr. Commun. 11, 21–39 (2019). https://doi.org/10.1007/s12095-018-0310-9

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Keywords

  • Boolean function
  • APN function
  • 2-to-1 function
  • APN permutation
  • Differential uniformity
  • S-box

Mathematics Subject Classification (2010)

  • 94A60
  • 06E30
  • 11T71