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Cryptography and Communications

, Volume 11, Issue 1, pp 21–39 | Cite as

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

  • Valeriya IdrisovaEmail author
Article
  • 86 Downloads
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications

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.

Keywords

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

Mathematics Subject Classification (2010)

94A60 06E30 11T71 

Notes

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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Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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