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Partially APN Boolean functions and classes of functions that are not APN infinitely often

Abstract

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field \(\mathbb {F}_{2}\), extending some earlier results of Leander and Rodier.

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Acknowledgements

The authors would like to thank the referees for their thorough reading and useful comments, and the editors for handling our manuscript very efficiently. The paper was started while the fourth named author visited Selmer center at UiB in the Summer of 2018. This author thanks the institution for the excellent working conditions. S.K. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1D1A1B03931912 and No. 2016R1A5A1008055). The research of the first two authors was supported by Trond Mohn foundation.

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Correspondence to Pantelimon Stănică.

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Budaghyan, L., Kaleyski, N.S., Kwon, S. et al. Partially APN Boolean functions and classes of functions that are not APN infinitely often. Cryptogr. Commun. 12, 527–545 (2020). https://doi.org/10.1007/s12095-019-00372-8

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  • DOI: https://doi.org/10.1007/s12095-019-00372-8

Keywords

  • Boolean function
  • Almost perfect nonlinear (APN)
  • Partial APN
  • Walsh-Hadamard coefficients

Mathematics Subject Classification (2010)

  • 94A60
  • 94C10
  • 06B30