Abstract
Browning et al. (2010) exhibited almost perfect nonlinear (APN) permutations on \(\mathbb {F}_{2^{6}}\). This was the first example of an APN permutation on an even degree extension of \(\mathbb {F}_{2}\). In their approach of finding an APN permutation, Browning et al. made use of a necessary and sufficient condition based on the Walsh transform. In this paper, we give an algorithm based on a related necessary condition which checks whether a vectorial Boolean function is CCZ-inequivalent to a permutation. Using this algorithm, we are able to show that no function belonging to a known family of APN functions is equivalent to a permutation on \(\mathbb {F}_{2^{2m}}\), where m ≤ 6 (except for the known case on \(\mathbb {F}_{2^{6}}\)). We also give an EA-invariant based on the condition. Finally, we give a theoretical proof of the fact that no member of a specific family of APN functions is equivalent to a permutation on doubly-even degree extensions of \(\mathbb {F}_{2}\).
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Notes
Note that Algorithm 1 is easily modified to list all vector spaces with dimension greater than some value. For instance, assume the maximum dimension of a vector space in NBF is n/2 and it returns only one such vector space (or it returns two such vector spaces U,V which do not satisfy \(U + V = \mathbb {F}_{2^{n}}\)), then one can conclude by Condition 2 that the function is not CCZ-equivalent to any permutation.
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Acknowledgements
The authors would like to thank the anonymous reviewers for their extensive reviews. During WCC 2019, we have become aware that Anne Canteaut and Leo Perrin independently devised an algorithm similar to Algorithm 1. This work was supported by the GAČR Grant 18-19087S - 301-13/201843.
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Göloğlu, F., Pavlu̇, J. On CCZ-inequivalence of some families of almost perfect nonlinear functions to permutations. Cryptogr. Commun. 13, 377–391 (2021). https://doi.org/10.1007/s12095-021-00476-0
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DOI: https://doi.org/10.1007/s12095-021-00476-0