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The Number of Almost Perfect Nonlinear Functions Grows Exponentially

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Abstract

Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least \(\frac{\varphi (m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil \) inequivalent APN functions on the finite field with \({2^{2m}}\) elements, where \(\varphi \) denotes Euler’s totient function. This is a great improvement of previous results: for even m, the best known lower bound has been \(\frac{\varphi (m)}{2}\left( \lfloor \frac{m}{4}\rfloor +1\right) \); for odd m, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions.

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Acknowledgements

We thank the anonymous reviewers for their useful comments and suggestions, and we thank Satoshi Yoshiara and Ulrich Dempwolff for their helpful comments on Lemma 2.4 and the connection of the automorphism groups of quadratic APN functions under EA- and under CCZ-equivalence.

This work is partially supported by National Key R&D Program of China (No. 2017YFB0802000) and Training Program for Excellent Young Innovators of Changsha (No. kq1905052).

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Authors and Affiliations

  1. Institute for Algebra and Geometry, Otto von Guericke University Magdeburg, 39106, Magdeburg, Germany

    Christian Kaspers

  2. Department of Mathematics, National University of Defense Technology, Changsha, 410073, China

    Yue Zhou

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  1. Christian Kaspers
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  2. Yue Zhou
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Correspondence to Yue Zhou.

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Communicated by Kaisa Nyberg

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Kaspers, C., Zhou, Y. The Number of Almost Perfect Nonlinear Functions Grows Exponentially. J Cryptol 34, 4 (2021). https://doi.org/10.1007/s00145-020-09373-w

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  • DOI: https://doi.org/10.1007/s00145-020-09373-w

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