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On generalized hyper-bent functions

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Abstract

Hyper-bent Boolean functions were introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems but no cryptographic attack has been identified until the one on the filtered LFSRs made by Canteaut and Rotella in 2016. Hyper-bent functions have properties still stronger than the well-known bent functions which were introduced by Rothaus and already studied by Dillon and next by several researchers in more than four decades. Hyper-bent functions are very rare and whose classification is still elusive. Therefore, not only their characterization, but also their generation are challenging problems. Recently, an important direction in the theory of hyper-bent functions was the extension of Boolean hyper-bent functions to whose codomain is the ring of integers modulo a power of a prime, that is, generalized hyper-bent functions. In this paper, we synthesize previous studies on generalized hyper-bent functions in a unified framework. We provide two characterizations of generalized hyper-bent functions in terms of their digits. We establish a complete characterization of a family of generalized hyper-bent functions defined over spreads and establish a link between vectorial hyper-bent functions found recently and that family.

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Acknowledgments

The author deeply thanks the program co-chairs (Tor Helleseth and Zhengchun Zhou) and the general co-chairs (Wai Ho Mow and Maosheng Xiong) of the conference SETA 2018 for their very nice invitation. She also thanks Cunsheng Ding and Chunming Tang for many interesting discussions in Hong-Kong. Many thanks to the Assoc. Edit. and the anonymous reviewers for their valuable comments which have highly improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  2. University of Paris XIII, CNRS, LAGA UMR 7539, Sorbonne Paris Cité, 93430, Villetaneuse, France

    Sihem Mesnager

  3. Telecom ParisTech, 75013, Paris, France

    Sihem Mesnager

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  1. Sihem Mesnager
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Correspondence to Sihem Mesnager.

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Mesnager, S. On generalized hyper-bent functions. Cryptogr. Commun. 12, 455–468 (2020). https://doi.org/10.1007/s12095-019-00390-6

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