Abstract
This work extends the idea introduced by Hou and Langevin (J. Combin. Theory, Ser. A, 80:232–246, 1997) of applying nonlinear permutations to (a portion of) the input variable space of a given Boolean function so that the resulting function is bent. Applying such a permutation to a bent function that can be represented in a suitable form then gives an affine inequivalent bent function which potentially does not belong to the same class as the original one. While Hou and Langevin only provided two sporadic examples of bent functions that can be turned into affine inequivalent ones, in this article we identify two generic families of bent functions suitable for generating such affine inequivalent counterparts. The same method when applied to the Marioana-McFarland class of bent functions, depending on the subset of inputs to which a nonlinear action is applied, either lead to bent functions that are provably within the same class or to bent functions that are potentially outside this class. The problem of finding suitable permutations that act nonlinearly on more than two input variables of the initial function and ensure the bentness of the resulting function appears to be generally hard. In this direction, we only slightly extend the approach of Hou and Langevin by identifying suitable permutations that act nonlinearly on three input variabl es. Most notably, the existence of nonlinear permutations that act without strict separation of the input space in terms of linear and nonlinear action is also confirmed. Finally, we show a direct correspondence between (some classes of) bent functions and permutations by providing an efficient method to define permutations using the derivatives of a given bent function. This not only gives a relationship between two seemingly different algebraic objects, but also provides us with a new infinite family of permutations over finite fields.
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Acknowledgments
The authors are thankful to PhD student Nastja Cepak for computer simulations related to permutations in [16]. Enes Pasalic is partly supported by the Slovenian Research Agency (research program P3- 0384 and research project J1-6720). Samir Hodžić is supported in part by the Slovenian Research Agency (research program P3-0384 and Young Researchers Grant). Fengrong Zhang is supported in part by National Science Foundation of China(61303263), and and in part by the Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS086), and in part by the China Postdoctoral Science Foundation funded project (Grant No. 2015T80600). Yongzhuang Wei is supported in part by the Natural Science Foundation of China (61572148), in part by the Guangxi Natural Science Found (2015GXNSFGA139007), in part by the project of Outstanding Young Teachers Training in Higher Education Institutions of Guangxi.
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Pasalic, E., Hodžić, S., Zhang, F. et al. Bent functions from nonlinear permutations and conversely. Cryptogr. Commun. 11, 207–225 (2019). https://doi.org/10.1007/s12095-018-0282-9
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DOI: https://doi.org/10.1007/s12095-018-0282-9