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Inverse function is not component-wise uniform

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Abstract

Recently, Carlet introduced (Carlet Finite. Fields Appl. 53, 226–253 (2018)) the concept of component-wise uniform (CWU) functions which is a stronger notion than being almost perfect nonlinear (APN). Carlet showed that crooked functions (in particular quadratic functions, including Gold functions) and the compositional inverse of a specific Gold function are CWU. Carlet also compiled a table on the component-wise uniformity of APN power functions, where for the extension degrees less than 12, the APN power functions which are always CWU are Gold, the compositional inverse of the specific Gold function and Kasami. In this paper, we show that the Inverse function is not CWU if n > 5. We also show that any additive shift of a derivative of the Inverse function is not a cyclic difference set.

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Acknowledgements

The author would like to thank the Editor, Associate Editor and the Reviewers for very useful comments. This work was supported by the GAČR Grant 18-19087S - 301-13/201843.

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  1. Department of Algebra, Charles University, Prague, Czech Republic

    Faruk Göloğlu

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  1. Faruk Göloğlu
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Correspondence to Faruk Göloğlu.

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Göloğlu, F. Inverse function is not component-wise uniform. Cryptogr. Commun. 12, 1179–1194 (2020). https://doi.org/10.1007/s12095-020-00441-3

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