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On the Algebraic Immunity of Weightwise Perfectly Balanced Functions

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Progress in Cryptology – LATINCRYPT 2023 (LATINCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14168))

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Abstract

In this article we study the Algebraic Immunity (AI) of Weightwise Perfectly Balanced (WPB) functions. After showing a lower bound on the AI of two classes of WPB functions from the previous literature, we prove that the minimal AI of a WPB n-variables function is constant, equal to 2 for \(n\ge 4\). Then, we compute the distribution of the AI of WPB function in 4 variables, and estimate the one in 8 and 16 variables. For these values of n we observe that a large majority of WPB functions have optimal AI, and that we could not obtain a WPB function with AI 2 by sampling at random. Finally, we address the problem of constructing WPB functions with bounded algebraic immunity, exploiting a construction from [12]. In particular, we present a method to generate multiple WPB functions with minimal AI, and we prove that the WPB functions with high nonlinearity exhibited in [12] also have minimal AI. We conclude with a construction giving WPB functions with lower bounded AI, and give as example a family with all elements with AI at least \(n/2-\log (n)+1\).

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Authors and Affiliations

  1. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Agnese Gini & Pierrick Méaux

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  1. Agnese Gini
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  2. Pierrick Méaux
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Correspondence to Agnese Gini .

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Editors and Affiliations

  1. Technology Innovation Institute, Abu Dhabi, United Arab Emirates

    Abdelrahaman Aly

  2. NTT Social Informatics Laboratories, Tokyo, Japan

    Mehdi Tibouchi

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Gini, A., Méaux, P. (2023). On the Algebraic Immunity of Weightwise Perfectly Balanced Functions. In: Aly, A., Tibouchi, M. (eds) Progress in Cryptology – LATINCRYPT 2023. LATINCRYPT 2023. Lecture Notes in Computer Science, vol 14168. Springer, Cham. https://doi.org/10.1007/978-3-031-44469-2_1

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  • DOI: https://doi.org/10.1007/978-3-031-44469-2_1

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  • Online ISBN: 978-3-031-44469-2

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