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Explicit values of the DDT, the BCT, the FBCT, and the FBDT of the inverse, the gold, and the Bracken-Leander S-boxes

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Abstract

The inverse, the Gold, and the Bracken-Leander functions are crucial for building S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. These functions have been intensively studied, and various properties related to standard attacks have been investigated. Thanks to novel advances in symmetric cryptography and, more precisely, those pertaining to boomerang cryptanalysis, this article continues to follow this momentum and further examine these functions. More specifically, we revisit and bring new results about their Difference Distribution Table (DDT), their Boomerang Connectivity Table (BCT), their Feistel Boomerang Connectivity Table (FBCT), and their Feistel Boomerang Difference Table (FBDT). For each table, we give explicit values of all entries by solving specific systems of equations over the finite field \(\mathbb {F}_{2^n}\) of cardinality \(2^n\) and compute the cardinalities of their corresponding sets of such values. The explicit values of the entries of these tables and their cardinalities are crucial tools to test the resistance of block ciphers based on variants of the inverse, the Gold, and the Bracken-Leander functions against cryptanalytic attacks such as differential and boomerang attacks. The computation of these entries and the cardinalities in each table aimed to facilitate the analysis of differential and boomerang cryptanalysis of S-boxes when studying distinguishers and trails.

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, F-93526 Saint-Denis, Laboratory Geometry, Analysis and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France

    Said Eddahmani & Sihem Mesnager

  2. Telecom Paris, Polytechnic institute, 91120, Palaiseau, France

    Sihem Mesnager

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  1. Said Eddahmani
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  2. Sihem Mesnager
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Correspondence to Sihem Mesnager.

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Eddahmani, S., Mesnager, S. Explicit values of the DDT, the BCT, the FBCT, and the FBDT of the inverse, the gold, and the Bracken-Leander S-boxes. Cryptogr. Commun. 14, 1301–1344 (2022). https://doi.org/10.1007/s12095-022-00581-8

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