Article
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Further Study $c$-Differential-Linear Connectivity Table of Vectorial Boolean Functions toward Promising Applications in Coding Theory
Version 1
: Received: 16 January 2024 / Approved: 17 January 2024 / Online: 17 January 2024 (11:28:13 CET)
A peer-reviewed article of this Preprint also exists.
Eddahmani, S.; Mesnager, S. The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions. Entropy 2024, 26, 188. Eddahmani, S.; Mesnager, S. The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions. Entropy 2024, 26, 188.
Abstract
Vectorial Boolean functions and codes are closely related, and each corresponding domain gives been to each other. On the one hand, various requirements of binary linear codes are needed for their theatrical interests but, more importantly, for their practical applicants (such as few-weight codes or minimal codes for secret sharing, locally recoverable codes for storage, etc.). On the other hand, various criteria and tables have been introduced to analyse the security of S-boxes that are related to vectorial Boolean functions, such as the Differential Distribution Table (DDT), the Boomerang Connectivity Table (BCT), and the Differential-Linear Connectivity Table (DLCT). In previous years, two new tables have been proposed for which the literature was pretty abundant: the $c$-DDT to extend the DDT and the $c$-BCT to extend the BCT. In the same vein, we propose extended concepts to study further the security of vectorial Boolean functions, especially the $c$-Walsh transform, the $c$-autocorrelation, and the $c$-differential-linear uniformity, and its accompanying table, the $c$-Differential-Linear Connectivity Table ($c$-DLCT). We study these properties of the novel functions at their optimal level concerning these concepts and describe the $c$-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, the derived functions could lead to new families of binary minimal codes, as the recent achievements on minimal codes from low differential uniformity. We draw new avenues for future research toward linear code designs.
Keywords
Differential uniformity; Vectorial Function; S-Box; Linear Codes; Minimal Codes
Subject
Engineering, Other
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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