Investigations of c-differential uniformity of permutations with Carlitz rank 3

Jeong, Jaeseong; Koo, Namhun; Kwon, Soonhak

doi:10.1016/j.ffa.2022.102145

Cited by 5 publications

(2 citation statements)

References 23 publications

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“…Likewise, regarding the original notion of differential uniformity leading to optimal functions, Perfect Nonlinear (PN) and Almost Perfect Nonlinear (APN) over finite fields in odd and even characteristics, respectively, optimal functions having the lowest possible values of a c-differential uniformity have also been introduced. One can refer to [17,19,22,25,48,51,52] and the references therein. Some of those functions with low c-differential uniformity have been investigated.…”

Section: Of 17mentioning

confidence: 99%

Further Study $c$-Differential-Linear Connectivity Table of Vectorial Boolean Functions toward Promising Applications in Coding Theory

Eddahmani,

Mesnager

2024

Preprint

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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.

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Section: Of 17mentioning

confidence: 99%

Further Study $c$-Differential-Linear Connectivity Table of Vectorial Boolean Functions toward Promising Applications in Coding Theory

Eddahmani,

Mesnager

2024

Preprint

View full text Add to dashboard Cite

show abstract

“…Likewise, regarding the original notion of differential uniformity leading to optimal functions Perfect Nonlinear (PN) and Almost Perfect Nonlinear (APN) over finite fields in odd and even characteristics, respectively, optimal functions having the lowest possible values of a c -differential uniformity have also been introduced. One can refer to [ 19 , 22 , 23 , 24 , 25 , 26 , 27 ] and the references therein. Some of those functions with low c -differential uniformity have been investigated.…”

Section: Introductionmentioning

confidence: 99%

The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions

Eddahmani,

Mesnager

2024

Entropy

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Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (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 the properties of these novel functions at their optimal level concerning these concepts and describe the c-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, we draw new ideas for future research toward linear code designs.

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