Minimal Linear Codes From Characteristic Functions

Mesnager, Sihem; Qi, Yanfeng; Ru, Hongming; Tang, Chunming

doi:10.1109/tit.2020.2978387

Cited by 39 publications

(29 citation statements)

References 35 publications

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“…In Table 1, some well-known constructions for minimal codes are summarized. Recently, Mesnager et al presented a generalized construction tool including several existing design methods [23]. Since most of known minimal codes are designed using the structure of GF(p m ), their lengths are restricted to p m − 1 for a prime p.…”

Section: Preliminariesmentioning

confidence: 99%

“…Using the minimal code, one user's information is not subordinate to other users' information. This has been constantly studied as one of the mathematical structures that can be used in secret-sharing schemes [16][17][18][19][20][21][22][23]. Furthermore, a minimal code can be used in federated learning due to its distributed characteristics of secret information.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, for binary cases, several design methods have been proposed [16][17][18][19][20]. On the other hand, non-binary minimal code has been investigated recently [21][22][23]. Research on previously known minimal codes has focused only on the weight distribution of the codes.…”

Section: Introductionmentioning

confidence: 99%

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Design and Application of Secret Codes for Learning Medical Data

Chung

2022

Applied Sciences

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In distributed learning for data requiring privacy preservation, such as medical data, the distribution of secret information is an important problem. In this paper, we propose a framework for secret codes in application to distributed systems. Then, we provide new methods to construct such codes using the synthesis or decomposition of previously known minimal codes. The numerical results show that new constructions can generate codes with more flexible parameters than original constructions in the sense of the number of possible weights and the range of weights. Thus, the secret codes from new constructions may be applied to more general situations or environments in distributed systems.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Design and Application of Secret Codes for Learning Medical Data

Chung

2022

Applied Sciences

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“…This behavior of well-known cryptographic vectorial Boolean functions might indicate a new weakness. Also, despite the lack of a cryptanalysis method that could highlight further the notions of the c-differential and c-boomerang uniformities, we emphasize that optimal binary codes deriving from c-differential uniform or c-boomerang uniformity multioutput Boolean functions (where c is a non-zero element in a finite field of order p n ) could be constructed [82,83]. Moreover, such constructions could be naturally extended to design codes defined over finite fields with any characteristic.…”

Section: Introductionmentioning

confidence: 99%

Survey on recent trends towards generalized differential and boomerang uniformities

2021

Self Cite

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Differential cryptanalysis is a general form of cryptanalysis applicable primarily to block and stream ciphers and cryptographic hash functions. The discovery of differential cryptanalysis is generally attributed to Biham and Shamir in the late 1980s, who published several attacks against various block ciphers and hash functions, including a theoretical weakness in the Data Encryption Standard (DES). Boomerang cryptanalysis is a method for the cryptanalysis of block ciphers based on differential cryptanalysis. It was invented by Wagner in (FSE, LNCS 1636, 156-170, 1999 and has allowed new avenues of attack for many ciphers previously deemed safe from differential cryptanalysis. Differential and boomerang uniformities are crucial tools to handle and analyze vectorial functions (designated by substitution boxes, or briefly S-boxes in the context of symmetric cryptography) to resist differential and boomerang attacks, respectively. Ellingsen et al. (IEEE Transactions on Information Theory 66(9), 2020) introduced a new variant of differential uniformity, called c-differential uniformity (where c is a non-zero element of a finite field of characteristic p), of p-ary (n, m)-function for any prime p obtained by extending the well-known derivative of vectorial functions into the (multiplicative) c-derivative. Later, Stănică [Discrete Applied Mathematics, 2021] introduced the notion of c-boomerang uniformity. Both c-differential and c-boomerang uniformities have been extended to the idea of simple differential and boomerang uniformities, respectively, which are recovered when c equals 1.This survey paper combines the known results on this new concept of differential and boomerang uniformities and analyzes their possible cryptographic applications. This survey presents an overview of these significant concepts that might have greater implications for future theoretical research on this subject and applied perspectives in symmetric cryptography and related topics. Along with the paper, we analyze these discoveries and the results provided synthetically. The article intends to help readers explore further avenues in this promising and emerging direction of research. At the end of the article, we present more than nine lines of perspectives and research directions to benefit symmetric cryptography and other related domains such as combinatorial theory (namely, graph theory).

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“…For example, some minimal linear codes have been successfully employed in secret sharing and secure two-party computations. The recent progress on this kind of codes can be found in [3,17,18,23]. Let Tr q/p denote the trace function from F q to F p .…”

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confidence: 99%

Two-weight and three-weight linear codes constructed from Weil sums

Zhang¹,

Lü²,

Yang³

2022

MFC

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<p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.</p>

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Minimal Linear Codes From Characteristic Functions

Cited by 39 publications

References 35 publications

Design and Application of Secret Codes for Learning Medical Data

Design and Application of Secret Codes for Learning Medical Data

Survey on recent trends towards generalized differential and boomerang uniformities

Two-weight and three-weight linear codes constructed from Weil sums

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