Linear codes from weakly regular plateaued functions and their secret sharing schemes

Mesnager, Sihem; Özbudak, Ferruh; Sınak, Ahmet

doi:10.1007/s10623-018-0556-4

Cited by 64 publications

(34 citation statements)

References 19 publications

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“…In [12], Bonini and Borello investigated the geometric generalization of the construction in [8], highlighting a first link between minimal codes and cutting blocking sets. Moreover, different types of recent constructions of minimal codes based on weakly regular bent and plateaued functions have been also presented in [31,32,33].…”

Section: Introductionmentioning

confidence: 99%

A geometric characterization of minimal codes and their asymptotic performance

Alfarano¹,

Borello²,

Neri³

2022

AMC

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In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes as cutting blocking sets.

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Section: Introductionmentioning

confidence: 99%

A geometric characterization of minimal codes and their asymptotic performance

Alfarano¹,

Borello²,

Neri³

2022

AMC

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“…M. Borello is with LAGA, UMR 7539, CNRS, Université Paris 13 -Sorbonne Paris Cité, Université Paris 8, F-93526, Saint-Denis, France. 1 where w min and w max respectively denote the minimum and maximum nonzero weights in C. This sufficient condition (which we will call AB condition) provides an easy criterion to construct minimal linear codes, especially in the case of codes with few nonzero weights (see for example [8,16]).…”

Section: Introductionmentioning

confidence: 99%

Minimal linear codes arising from blocking sets

Bonini

Borello

2020

J Algebr Comb

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Minimal linear codes are algebraic objects which gained interest in the last twenty years, due to their link with Massey's secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy to handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes.In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin-Barg's condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes arising from particular blocking sets, which are well-studied combinatorial objects. In this context, we will need to define cutting blocking sets and to prove some of their relations with other notions in blocking sets' theory. At the end of the paper, we provide one explicit family of cutting blocking sets and related minimal linear codes, showing that infinitely many of its members do not satisfy the Ashikhmin-Barg's condition.

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“…McEliece and Sarwate noted the relationship between Shamir-Blakley's secret-sharing scheme and Reed-Solomon codes in [13]. Since then, several secretsharing schemes have been constructed in terms of linear error-correcting codes [14][15][16]. To construct a secret-sharing scheme from linear codes, Massey pointed out the relationship between the access structure and the minimal codewords of the dual code of the underlying code [17,18].…”

Section: Secret-sharing Scheme From Linear Codesmentioning

confidence: 99%

A Group Identification Protocol with Leakage Resilience of Secret Sharing Scheme

Yan

et al. 2020

Complexity

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Secret sharing has been study for many years and has had a number of real-word applications. ere are several methods to construct the secret-sharing schemes. One of them is based on coding theory. In this work, we construct a secret-sharing scheme that realizes an access structure by using linear codes, in which any element of the access structure can reconstruct the secret key. We prove that our scheme is a multiprover zero-knowledge proof system in the random oracle model, which shows that a passive adversary gains no information about the secret key. Our scheme is also a leakage-resilient secret-sharing scheme (LRSS) in the bounded-leakage model, which remain provably secure even if the adversary learns a bounded amount of leakage information about their secret key. As an application, we propose a new group identification protocol (GID-scheme) from our LRSS. We prove that our GID-scheme is a leakage-resilient scheme. In our leakage-resilient GID-scheme, the verifier believes the validity of qualified group members and tolerates l bits of adversarial leakage in the distribution protocol, whereas for unqualified group members, the verifier cannot believe their valid identifications in the proof protocol.

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Linear codes from weakly regular plateaued functions and their secret sharing schemes

Cited by 64 publications

References 19 publications

A geometric characterization of minimal codes and their asymptotic performance

A geometric characterization of minimal codes and their asymptotic performance

Minimal linear codes arising from blocking sets

A Group Identification Protocol with Leakage Resilience of Secret Sharing Scheme

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