Representing Small Identically Self‐Dual Matroids by Self‐Dual Codes

Abstract-Error-correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, the connections between codes, matroids, and a special class of secret sharing schemes, namely, multiplicative linear secret sharing schemes (LSSSs), are studied. Such schemes are known to enable multiparty computation protocols secure against general (nonthreshold) adversaries.Two open problems related to the complexity of multiplicative LSSSs are considered in this paper. The first one deals with strongly multiplicative LSSSs. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. A property of strongly multiplicative LSSSs that could be useful in solving this problem is proved. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, it is shown that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults. The second one is to characterize the access structures of ideal multiplicative LSSSs. Specifically, the considered open problem is to determine whether all self-dual vector space access structures are in this situation. By the aforementioned connection, this in fact constitutes an open problem about matroid theory, since it can be restated in terms of representability of identically self-dual matroids by self-dual codes. A new concept is introduced, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. It is proved that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids.

show abstract

“…It has been proved recently that this property also holds for the identically self-dual matroids on at most eight points [26].…”

Section: Sincementioning

confidence: 91%

“…Finally, all identically self-dual matroids with rank at most four, that is, on at most eight points, are self-dually representable [26].…”

Section: Known Families Of Self-dually Representable Matroidsmentioning

confidence: 99%

On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

Cramer

Daza²,

Gracia³

et al. 2005

Advances in Cryptology – CRYPTO 2005

Self Cite

View full text Add to dashboard Cite

show abstract

“…A special case of this problem is the characterization of access structures that admit ideal multiplicative LSSS. This problem was considered in [37,91] for some self-dual access structures. Many authors also considered the construction of ideal multiplicative LSSS [32,37,68,73,87].…”

Section: Secret Sharing Schemes Based On Graphical Codesmentioning

confidence: 99%

“…. Given a self-dual access structure which can be realized by an ideal LSSS, the question of whether it admits an ideal multiplicative scheme was studied in [37,91]. Self-dual access structures are the minimally Q 2 access structures.…”

Section: Shamir Scheme)mentioning

confidence: 99%

See 1 more Smart Citation

On secret sharing schemes and linear codes

Cruz.¹

View full text Add to dashboard Cite

A secret sharing scheme is a method of distributing a secret among a group of participants in such a way that only certain subgroups are authorized to know the secret. Secret sharing schemes have been used in many areas of cryptography such as secure information storage, threshold cryptographic protocols, secure message transmission, secure multiparty computation, and generalized oblivious transfer. This thesis deals with secret sharing schemes constructed using linear codes. First, we consider the secret sharing schemes based on linear codes constructed using the approach by Karnin, Greene and Hellman, by Bertilsson and Ingemarsson and by Massey. In this approach, the minimal authorized groups correspond to certain minimal codewords of the dual of the underlying code. We study three topics related to secret sharing schemes based on linear codes. The first topic is on the access structure and multiplicativity of linear secret sharing schemes based on codes from complete graphs. Multiplicative linear secret sharing schemes are used in the construction of secure multiparty computation protocols. The characterization of access structures of ideal multiplicative schemes has not been solved completely. We describe the access structure of the schemes based on cut-set and cycle codes and show that these schemes cannot be multiplicative. We then show v that the class of access structures based on odd cycles cannot be realized by ideal multiplicative linear secret sharing schemes over any finite field. This can be seen as a contribution to the characterization of access structures of ideal multiplicative schemes. The access structure based on odd cycles corresponds to the scheme based on the dual of the extended cycle code. Finally, we show that we can obtain ideal multiplicative linear secret sharing scheme based on the dual of an augmented extended cycle code. The second topic is a generalization of secret sharing schemes based on linear codes. In this generalization, the secret consists of multiple field elements. We give a characterization of the groups in the access structure using the dual code. We then use a generalized joint weight enumerator to describe the access structure. The third topic is on the maximum number of minimal codewords in binary linear codes. This subject has been studied before but in the setting of cycles in graphs and of circuits in matroids. We prove several upper and lower bounds on the maximum number of minimal codewords in binary linear codes given the length and dimension. We make the bounds explicit for code of small length and dimension. Exact values are given for small cycle codes of graphs. Some asymptotic results are presented including the asymptotic behavior for cycle codes of graphs. We also consider another type of secret sharing schemes constructed using linear codes. These are the so-called cheating-immune secret sharing schemes. A secret sharing scheme that is cheating-immune prevents a cheater, who submits a corrupted share, from gaining an advantage in knowing the secret ove...

show abstract

Secret sharing schemes based on graphical codes

Gao

Cruz

2013

Cryptogr. Commun.

View full text Add to dashboard Cite

Representing Small Identically Self‐Dual Matroids by Self‐Dual Codes

Cited by 7 publications

References 10 publications

On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

On Codes, Matroids and Secure Multi-party Computation from Linear Secret Sharing Schemes

On secret sharing schemes and linear codes

Secret sharing schemes based on graphical codes

Contact Info

Product

Resources

About