STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Multiparty unconditionally secure protocols
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
On decoding by error location and dependent sets of error positions
Discrete Mathematics - A collection of contributions in honour of Jack van Lint
An explication of secret sharing schemes
Designs, Codes and Cryptography
Geometric secret sharing schemes and their duals
Designs, Codes and Cryptography
Adaptively secure multi-party computation
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
PODC '97 Proceedings of the sixteenth annual ACM symposium on Principles of distributed computing
Designs, Codes and Cryptography
Communications of the ACM
On the Composition of Matroids and Ideal Secret Sharing Schemes
Designs, Codes and Cryptography
On Some Polynomials Related to Weight Enumerators of Linear Codes
SIAM Journal on Discrete Mathematics
A Representation of a Family of Secret Sharing Matroids
Designs, Codes and Cryptography
General secure multi-party computation from any linear secret-sharing scheme
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Secret sharing schemes on access structures with intersection number equal to one
SCN'02 Proceedings of the 3rd international conference on Security in communication networks
Characterizing ideal weighted threshold secret sharing
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
Secret sharing schemes with bipartite access structure
IEEE Transactions on Information Theory
Secure Computation from Random Error Correcting Codes
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
Strongly Multiplicative and 3-Multiplicative Linear Secret Sharing Schemes
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Strongly multiplicative ramp schemes from high degree rational points on curves
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
Cryptography and Communications
Efficient reductions for non-signaling cryptographic primitives
ICITS'11 Proceedings of the 5th international conference on Information theoretic security
Algebraic geometric secret sharing schemes and secure multi-party computations over small fields
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
On matroids and non-ideal secret sharing
TCC'06 Proceedings of the Third conference on Theory of Cryptography
Coset bounds for algebraic geometric codes
Finite Fields and Their Applications
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Error correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, we study the connections between codes, matroids and a special class of secret sharing schemes, namely multiplicative linear secret sharing schemes. Such schemes are known to enable multi-party computation protocols secure against general (non-threshold) 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. We prove a property of strongly multiplicative LSSSs that could be useful in solving this problem. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, we show 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, we wonder 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 re-stated in terms of representability of identically self-dual matroids by self-dual codes. We introduce a new concept, 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. We prove 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.