Volume 4, Issue 2

The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. The optimization of this parameter for general access structures is an important and very difficult open problem in secret sharing. We explore in this paper the connections of this open problem with matroids and polymatroids. Matroid ports were introduced by Lehman in 1964. A forbidden minor characterization of matroid ports was given by Seymour in 1976. These results precede the invention of secret sharing by Shamir in 1979. Important connections between ideal secret sharing schemes and matroids were discovered by Brickell and Davenport in 1991. Their results can be restated as follows: every ideal secret sharing scheme defines a matroid, and its access structure is a port of that matroid. Our main result is a lower bound on the optimal complexity of access structures that are not matroid ports. Namely, by using the aforementioned characterization of matroid ports by Seymour, we generalize the result by Brickell and Davenport by proving that, if the length of every share in a secret sharing scheme is less than 3/2 times the length of the secret, then its access structure is a matroid port. This generalizes and explains a phenomenon that was observed in several families of access structures. In addition, we introduce a new parameter to represent the best lower bound on the optimal complexity that can be obtained by taking into account that the joint Shannon entropies of a set of random variables define a polymatroid. We prove that every bound that is obtained by this technique for an access structure applies to its dual as well. Finally, we present a construction of linear secret sharing schemes for the ports of the Vamos and the non-Desargues matroids. In this way new upper bounds on their optimal complexity are obtained, which are a contribution on the search of access structures whose optimal complexity lies between 1 and 3/2.

Message authentication codes (MACs) based on universal hash-function families are becoming increasingly popular due to their fast implementation. In this paper, we investigate a family of universal hash functions that has been appeared repeatedly in the literature and provide a detailed algebraic analysis for the security of authentication codes based on this universal hash family. In particular, the universal hash family under analysis, as appeared in the literature, uses operation in the finite field ℤ p . No previous work has studied the extension of such universal hash family when computations are performed modulo a non-prime integer n . In this work, we provide the first such analysis. We investigate the security of authentication when computations are performed over arbitrary finite integer rings ℤ n and derive an explicit relation between the prime factorization of n and the bound on the probability of successful forgery. More specifically, we show that the probability of successful forgery against authentication codes based on such a universal hash-function family is bounded by the reciprocal of the smallest prime factor of the modulus n .

We analyze the Sibert et al. group-based (Feige–Fiat–Shamir type) authentication protocol and show that the protocol is not computationally zero-knowledge. In addition, we provide experimental evidence that our approach is practical and can succeed even for groups with no efficiently computable length function such as braid groups. The novelty of this work is that we are not attacking the protocol by trying to solve an underlying complex algebraic problem, namely, the conjugacy search problem, but use a probabilistic approach, instead.

In this paper we investigate the asymptotic effectiveness of the Gaudry–Hess–Smart Weil descent attack and its generalization to the discrete logarithm problem for elliptic curves over characteristic-two finite fields. In particular we obtain nontrivial lower and upper bounds on the smallest possible genus to which it can lead.

The paper provides a cryptanalysis of the AA β -cryptosystem recently proposed by Ariffin and Abu. The scheme is in essence a key agreement scheme whose security is based on a discrete logarithm problem in the infinite (additive) group ℝ/ℤ (the reals modulo 1). The paper breaks the AA β -cryptosystem (in a passive adversary model) by showing that this discrete logarithm problem can be efficiently solved in practice.