Matroids and Quantum Secret Sharing Schemes

A secret sharing scheme is a cryptographic protocol to distribute a secret state in an encoded form among a group of players such that only authorized subsets of the players can reconstruct the secret. Classically, efficient secret sharing schemes have been shown to be induced by matroids. Furthermore, access structures of such schemes can be characterized by an excluded minor relation. No such relations are known for quantum secret sharing schemes. In this paper we take the first steps toward a matroidal characterization of quantum secret sharing schemes. In addition to providing a new perspective on quantum secret sharing schemes, this characterization has important benefits. While previous work has shown how to construct quantum secret sharing schemes for general access structures, these schemes are not claimed to be efficient. In this context the present results prove to be useful; they enable us to construct efficient quantum secret sharing schemes for many general access structures. More precisely, we show that an identically self-dual matroid that is representable over a finite field induces a pure state quantum secret sharing scheme with information rate one

Secret sharing schemes: Optimizing the information ratio

Secret sharing refers to methods used to distribute a secret value among a set of participants. This work deals with the optimization of two parameters regarding the efficiency of a secret sharing scheme: the information ratio and average information ratio. Only access structures (a special family of sets) on 5 and 6 participants will be considered. First, access structures with 5 participants will be studied, followed by the ones on 6 participants that are based on graphs. The main goal of the paper is to check existing lower bounds (and improve some of them) by using linear programs with the sage solver. Shannon information inequalities have been used to translate the polymatroid axioms into linear constraints

On the information ratio of non-perfect secret sharing schemes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-016-0217-9A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values.Peer ReviewedPostprint (author's final draft

On the Information Ratio of Non-Perfect Secret Sharing Schemes

A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values

The Share Size of Secret-Sharing Schemes for Almost All Access Structures and Graphs

The share size of general secret-sharing schemes is poorly understood. The gap between the best known upper bound on the total share size per party of $2^{0.59n}$ (Applebaum and Nir, CRYPTO 2021) and the best known lower bound of $\Omega(n/\log n)$ (Csirmaz, J. of Cryptology 1997) is huge (where $n$ is the number of parties in the scheme). To gain some understanding on this problem, we study the share size of secret-sharing schemes of almost all access structures, i.e., of almost all collections of authorized sets. This is motivated by the fact that in complexity, many times almost all objects are hardest (e.g., most Boolean functions require exponential size circuits). All previous constructions of secret-sharing schemes were for the worst access structures (i.e., all access structures) or for specific families of access structures. We prove upper bounds on the share size for almost all access structures. We combine results on almost all monotone Boolean functions (Korshunov, Probl. Kibern. 1981) and a construction of (Liu and Vaikuntanathan, STOC 2018) and conclude that almost all access structures have a secret-sharing scheme with share size $2^{\tilde{O}(\sqrt{n})}$. We also study graph secret-sharing schemes. In these schemes, the parties are vertices of a graph and a set can reconstruct the secret if and only if it contains an edge. Again, for this family there is a huge gap between the upper bounds - $O(n/\log n)$ (Erdös and Pyber, Discrete Mathematics 1997) - and the lower bounds - $\Omega(\log n)$ (van Dijk, Des. Codes Crypto. 1995). We show that for almost all graphs, the share size of each party is $n^{o(1)}$. This result is achieved by using robust 2-server conditional disclosure of secrets protocols, a new primitive introduced and constructed in (Applebaum et al., STOC 2020), and the fact that the size of the maximal independent set in a random graph is small. Finally, using robust conditional disclosure of secrets protocols, we improve the total share size for all very dense graphs