László Csirmaz

The Size of a Share Must Be Large

Abstract. A secret sharing scheme permits a secret to be shared among participants of an n-element group in such a way that only qualified subsets of participants can recover the secret. If any nonqualified subset has absolutely no information on the secret, then the scheme is called perfect. The share in a scheme is the information that a participant must remember. In [3] it was proved that for a certain access structure any perfect secret sharing scheme must give some participant a share which is at least 50\percent larger than the secret size. We prove that for each n there exists an access structure on n participants so that any perfect sharing scheme must give some participant a share which is at least about

The size of a share must be large

n/\log n

An impossibility result on graph secret sharing

times the secret size.^1 We also show that the best possible result achievable by the information-theoretic method used here is n times the secret size. ^1 All logarithms in this paper are of base 2.

Entropy Region and Convolution

A secret sharing scheme permits a secret to be shared among participants of an n-element group in such a way that only qualified subsets of participants can recover the secret. If any non-qualified subset has absolutely no information on the secret, then the scheme is called perfect. The share in a scheme is the information what a participant must remember. We prove that for each n there exists an access structure on n participants so that any perfect sharing scheme must give some participant a share which is at least about n/log n times the secret size. We also show that the best possible result achievable by the information theoretic method used here is n times the secret size.

Using multiobjective optimization to map the entropy region

A perfect secret sharing scheme based on a graph G is a randomized distribution of a secret among the vertices of the graph so that the secret can be recovered from the information assigned to vertices at the endpoints of any edge, while the total information assigned to an independent set of vertices is independent (in statistical sense) of the secret itself. The efficiency of a scheme is measured by the amount of information the most heavily loaded vertex receives divided by the amount of information in the secret itself. The (worst case) information ratio of G is the infimum of this number. We calculate the best lower bound on the information ratio for an infinite family of graphs the entropy method can give. We argue that almost all existing constructions for secret sharing schemes are special cases of the generalized vector space construction. We give direct constructions of this type for the first two members of the family, and show that for the other members no such construction exists which would match the bound yielded by the entropy method.

Secret sharing on the d-dimensional cube

The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under self-adhesivity is clarified. Results are specialized and extended to the region constructed from four tuples of random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. The four-atom conjecture on the minimal Ingleton score is refuted.

Optimal Information Rate of Secret Sharing Schemes on Trees

Mapping the structure of the entropy region of at least four jointly distributed random variables is an important open problem. Even partial knowledge about this region has far reaching consequences in other areas in mathematics, like information theory, cryptography, probability theory and combinatorics. Presently, the only known method of exploring the entropy region is, or equivalent to, the one of Zhang and Yeung from 1998. Using some non-trivial properties of the entropy function, their method is transformed to solving high dimensional linear multiobjective optimization problems. Benson’s outer approximation algorithm is a fundamental tool for solving such optimization problems. An improved version of Benson’s algorithm is presented, which requires solving one scalar linear program in each iteration rather than two or three as in previous versions. During the algorithm design, special care is taken for numerical stability. The implemented algorithm is used to verify previous statements about the entropy region, as well as to explore it further. Experimental results demonstrate the viability of the improved Benson’s algorithm for determining the extremal set of medium-sized numerically ill-posed optimization problems. With larger problem sizes, two limitations of Benson’s algorithm is observed: the inefficiency of the scalar LP solver, and the unexpectedly large number of intermediate vertices.

On the strength of sometimes and always in program verification

We prove that for

On-line secret sharing

d>1