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World Congress on Software Engineering (WCSE)

The Existence of Multi-secret Sharing Schemes Based on Random Walks on Graphs

Xiaomin Zha, Yongxuan Sang, Jiwen Zen, Jun Xu

Published Jan 1, 2009• 0 citations
Publisher

Abstract

A multi-secret sharing scheme is the extension of a secret sharing scheme to the case in which there are many secrets need to be shared, and each secret may with different qualified subsets of participants. We study multi-secret sharing schemes based on random walks on graphs. Since many problems of secret sharing can be seen as having structure of the graphs. Different access structures corresponding to different walks for a graph with n vertices can be defined. So considering multi-secret sharing scheme based on random walks on graphs is meaningful. We prove that there exists multi-secret sharing schemes based on random walks on graphs by studying monotone span programs computing monotone Boolean function and secret sharing schemes realizing these classes of access structure.

DOI
10.1109/wcse.2009.294

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A multi-secret sharing scheme is the extension of a secret sharing scheme to the case in which there are many secrets need to be shared, and each secret may with different qualified subsets of participants. We study multi-secret sharing schemes based on random walks on graphs. Since many problems of secret sharing can be seen as having structure of the graphs. Different access structures corresponding to different walks for a graph with n vertices can be defined. So considering multi-secret sharing scheme based on random walks on graphs is meaningful. We prove that there exists multi-secret sharing schemes based on random walks on graphs by studying monotone span programs computing monotone Boolean function and secret sharing schemes realizing these classes of access structure.
: A multi--secret sharing scheme is a protocol to share a number of (arbitrarily related) secrets among a set of participants in such a way that only qualified sets of participants can recover the secrets, whereas non-qualified sets of participants might have partial information about them. In this paper we analyze the amount of randomness needed by multi--secret sharing schemes. Given an m-tuple of access structures, we give a lower bound on the number of random bits needed by multi--secret sharing schemes; the lower bound is expressed in terms of a combinatorial parameter that depends only upon the access structures and not on the particular multi--secret sharing scheme used. Key Words: Data Security, Cryptography, Randomness, Secret Sharing Schemes. Category: E.3 1 Introduction There are many situations in cryptography in which it is important to be able to generate random numbers, random bit strings, etc. For example, cryptographic keys are to be generated at random from a speci...
With respect to a special class of access structures based on connectivity of graphs, we start from a linear secret sharing scheme and turn it into a secret sharing scheme with perfect security and exponentially small error probability by randomizing the reconstruction algorithm through random walks on graphs. It reduces the polynomial work space to logarithmic. Then we build the corresponding statistical multiparty computation protocol by using the secret sharing scheme. The results of this paper also imply the inherent connections and influences among secret sharing, randomized algorithms, and secure multiparty computation.
The study of probabilistic secret sharing schemes using arbitrary probability spaces and possibly infinite number of participants lets us investigate abstract properties of such schemes. It highlights important properties, explains why certain definitions work better than others, connects this topic to other branches of mathematics, and might yield new design paradigms. A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, we define four scheme types: perfect, almost perfect, ramp, and almost ramp. Our main results characterize the access structures which can be realized by schemes of these types. We show that every access structure can be realized by a non-measurable perfect probabilistic scheme. The construction is based on a paradoxical pair of independent random variables which determine each other. For measurable schemes we have the following complete characterization. An access structure can be realized by a (measurable) perfect, or almost perfect scheme if and only if the access structure, as a subset of the Sierpiński space {0, 1} P, is open, if and only if it can be realized by a span program. The access structure can be realized by a (measurable) ramp or almost ramp scheme if and only if the access structure is a Gδ set (intersection of countably many open sets) in the Sierpiński topology, if and only if it can be realized by a Hilbert-space program.
A secret-sharing scheme realizes the forbidden graph access structure determined by a graphG=(V,E)G=(V,E)G=(V,E)if the parties are the vertices of the graph and the subsets that can reconstruct the secret are the pairs of vertices inEEE(i.e., the edges) and the subsets of at least three vertices. Secret-sharing schemes for forbidden graph access structures defined by bipartite graphs are equivalent to conditional disclosure of secrets (CDS) protocols. We study the complexity of realizing a forbidden graph access structure by linear secret-sharing schemes, which are schemes in which the secret can be reconstructed from the shares by a linear mapping. We provide efficient constructions and lower bounds on the share size of linear secret-sharing schemes for sparse and very dense graphs, closing the gap between upper and lower bounds. Given a sparse (resp. very dense) graph withnnnvertices and at mostn1+βn^{1+\beta }n1+βedges (resp. at least(n2)−n1+β\binom {n}{2} - n^{1+\beta }(2n​)−n1+βedges), for some0≤β<10 \leq \beta < 10≤β<1, we construct a linear secret-sharing scheme realizing its forbidden graph access structure with total share sizeO~(n1+β/2)\tilde {O} (n^{1+\beta /2})O~(n1+β/2). Furthermore, we construct linear secret-sharing schemes realizing these access structures in which the size of each share isO~(n1/4+β/4)\tilde {O} (n^{1/4+\beta /4})O~(n1/4+β/4). We also provide constructions achieving different trade-offs between the size of each share and the total share size. We prove that almost all forbidden graph access structures require linear secret-sharing schemes with total share sizeΩ(n3/2)\Omega (n^{3/2})Ω(n3/2); this shows that the construction of Gay, Kerenidis, and Wee [CRYPTO 2015] is optimal. Furthermore, we show that for every0≤β<10 \leq \beta < 10≤β<1there exist a graph with at mostn1+βn^{1+\beta }n1+βedges and a graph with at least(n2)−n1+β\binom {n}{2}-n^{1+\beta }(2n​)−n1+βedges such that the total share size in any linear secret-sharing scheme realizing the associated forbidden graph access structures isΩ(n1+β/2)\Omega (n^{1+\beta /2})Ω(n1+β/2). Finally, we show that for every0≤β<10 \leq \beta < 10≤β<1there exist a graph with at mostn1+βn^{1+\beta }n1+βedges and a graph with at least(n2)−n1+β\binom {n}{2}-n^{1+\beta }(2n​)−n1+βedges such that the size of the share of at least one party in any linear secret-sharing scheme realizing these forbidden graph access structures isΩ(n1/4+β/4)\Omega (n^{1/4+\beta /4})Ω(n1/4+β/4). This shows that our constructions are optimal (up to poly-logarithmic factors).
Abstract How to efficiently share secrets among multiple participants is a very important problem in key management. In this paper, we propose a multi-secret sharing scheme based on the Greenberger Horne Zeilinger (GHZ) state. First, the distributor uses monotone span program to encode the secrets and generate the corresponding secret shares to send to the participants. Then, each participant uses the generalized Pauli operator to embed its own secret share into the transmitted particle. The participant who wants to get the secrets can get multiple secrets at the same time by performing a GHZ-state joint measurement. Furthermore, since this scheme is based on a monotone span program, its access structure is more general than the access structure (t, n) threshold. Compared with other schemes, our proposed scheme is more efficient, less computational cost.