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arXiv:2312.05737

Multiple Threshold Schemes Under The Weak Secure Condition

Jiahong Wu, Nan Liu, Kang, Wei

Published Dec 10, 2023• 0 citations
arXivHTMLPDF

Abstract

In this paper, we consider the case that sharing many secrets among a set of participants using the threshold schemes. All secrets are assumed to be statistically independent and the weak secure condition is focused on. Under such circumstances we investigate the infimum of the (average) information ratio and the (average) randomness ratio for any structure pair which consists of the number of the participants and the threshold values of all secrets. For two structure pairs such that the two numbers of the participants are the same and the two arrays of threshold values have the subset relationship, two leading corollaries are proved following two directions. More specifically, the bound related to the lengths of shares, secrets and randomness for the complex structure pair can be truncated for the simple one; and the linear schemes for the simple structure pair can be combined independently to be a multiple threshold scheme for the complex one. The former corollary is useful for the converse part and the latter one is helpful for the achievability part. Three new bounds special for the case that the number of secrets corresponding to the same threshold value $ t $ is lager than $ t $ and two novel linear schemes modified from the Vandermonde matrix for two similar cases are presented. Then come the optimal results for the average information ratio, the average randomness ratio and the randomness ratio. We introduce a tiny example to show that there exists another type of bound that may be crucial for the information ratio, to which we only give optimal results in three cases.

DOI
10.48550/arxiv.2312.05737

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In this paper, we consider the case that sharing many secrets among a set of participants using the threshold schemes. All secrets are assumed to be statistically independent and the weak secure condition is focused on. Under such circumstances we investigate the infimum of the (average) information ratio and the (average) randomness ratio for any structure pair which consists of the number of the participants and the threshold values of all secrets. For two structure pairs such that the two numbers of the participants are the same and the two arrays of threshold values have the subset relationship, two leading corollaries are proved following two directions. More specifically, the bound related to the lengths of shares, secrets and randomness for the complex structure pair can be truncated for the simple one; and the linear schemes for the simple structure pair can be combined independently to be a multiple threshold scheme for the complex one. The former corollary is useful for the converse part and the latter one is helpful for the achievability part. Three new bounds special for the case that the number of secrets corresponding to the same threshold value ttt is lager than ttt and two novel linear schemes modified from the Vandermonde matrix for two similar cases are presented. Then come the optimal results for the average information ratio, the average randomness ratio and the randomness ratio. We introduce a tiny example to show that there exists another type of bound that may be crucial for the information ratio, to which we only give optimal results in three cases.
A perfect secret-sharing scheme is a method to distribute a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the joint share of participants in any unqualified subset is statistically independent of the secret. The collection of all qualified subsets is called the access structure of the perfect secret-sharing scheme. In a graph-based access structure, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme realizing the access structure based on G is defined as AR = (Pv2V (G) H(v))/(|V (G)|H(s)), where s is the secret and v is the share of v, both are random variables from and H is the Shannon entropy. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing a given access structure is called the optimal average information ratio of that access structure. Most known results about the optimal average information ratio give upper bounds or lower bounds on it. In this present structures based on bipartite graphs and determine the exact values of the optimal average information ratio of some infinite classes of them.
In a perfect secret-sharing scheme, a dealer distributes a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret and the joint share of the participants in any unqualified subset is statistically independent of the secret. The access structure of the scheme refers to the collection of all qualified subsets. In a graph-based access structures, each vertex of a graph G represents a participant and each edge of G represents a minimal qualified subset. The average information ratio of a perfect secret-sharing scheme realizing a given access structure is the ratio of the average length of the shares given to the participants to the length of the secret. The infimum of the average information ratio of all possible perfect secret-sharing schemes realizing an access structure is called the optimal average information ratio of that access structure. We study the optimal average information ratio of the access structures based on bipartite graphs. Based on some previous results, we give a bound on the optimal average information ratio for all bipartite graphs of girth at least six. This bound is the best possible for some classes of bipartite graphs using our approach.
Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.
Abstract The efficiency of a secret sharing scheme is measured via its information ratio. The optimal information ratio (OIR) of an access structure is the infimum of the information ratios of all secret sharing schemes realizing it. When the infimum is taken over all linear secret sharing schemes, it is called the optimal linear information ratio (OLIR). We review the problem of finding the OIRs and OLIRs of graph access structures on six participants. Study of such access structures was initiated by van Dijk et al. (1998) [43] . In a sequence of works, the OIRs of nine access structures, out of the 18 initially unsolved non-isomorphic ones, were determined. Very recently, the known lower bounds on the OLIRs of the other nine cases have been improved (Farras et al., 2018 [20] ), determining the OLIRs for another two cases. In this paper, for each of the remaining seven cases, we provide a new upper bound on the OIR which matches that of the recently found lower bound on the OLIR. Improved upper bounds are achieved by constructing a linear secret sharing scheme, using a suitable decomposition technique for each access structure. A new decomposition technique, called ( λ , ω ) -weighted decomposition, is also introduced which is a generalization of all known decomposition methods.
: 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...
We consider the class of secret sharing schemes where there is no a priori bound on the number of players n but where each of the n share-spaces has fixed cardinality q. We show two fundamental lower bounds on the threshold gap of such schemes. The threshold gap g is defined as r-t, where r is minimal and t is maximal such that the following holds: for a secret with arbitrary a priori distribution, each r-subset of players can reconstruct this secret from their joint shares without error ( r-reconstruction) and the information gain about the secret is nil for each t-subset of players jointly ( t-privacy). Our first bound, which is completely general, implies that if , then g ≥ [( n-t+1)/q] independently of the cardinality of the secret-space. Our second bound pertains to \BBF q-linear schemes with secret-space \BBF qk( k ≥ 2). It improves the first bound when k is large enough. Concretely, it implies that g ≥ [( n-t+1)/ q]+f(q,k,t,n), for some function f that is strictly positive when k is large enough. Moreover, also in the \BBF q-linear case, bounds on the threshold gap independent of t or r are obtained by additionally employing a dualization argument. As an application of our results, we answer an open question about the asymptotics of arithmetic secret sharing schemes and prove that the asymptotic optimal corruption tolerance rate is strictly smaller than 1.