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arXiv:1909.13663 · Journal of Mathematical Cryptology

Secret sharing and duality

László Csirmaz

Published Nov 25, 2020• 1 citation
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Abstract

Abstract Secret sharing is an important building block in cryptography. All explicit secret sharing schemes which are known to have optimal complexity are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give a partial answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual.

DOI
10.1515/jmc-2019-0045

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Secret sharing is an important building block in cryptography. All explicit secret sharing schemes which are known to have optimal complexity are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give a partial answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual.

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IEEE Transactions on Information Theory•May 29, 2008•15 citations
Abstract Secret sharing is an important building block in cryptography. All explicit secret sharing schemes which are known to have optimal complexity are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give a partial answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual.
Secret sharing is an important building block in cryptography. All explicit secret sharing schemes which are known to have optimal complexity are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give a partial answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual.
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.
Secret sharing schemes are a fundamental cryptographic tool for securely distributing sensitive information among multiple parties according to a specified access structure. For a given access policy, several schemes with different levels of efficiency may exist. One of the main parameters in this analysis is the share size and a central goal is to minimize it. This thesis studies the complexity of secret sharing schemes by focusing on lower and upper bounds on the share size for arbitrary and explicit access structures. The main objective is to collect the most relevant results in the literature, ranging from early lower bounds due to Karnin et al. and Csirmaz and upper-bound of Liu and Vaikuntanathan to recent advances that have significantly improved known bounds. Despite this progress, a significant gap between lower and upper bounds remains, leaving the determination of the optimal share size an open problem.
For more than two decades, proving or refuting the following statement has remained a challenging open problem in the theory of secret sharing schemes (SSSs): every ideal access structure admits an ideal perfect multi-linear SSS. The class of group-characterizable (GC) SSSs include the multi-linear ones. Hence, if the above statement is true, then so is the following weaker statement: every ideal access structure admits an ideal perfect GC SSS. One contribution of this paper is to show that ideal SSSs are not necessarily GC. Our second contribution is to study the above two statements with respect to several variations of weakly-ideal access structures. Recently, Mejia and Montoya studied ideal access structures that admit ideal multi-linear schemes and provided a classification-like theorem for them. We additionally present some tools that are useful to extend their result.
The following standard relaxations of perfect security for secret sharing schemes (SSSs) exist in the literature:quasi-perfect,almost-perfect, andstatistical. Understanding the power of these relaxations on the efficiency of SSSs, measured via a parameter calledinformation ratio, is a long-standing open problem. In this article, we introduce and study an extremely relaxed security notion, calledpartial security, for which it is only required that any qualified set gains strictly more information about the secret than any unqualified one. To get a meaningful efficiency measure, we normalize the (standard) information ratio of such schemes by an appropriate parameter and refer to the new measure aspartial information ratio. We present three main results in this paper.First, we prove that partial and perfect information ratios coincide for the class of linear SSSs.Second, we prove that for the general (i.e., non-linear) class of SSSs, partial and statistical information ratios are equal.Third, we show that partial and almost-perfect information ratios do not coincide for the class of mixed-linear schemes (i.e., schemes constructed by combining linear schemes with different underlying finite fields). We also use the notion of partial secret sharing to strengthen and unify the previousdecompositiontheorems for constructing SSSs.
Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (Journal of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal-the access structure must be induced by a matroid. Seymour (Journal of Combinatorial Theory B, 1992) has proved that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. The research on access structures induced by matroids is continued in this work. The main result in this paper is strengthening the result of Seymour. It is shown that in any secret-sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Omega(radic(k)). The second result considers nonideal secret-sharing schemes realizing access structures induced by matroids. It is proved that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (as long as the shares are still relatively short). This generalized results of Brickell and Davenport for ideal schemes. Finally, an example of a nonideal access structure that is nearly ideal is presented.