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IEEE Transactions on Information Theory

Secret Sharing, Rank Inequalities, and Information Inequalities

Sebastià Martı́n, Carles Padró, Yang An

Published Nov 12, 2015• 16 citations
Publisher

Abstract

Beimel and Orlov proved that all information inequalities on four or five variables, together with all information inequalities on more than five variables that are known to date, provide lower bounds on the size of the shares in secret sharing schemes that are at most linear on the number of participants. We present here another two negative results about the power of information inequalities in the search for lower bounds in secret sharing. First, we prove that all information inequalities on a bounded number of variables can only provide lower bounds that are polynomial on the number of participants. Second, we prove that the rank inequalities that are derived from the existence of two common informations can provide only lower bounds that are at most cubic in the number of participants.

DOI
10.1109/tit.2015.2500232

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Beimel and Orlov proved that all information inequalities on four or five variables, together with all information inequalities on more than five variables that are known to date, provide lower bounds on the size of the shares in secret sharing schemes that are at most linear on the number of participants. We present here another two negative results about the power of information inequalities in the search for lower bounds in secret sharing. First, we prove that all information inequalities on a bounded number of variables can only provide lower bounds that are polynomial on the number of participants. Second, we prove that the rank inequalities that are derived from the existence of two common informations can provide only lower bounds that are at most cubic in the number of participants.
The known secret-sharing schemes for most access structures are not efficient; even for a one-bit secret the length of the shares in the schemes is 2O(n), wherenis the number of participants in the access structure. It is a long standing open problem to improve these schemes or prove that they cannot be improved. The best known lower bound is by Csirmaz, who proved that there exist access structures withnparticipants such that the size of the share of at least one party isn/logntimes the secret size. Csirmaz's proof uses Shannon information inequalities, which were the only information inequalities known when Csirmaz published his result. On the negative side, Csirmaz proved that by only using Shannon information inequalities one cannot prove a lower bound of ω(n) on the share size. In the last decade, a sequence of non-Shannon information inequalities were discovered. In fact, it was proved that there are infinity many independent information inequalities even in four variables. This raises the hope that these inequalities can help in improving the lower bounds beyondn. However, we show that any information inequality with four or five variables cannot prove a lower bound of ω(n) on the share size. In addition, we show that the same negative result holds for all information inequalities with more than five variables that are known to date.
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.
There is a huge gap between the upper and lower bounds on the share size of secret-sharing schemes fornnn-party access structures; consistent with our current knowledge the optimal share size can be anywhere between polynomial and exponential innnn. For linear secret-sharing schemes, the share size for almost allnnn-party access structures is exponential innnn. We would like to study larger classes of secret-sharing schemes with two goals: 1) prove lower bounds for larger classes of secret-sharing schemes; and 2) construct efficient secret-sharing schemes. Given this motivation, Paskin-Cherniavsky and Radune (ITC’20) introduced a new class of secret-sharing schemes in which the shares are generated by applying degree-dddpolynomials to the secret and some random field elements. We define and study two additional classes of polynomial secret-sharing schemes: 1) schemes in which the reconstruction of the secret is done using polynomials; and 2) schemes in which both sharing and reconstruction are done by polynomials. Our main result is a construction of secret-sharing schemes and conditional disclosure of secrets protocols with quadratic sharing and reconstruction that are more efficient than linear secret-sharing schemes. To complement our results, we prove lower bounds on the share size for schemes with polynomial reconstruction. Finally, we give an evidence that schemes with polynomial sharing are probably stronger than schemes with polynomial reconstruction.
Secret sharing is a central topic in information-theoretical cryptography. In a secret sharing scheme a random secret is distributed among participants so that only the qualified subsets of parties can recover the secret. In this paper we improve previously known [13] lower bounds on the size of the largest share for several specific access structures. Our approach is based on non-Shannon-type information inequalities. We employ the linear programming (LP) technique that permits to apply new information inequalities indirectly. To reduce computational complexity of the involved linear programs we extensively use symmetry considerations.
We present a new improvement in the linear programming technique to derive lower bounds on the information ratio of secret sharing schemes. We obtain non-Shannon-type bounds without using information inequalities explicitly. Our new technique makes it possible to determine the optimal information ratio of linear secret sharing schemes for all access structures on 5 participants and all graph-based access structures on 6 participants. In addition, new lower bounds are presented also for some small matroid ports and, in particular, the optimal information ratios of the linear secret sharing schemes for the ports of the Vamos matroid are determined.
In a secret sharing scheme, a secret s is divided into multiple shares and distributed among n participants. As more shares are collected, the mutual information between the secret and the shares increases monotonically. In the perfect case, the best known explicit lower bound on the share size is given by 2k−1k⋅H(s)\frac{{{2^k} - 1}}{k} \cdot H(s)k2k−1​⋅H(s), where H(s) is the entropy of the secret and k is the largest integer such that n ≥ 2k+k−2. This paper considers the non-perfect case, in which each share reveals at most a 1L\frac{1}{L}L1​ fraction of the secret's entropy, for an integer L > 1. A prior result in this setting provides a lower bound of 2k−1kL⋅H(s)\frac{{{2^k} - 1}}{{kL}} \cdot H(s)kL2k−1​⋅H(s), under the condition n ≥ 2k+k+L−3. We provide a new lower bound that approaches the perfect-case bound, namely 2k−2+1/Lk⋅H(s)\frac{{{2^k} - 2 + 1/L}}{k} \cdot H(s)k2k−2+1/L​⋅H(s), which holds under the condition n ≥ 2k+k+2L−4. The proof generalizes perfect-case techniques by increasing the number of participants by 2(L−1). These results refine our understanding of the trade-off between share size and information leakage in non-perfect secret sharing. In particular, we construct non-perfect access structures that nearly attain the perfect lower bound with fewer participants than previous non-perfect bounds, thus complementing and extending prior work.