Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Transactions on Information Theory

New bounds on the information rate of secret sharing schemes

Carlo Blundo, Alfredo De Santis, Antonio Giorgio Gaggia, Ugo Vaccaro

Published Mar 1, 1995• 30 citations
Publisher

Abstract

/spl acute/A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of participants can recover the secret, but any nonqualified subset has absolutely no information on the secret. We derive new limitations on the information rate of secret sharing schemes, that measures how much information is being distributed as shares as compared to the size of the secret key, and the average information rate, that is the ratio between the secret size and the arithmetic mean of the size of the shares. By applying the substitution technique, we are able to construct many new examples of access structures where the information rate is bounded away from 1. The substitution technique is a method used to obtain a new access structure by replacing a participant in a previous structure with a new access structure.>

DOI
10.1109/18.370169

Related papers

A New Bound on the Average Information Ratio of Perfect Secret-Sharing Schemes for Access Structures Based on Bipartite Graphs of Larger Girth

Hui-Chuan Lu

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.

•Jul 5, 2014•0 citations

A Study On The Average Information Ratio Of Perfect Secret-Sharing Schemes For Access Structures Based On Bipartite Graphs

Hui-Chuan Lu

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.

•Sep 29, 2012•1 citation

Symmetries in Linear Programming for Information Inequalities

Emirhan Gürpınar

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.

arXiv:2201.11656•IEEE International Symposium on Information Theory (ISIT)•Jun 26, 2022•1 citation

On the Size of a Share for Non-perfect Secret Sharing Schemes

Maki Yoshida

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 $\frac{{{2^k} - 1}}{k} \cdot H(s)$, 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 $\frac{1}{L}$ fraction of the secret's entropy, for an integer L > 1. A prior result in this setting provides a lower bound of $\frac{{{2^k} - 1}}{{kL}} \cdot H(s)$, under the condition n ≥ 2k+k+L−3. We provide a new lower bound that approaches the perfect-case bound, namely $\frac{{{2^k} - 2 + 1/L}}{k} \cdot H(s)$, 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.

IEEE Information Theory Workshop (ITW)•Sep 29, 2025•0 citations

ON THE DEALER'S RANDOMNESS REQUIRED IN PERFECT SECRET SHARING SCHEMES WITH ACCESS STRUCTURES OF CONSTANT RANK

Hung–Min Sun

A secret sharing scheme is a method which allows a dealer to share a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret. The collection of subsets of participants that can reconstruct the secret in this way is called access structure. The rank of an access structure is the maximum cardinality of a minimal qualified subset. A secret sharing scheme is perfect if unqualified subsets of participants obtain no information regarding the secret. The dealer's randomness is the number of random bits required by the dealer to setup a secret sharing scheme. The efficiency of the dealer's randomness is the ratio between the amount of the dealer's randomness and the length of the secret. Because random bits are a natural computational resource, it is important to reduce the amount of randomness used by the dealer to setup a secret sharing scheme. In this paper, we propose some decomposition constructions for perfect secret sharing schemes with access structures of constant rank. Compared with the best previous results, our constructions have some improved upper bounds on the dealer's randomness and on the efficiency of the dealer's randomness.

International Journal of Foundations of Computer Science•Jun 1, 2000•0 citations

NEW METHOD TO COMPUTE BOUNDS ON THE OPTIMAL INFORMATION RATE OF SECRET SHARING SCHEMES

Abbas Cheraghi

In a bipartite secret sharing scheme, the set of participants is divided into two parts, and all participants in each part play an equivalent role. The ideal bipartite access structures were characterized by Padr´o and S´aez, but it is not known which is the optimal information rate of non ideal bipartite access structures. Determining the optimal information rate of general access structures is one of the major problems in secret sharing. We study this open problem restricted to the bipartite access structures, obtaining a new method to compute bounds on the optimal information rate. Namely, by using the connection between secret sharing schemes and polymatroids, we show a linear programming problem that determines, for each access structure, an upper bound on the information rate. Moreover, we show new optimal constructions for certain bipartite access structures.

Facta Universitatis Series Mathematics and Informatics•Nov 21, 2025•0 citations
/spl acute/A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of participants can recover the secret, but any nonqualified subset has absolutely no information on the secret. We derive new limitations on the information rate of secret sharing schemes, that measures how much information is being distributed as shares as compared to the size of the secret key, and the average information rate, that is the ratio between the secret size and the arithmetic mean of the size of the shares. By applying the substitution technique, we are able to construct many new examples of access structures where the information rate is bounded away from 1. The substitution technique is a method used to obtain a new access structure by replacing a participant in a previous structure with a new access structure.>
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.
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.
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.
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.
A secret sharing scheme is a method which allows a dealer to share a secret among a set of participants in such a way that only qualified subsets of participants can recover the secret. The collection of subsets of participants that can reconstruct the secret in this way is called access structure. The rank of an access structure is the maximum cardinality of a minimal qualified subset. A secret sharing scheme is perfect if unqualified subsets of participants obtain no information regarding the secret. The dealer's randomness is the number of random bits required by the dealer to setup a secret sharing scheme. The efficiency of the dealer's randomness is the ratio between the amount of the dealer's randomness and the length of the secret. Because random bits are a natural computational resource, it is important to reduce the amount of randomness used by the dealer to setup a secret sharing scheme. In this paper, we propose some decomposition constructions for perfect secret sharing schemes with access structures of constant rank. Compared with the best previous results, our constructions have some improved upper bounds on the dealer's randomness and on the efficiency of the dealer's randomness.
In a bipartite secret sharing scheme, the set of participants is divided into two parts, and all participants in each part play an equivalent role. The ideal bipartite access structures were characterized by Padr´o and S´aez, but it is not known which is the optimal information rate of non ideal bipartite access structures. Determining the optimal information rate of general access structures is one of the major problems in secret sharing. We study this open problem restricted to the bipartite access structures, obtaining a new method to compute bounds on the optimal information rate. Namely, by using the connection between secret sharing schemes and polymatroids, we show a linear programming problem that determines, for each access structure, an upper bound on the information rate. Moreover, we show new optimal constructions for certain bipartite access structures.