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IEEE Transactions on Information Theory

Coding Theorems on the Threshold Scheme for a General Source

Hiroki Koga

Published May 29, 2008• 9 citations
Publisher

Abstract

In this paper, coding theorems on the (t, m) -threshold scheme for a general source are discussed, where m means the number of the shares and t means a threshold. The (t,m) -threshold scheme treated in this paper encrypts n source outputs Xnto m shares at once and is required to satisfy the two conditions that 1) Xnis reproduced from arbitrary t shares, and 2) almost no information of Xnis revealed from any t - 1 shares. It is shown that the (t,m) -threshold scheme must satisfy certain inequalities including the limit inferiors in probability. One of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the (t, m) -threshold scheme. In addition, it is shown that a certain variation of Shamir's threshold scheme meets the two conditions. The same approach can be taken to the problems of Shannon's cipher system with the perfect secrecy and fixed-length source coding with vanishing decoding error probability. It is shown that the same kind of inequalities, which indicate the converse coding theorems, hold in both two cases.

DOI
10.1109/tit.2008.921860

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In this paper, coding theorems on the (t, m) -threshold scheme for a general source are discussed, where m means the number of the shares and t means a threshold. The (t,m) -threshold scheme treated in this paper encrypts n source outputs Xnto m shares at once and is required to satisfy the two conditions that 1) Xnis reproduced from arbitrary t shares, and 2) almost no information of Xnis revealed from any t - 1 shares. It is shown that the (t,m) -threshold scheme must satisfy certain inequalities including the limit inferiors in probability. One of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the (t, m) -threshold scheme. In addition, it is shown that a certain variation of Shamir's threshold scheme meets the two conditions. The same approach can be taken to the problems of Shannon's cipher system with the perfect secrecy and fixed-length source coding with vanishing decoding error probability. It is shown that the same kind of inequalities, which indicate the converse coding theorems, hold in both two cases.
This paper is concerned with basic properties of the (t, m)-threshold scheme, where m denotes the number of shares and t a threshold. We are interested in the case where a secret information is generated from a source with no assumption on its probabilistic structure. We first give conditions that the (t, m)-threshold scheme must satisfy. Then, we give a theorem which is related to lower bounds on the sizes of shares and the length of random bits needed to a dealer. In addition, we show that the (t, m)-threshold scheme is realized under a certain mild assumption on the source
In this paper, we propose a framework of source encryption, where cryptographic processing is applied to a prescribed fixed length source code. The proposed source encryption framework is based on the secure communication framework of the Shannon cipher system. In the proposed framework, we use the mutual information as a measure of information leakage to an adversary. For the proposed framework, we explicitly establish the necessary and sufficient condition for reliable and secure communication under the condition that error probability and information leakage, respectively, are upper bounded by prescribed constants ε∈(0,1)ε\in (0,1)ε∈(0,1) and δ∈(0,∞)δ\in (0,\infty)δ∈(0,∞). We also show that the obtained necessary and sufficient condition does not depend on the constants ε∈(0,1)ε\in (0,1)ε∈(0,1) and δ∈(0,∞)δ\in (0,\infty)δ∈(0,∞), demonstrating that we have the strong converse theorem for the proposed framework of source encryption. We further prove the existence of encryption/decryption schemes, which are universal in the sense that they work effectively for any distributions of the plain text and those of the key used for the encryption.
It is known that a secret sharing scheme (SSS) with perfect cheating detection cannot be realized because such a SSS requires infinite share rates. However, this impossibility comes from the fact that block coding is not used and any decoding error is not allowed in the SSS. Hence, in this paper, we consider a SSS constructed by block coding with an arbitrarily small decoding error probability. It is shown that the perfect cheating detection with finite rates is possible for the 2-out-of-2 SSS in a certain asymptotic sense. Furthermore, the supremum of the achievable exponent in the maximum success probability of impersonation attack turns out to be the mutual information between the two shares.
We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.
Shamir's (n, k) threshold secret sharing is an important component of several cryptographic protocols, such as those for secure multiparty-computation. These protocols typically assume the presence of direct communication links from the dealer to all participants, in which case the dealer can directly pass the shares of the secret to every participant. In this paper, we consider the problem of secret sharing when the dealer does not have direct communication links to all participants, and instead, they form a general network. We present an algorithm for secret sharing over networks that satisfy what we call the k-propagating-dealer condition. The algorithm is communication-efficient, distributed and deterministic. Interestingly, the solution constitutes an instance of a network coding problem admitting a distributed and deterministic solution, and furthermore, handles the case of nodal-eavesdropping, about which very little appears to be known in the literature. In the second part of the paper, we derive information-theoretic lower bounds on the communication complexity of secret sharing over any network, which may also be of independent interest. We show that for networks satisfying the k-propagating-dealer condition, the communication complexity of our algorithm is Θ(n), and furthermore, is always within a constant factor of the lower bound. We also show that, in contrast, existing solutions in the literature entail a communication-complexity that is superlinear for a wide class of networks, and is Θ(n2) in the worst case. Our algorithm thus allows for efficient generalization of several cryptographic protocols to a large class of networks.
Suppose we are given a random source and want to use it as a random number generator; at what rate can we generate fair bits from it? We address this question in an information-theoretic setting by allowing for some arbitrarily small but nonzero deviation from "ideal" random bits. We prove our results with three different measures of approximation between the ideal and the obtained probability distributions: the variational distance, the d-bar distance, and the normalized divergence. Two different contexts are studied: fixed-length and variable-length random number generation. The fixed-length results of this paper provide an operational characterization of the inf-entropy rate of a source, defined in Han and Verdu (see ibid., vol.39, no.3, p.752-772, 1993) and the variable-length results characterize the liminf of the entropy rate, thereby establishing a pleasing duality with the fundamental limits of source coding. A feature of our results is that we do not restrict ourselves to ergodic or to stationary sources.>