Title: A Study on (t, n) Threshold Secret Sharing Schemes Based on the Generalized Vector Space Construction
Abstract: A secret sharing scheme is a technique of sharing a secret s into n pieces, called shares, and distributing them to a set of users P , {P1, .., Pn} in such a way that only certain qualified subsets of users can recover the secret by combining their shares. As a special class of secret sharing schemes, threshold secret sharing schemes were introduced. A secret sharing scheme is called a (t, n) threshold secret sharing scheme if knowledge of any t or more shares makes the secret s computable and the knowledge of any t − 1 or fewer shares leaves s completely undetermined (in the sense that all its possible values are equally likely). Threshold secret sharing schemes are an important tool in the information security. In many of the real applications of the threshold secret sharing schemes it is important to be able to share the secret among a large number of users. In this thesis we consider the scenario that the secret information is assumed to be very important but can be represented as a symbol and there are needs to be shared among a really large number of users. Therefore we focus on (t, n) threshold secret sharing schemes such that the number of users n is to be increased as much as possible under the condition that a a secret s is chosen from a finite field with q elements, where q is a prime power. In fact, a secret of one bit would be very important in many applications in the real file where the secret might be the ”yes” or ”no” answer to a really important question (e. g. stock market, a company keeps excellent condition or not, and so on). The efficiency of a given secret sharing scheme is measured by its information rate ρ, which is the ratio between the size of the secret and the size of the shares. It has been proven that for perfect secret sharing schemes 0 ≤ ρ ≤ 1. Depending on their information rate, secret sharing schemes can be divided into two classes of ideal and non-ideal secret sharing schemes. A secret sharing scheme is called ideal if it is perfect and has information rate ρ = 1 and non-ideal if it has information rate 0 ≤ ρ 3. The power k is a function of the size of the shares for each user and if an appropriate value of k has been chosen then (2, n) and (3, n) threshold secret sharing schemes for any arbitrary n can be constructed. Thus, it is possible to increase the number of the users in the schemes for a fixed alphabet size q by increasing the number of the columns in the matrices corresponding to each user in the generalized vector space construction. Moreover, we present a recursive algorithm for constructing general (t, n) threshold secret sharing schemes for t ≥ 4 and any arbitrary n. The algorithm is based on a combination between the proposed (2, n) threshold secret sharing schemes and (t− 1, t− 1) threshold secret sharing schemes. Using the proposed algorithms, a (t, n) threshold secret sharing scheme can be constructed for any arbitrary number of users n and any threshold value t.
Publication Year: 2008
Publication Date: 2008-01-01
Language: en
Type: dissertation
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