BlackBox Secret Sharing from Primitive Sets in Algebraic Number Fields

01/2005; DOI: 10.1007/11535218_21
Source: DBLP

ABSTRACT A black-box secret sharing scheme (BBSSS) for a given access structure works in exactly the same way over any finite Abelian group, as it only requires black-box access to group operations and to random group elements. In particular, there is no dependence on e.g. the structure of the group or its order. The expansion factor of a BBSSS is the length of a vector of shares (the number of group elements in it) divided by the number of players n. At CRYPTO 2002 Cramer and Fehr proposed a threshold BBSSS with an asymptotically minimal expansion factor Θ(log n). In this paper we propose a BBSSS that is based on a new paradigm, namely, primitive sets in algebraic number fields. This leads to a new BB- SSS with an expansion factor that is absolutely minimal up to an additive term of at most 2, which is an improvement by a constant additive factor. We provide good evidence that our scheme is considerably more ef- ficient in terms of the computational resources it requires. Indeed, the number of group operations to be performed is ˜ O(n2) instead of ˜ O(n3)

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    ABSTRACT: Integer span program (ISP) is a monotone span program (MSP) over ZZ, which is introduced by Cramer and Fehr in CRYPTO 2002. ISP can be used to construct black-box secret sharing scheme (BBSSS) and linear integer secret sharing scheme (LISSS). The efficiency of ISPs is a very important research objective, as efficient ISP can be used to construct efficient BBSSS and efficient LISSS. Until now, only efficient ISPs that realize threshold access structure have been constructed, but not efficient ISPs realizing other access structures. The main contribution of this paper is that we propose an efficient ISP that realizes the hierarchical threshold access structure, to the best knowledge of the authors, which is the first efficient one for non-threshold access structure. Accordingly, with the proposed construction of efficient ISP realizing hierarchical threshold access structure, the construction of efficient non-threshold BBSSSs and LISSSs for a useful family of access structures is presented.
    Information Processing Letters 08/2013; 113(17):621–627. · 0.48 Impact Factor
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    ABSTRACT: This work deals with “MPC-friendly” linear secret sharing schemes (LSSS), a mathematical primitive upon which secure multi-party computation (MPC) can be based and which was introduced by Cramer, Damgaard and Maurer (EUROCRYPT 2000). Chen and Cramer proposed a special class of such schemes that is constructed from algebraic geometry and that enables efficient secure multi-party computation over fixed finite fields (CRYPTO 2006). We extend this in four ways. First, we propose an abstract coding-theoretic framework in which this class of schemes and its (asymptotic) properties can be cast and analyzed. Second, we show that for every finite field \mathbb Fq{\mathbb F}_q, there exists an infinite family of LSSS over \mathbb Fq{\mathbb F}_q that is asymptotically good in the following sense: the schemes are “ideal,” i.e., each share consists of a single \mathbb Fq{\mathbb F}_q-element, and the schemes have t-strong multiplication on n players, where the corruption tolerance \frac3tn-1\frac{3t}{n-1} tends to a constant ν(q) with 0 < ν(q) < 1 when n tends to infinity. Moreover, when |\mathbb Fq||{\mathbb F}_q| tends to infinity, ν(q) tends to 1, which is optimal. This leads to explicit lower bounds on [^(t)](q)\widehat{\tau}(q), our measure of asymptotic optimal corruption tolerance. We achieve this by combining the results of Chen and Cramer with a dedicated field-descent method. In particular, in the \mathbb F2{\mathbb F}_2-case there exists a family of binary t-strongly multiplicative ideal LSSS with \frac3tn-1 » 2.86%\frac{3t}{n-1}\approx 2.86\% when n tends to infinity, a one-bit secret and just a one-bit share for every player. Previously, such results were shown for \mathbb Fq{\mathbb F}_q with q ≥ 49 a square. Third, we present an infinite family of ideal schemes with t-strong multiplication that does not rely on algebraic geometry and that works over every finite field \mathbb Fq{\mathbb F}_q. Its corruption tolerance vanishes, yet still \frac3tn-1 = W(1/(loglogn)logn)\frac{3t}{n-1}= \Omega(1/(\log\log n)\log n). Fourth and finally, we give an improved non-asymptotic upper bound on corruption tolerance.
    08/2009: pages 466-486;

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