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: [9, 785, 73, 1273, 2039, 1318, 46, 2138, 2250, 275, 403, 1220, 820]. Abstraction [1483, 171, 1891, 2272, 2044]. Abstractions [1805, 1209, 656, 1557, 1558]. AC [803]. ACACIA [1960]. Academic [435]. Acceleration [1665]. Acceptance [1762, 919]. Accepted [1657]. Accepting [2121, 544]. Access [1020, 1342, 1614, 502, 1281, 1576, 1589, 301, 1874, 1889]. Accessible [1569]. accessing [1907]. Accounting [910, 1338]. Accumulators [844]. Accuracy [1601, 323, 1721]. Accurate [963, 1735]. acetabular [321]. Achieve [2080]. Achieving [243]. ACLP [1456]. Acoustic [741, 354]. Acoustic-labial [354]. Acoustics [994]. Acquiring [412]. Acquisition [1329, 609, 2298, 414, 1344, 1954, 1944]. across [1188]. Acting [104]. Action [784, 886, 93, 1857, 1604, 1833, 2152]. Action-Based [784]. Actions [520, 1879, 1867]. Activated [734]. Activation [698]. activations [1191]. Active [1535, 278, 757, 1507, 2247, 277, 2187, 2090, 9, 10, 306]. Activities [1026, 652, 1888]. Activity [951, 1738]. ACTL ...
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    ABSTRACT: The algebraic setting for threshold secret sharing scheme can vary, dependent on the application. This algebraic setting can limit the number of participants of an ideal secret sharing scheme. Thus it is im- portant to know for which thresholds one could utilize an ideal threshold sharing scheme and for which thresholds one would have to use non- ideal schemes. The implication is that more than one share may have to be dealt to some or all parties. Karnin, Greene and Hellman con- structed several bounds concerning the maximal number of participants in threshold sharing scheme. There has been a number of researchers who have noted the relationship between k-arcs in projective spaces and ideal linear threshold secret schemes, as well as between MDS codes and ideal linear threshold secret sharing schemes. Further, researchers have constructed optimal bounds concerning the size of k-arcs in projective spaces, MDS codes, etc. for various nite elds. Unfortunately, the ap- plication of these results on the Karnin, Greene and Hellamn bounds has not been widely disseminated. Our contribution in this paper is revisit- ing and updating the Karnin, Greene, and Hellman bounds, providing optimal bounds on the number of participants in ideal linear threshold secret sharing schemes for various nite elds, and constructing these bounds using the same tools that Karnin, Greene, and Hellman intro- duced in their seminal paper. We provide optimal bounds for the maximal number of players for a t out of n ideal linear threshold scheme when t = 3, for all possible nite elds. We also provide bounds for innitely many t and innitely many elds and a unifying relationship between this problem and the MDS (maximum distance separable) codes that shows that any improvement on bounds for ideal linear threshold secret sharing scheme will impact bounds on MDS codes, for which there is a number of conjectured (but open) problems.
    Information Theoretic Security, Third International Conference, ICITS 2008, Calgary, Canada, August 10-13, 2008, Proceedings; 01/2008
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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. · 0.49 Impact Factor

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