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Remarks on the Classical Threshold Secret Sharing Schemes

Spież S., Srebrny M., Urbanowicz J.

Fundamenta Informaticae

2012

Vol. 114, nr 3/4

345-357

We survey some results related to classical secret sharing schemes defined in Shamir [10] and Blakley [1], and developed in Brickell [2] and Lai and Ding [4]. Using elementary symmetric polynomials, we describe in a unified way which allocations of identities to participants define Shamir’s threshold scheme, or its generalization by Lai and Ding, with a secret placed as a fixed coefficient of the scheme polynomial. This characterization enabled proving in Schinzel et al. [8], [9] and Spie˙z et al. [13] some new and non-trivial properties of such schemes. Also a characterization of matrices corresponding to the threshold secret sharing schemes of Blakley and Brickell’s type is given. Using Gaussian elimination we provide an algorithm to construct all such matrices which is efficient in the case of relatively small matrices. The algorithm may be useful in constructing systems where dynamics is important (one may generate new identities using it). It can also be used to construct all possible MDS codes.

On the extension of certain maps with values in spheres

Biasi C., Libardi A. K., Pergher P. L., Spież S.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 2

177-182

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m - 2)-dimensional submanifold which is homologous to zero in E. Let Sn[sup]n-2 ⊂ S[sup]n be the standard inclusion, where S[sup]n is the n-sphere and n ≥ 3. We prove the following extension result: if h : V → S[sup]n-2 is a smooth map, then h extends to a smooth map g : E → S[sup]n transverse to S[sup]n-2 and with g[sup]-1(S[sup]n-2) = V. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism question, which asks whether, given a smooth closed n-dimensional manifold E and a smooth closed m-dimensional submanifold V ⊂ E, one can find a compact smooth (m + 1)-dimensional submanifold W ⊂ E such that the boundary of W is V.