Communications of the ACM
Cryptography: Theory and Practice
Cryptography: Theory and Practice
A computational introduction to number theory and algebra
A computational introduction to number theory and algebra
On proper secrets, (t, k)-bases and linear codes
Designs, Codes and Cryptography
Non-admissible tracks in Shamir's scheme
Finite Fields and Their Applications
Admissible tracks in Shamir's scheme
Finite Fields and Their Applications
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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 Spież 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. MSC: primary 94A62; secondary 11T71; 11C20