Communications of the ACM
Cryptography: Theory and Practice
Cryptography: Theory and Practice
Proceedings of the 1982 conference on Cryptography
A modular approach to key safeguarding
IEEE Transactions on Information Theory
Admissible tracks in Shamir's scheme
Finite Fields and Their Applications
Remarks on the Classical Threshold Secret Sharing Schemes
Fundamenta Informaticae - Cryptology in Progress: 10th Central European Conference on Cryptology, Będlewo Poland, 2010
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We consider Shamir@?s secret sharing schemes, with the secret placed as a"i in the scheme polynomial f(x)=a"0+...+a"k"-"1x^k^-^1, determined by sequences t=(t"1,...,t"n)@?F"q^n, called tracks, of pairwise different public identities assigned to shareholders. The shares are given by y"j=f(t"j), 1=0,k-1, it was proved that the number of privileged coalitions of maximal length is q^k^-^2+O(q^k^-^3), where the constant in the O-symbol depends on k and i. In this paper we characterize (k,i)-privileged coalitions of length r as common zeros of k-r elementary symmetric polynomials @t"j(s)=0, r-i=0,k-1 exist if and only if q=1(modk-1). Their number is q-1k-1 and they are permutations of the tracks (a,a@z,...,a@z^k^-^2) with a@?F"q^@? and @z@?F"q a primitive r-th root of unity.