Communications of the ACM
Cryptography: Theory and Practice
Cryptography: Theory and Practice
Non-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
| Hi-index | 0.00 |
We consider Shamir's secret sharing schemes, with the secret placed as a coefficient a"i of the scheme polynomial f(x)=a"0+...+a"k"-"1x^k^-^1, determined by a sequence t=(t"0,...,t"n"-"1)@?F"q^n pairwise different public identities, called a track. If t defines a k-out-of-n Shamir's scheme then the track t is called (k,i)-admissible. If t is not a (k,i)-admissible track, we obtain the scheme with some privileged coalitions of less than k shareholders who can reconstruct the secret by themselves. No (k,i)-admissible tracks contain privileged coalitions. In Spiez et al. [11] it is proved that the coalitions are common zeros of some elementary symmetric polynomials. We obtain some quantitative results on the tracks. Given i0,k-1 we prove that the number of (k,i)-admissible tracks of length n is q^n-((n2)+(nk-1))q^n^-^1+O(q^n^-^2), where the constant in the O-symbol depends on n, k and i. We also estimate the number of tracks being (k,i)-admissible for every i. We prove the existence and extendability of all tracks for sufficiently large q, giving algorithms for their constructing and extending. Furthermore, we investigate (k,i)-privileged coalitions of length k-1, which can reconstruct the secret, placed as a"i, by themselves. We prove that the number of such coalitions is q^k^-^2+O(q^k^-^3), where the constant in the O-symbol depends on k and i.