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Non-admissible tracks in Shamirs scheme. (English)
Finite Fields Appl. 17, No. 4, 329-342 (2011).
Summary: We consider Shamir’s secret-sharing schemes, with the secret placed as $a_i$ in the scheme polynomial $f(x)=a_0+\cdots+a_{k - 1}x^{k - 1}$, determined by sequences $\bold{t}=(t_1,\dots,t_n)\in \Bbb F^n_q$, called tracks, of pairwise different public identities assigned to shareholders. The shares are given by $y_j=f(t_j),\, 1 \leqslant j\leqslant n$. If a track $\bold t$ defines Shamir’s scheme with threshold $k$ then $\bold t$ is called $(k,i)$-admissible (cf. [{\it A. Schinzel} et al., Finite Fields Appl. 16, No. 6, 449‒462 (2010; Zbl 1209.94054); {\it S. Spież} et al., Fundam. Inform. 114, No. 3‒4, 345‒357 (2012; Zbl 06033533)]. If $\bold t$ is not $(k,i)$-admissible, then there is a coalition, called $(k,i)$-privileged, consisting of less than $k$ shareholders who can reconstruct the secret $a_i$ by themselves. In [Schinzel et al., loc. cit.], given $i\neq 0,\, k - 1$, 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 $τ_j(\bold {s})=0$, $r-i\leqslant j\leqslant k-1-i$. We prove that special coalitions being $(k,i)$-privileged for every $i\neq 0,\, k - 1$ exist if and only if $q\equiv 1(\mathrm{mod}\,\, k-1)$. Their number is ${q-1}\over{k-1}$ and they are permutations of the tracks $(a,aζ,\dots ,aζ^{k - 2})$ with $a\in\Bbb F^\ast_q$ and $ζ\in\Bbb F_q$ a primitive $r$-th root of unity.