Admissible tracks in Shamir's scheme

Abstract

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 − 1 x 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 Spież 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 i ≠ 0 , k − 1 we prove that the number of ( k , i ) -admissible tracks of length n is q n − ( ( n 2 ) + ( n k − 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.