Bounds on the Threshold Gap in Secret Sharing over Small Fields

We consider the class of secret sharing schemes where there is no a priori bound on the number of players n but where each of the n share-spaces has xed cardinality q. We show two fundamental lower bounds on the threshold gap of such schemes. The threshold gap g is de ned as r − t, where r is minimal and t is maximal such that the following holds: for a secret with arbitrary a priori distribution, each r-subset of players can reconstruct this secret from their joint shares without error (r-reconstruction) and the information gain about the secret is nil for each t-subset of players jointly (t-privacy). Our rst bound, which is completely general, implies that if 1 ≤ t < r ≤ n, then g ≥ n−t+1 q independently of the cardinality of the secret-space. Our second bound pertains to Fq-linear schemes with secret-space Fq (k ≥ 2). It improves the rst bound when k is large enough. Concretely, it implies that g ≥ n−t+1 q + f(q, k, t, n), for some function f that is strictly positive when k is large enough. Moreover, also in the Fq-linear case, bounds on the threshold gap independent of t or r are obtained by additionally employing a dualization argument. As an application of our results, we answer an open question about the asymptotics of arithmetic secret sharing schemes and prove that the asymptotic optimal corruption tolerance rate is strictly smaller than 1.