Determining the optimal contrast for secret sharing schemes in visual cryptography

Abstract

This paper shows that the largest possible contrast Ck, n in an k-out-of-n secret sharing scheme is approximately 4-(k-1). More precisely, we show that 4-(k-1) ≤ Ck, n ≤ 4 -(k-1) nk/(n (n - 1)⋯ (n - (k-1))). This implies that the largest possible contrast equals 4-(k-1) in the limit when n approaches infinity. For large n, the above bounds leave almost no gap. For values of n that come close to k, we will present alternative bounds (being tight for n = k). The proofs of our results proceed by revealing a central relation between the largest possible contrast in a secret sharing scheme and the smallest possible approximation error in problems occuring in Approximation Theory. © Springer-Verlag Berlin Heidelberg 2000.