On (K, N) Visual Cryptography Scheme With T Essential Parties

In visual cryptography schemes (VCS), we often denote the set of all parties by P = {1,2, ... ,n}. Arumugam et al. proposed a (k,n)-VCS with one essential party recently, in which only subset S of parties satisfying S subset of P and vertical bar S vertical bar >= k and 1 is an element of S can recover the secret. In this paper, we extend Arumugam et al.'s idea and propose a (k,n)-VCS with t essential parties, say (k, n, t)-VCS for brevity, in which only subset S of parties satisfying S subset of P and vertical bar S vertical bar >= k and {1,2, ... , t} is an element of S can recover the secret. Furthermore, some bounds for the optimal pixel expansion and optimal relative contrast of (k, n, t)-VCS are derived.