On constructing privileged coalitions in Shamirʼs type scheme

We consider Shamirʼs type secret sharing scheme with the secret placed as a coefficient ai of the scheme polynomial over Fq. A coalition of shareholders equipped with pairwise different public identities tj belonging to Fq is called (k,i)-authorized if the holders using their secret shares yj=f(tj) (1⩽j⩽n) are able to reconstruct the secret by themselves. A (k,i)-authorized coalition of k−1 or less shareholders is called a (k,i)-privileged coalition.We construct all (k,i)-privileged coalitions of k−1 shareholders if i≠0,k−1. This construction allows us to deduce a new estimate for the number of such coalitions, which implies that they exist if q>2k−1. We also give a method of extending a given minimal (k,i)-privileged coalition of r shareholders to a coalition of n shareholders (n⩾r) each of whose (k,i)-privileged subcoalitions contains the given one provided .