A secret sharing scheme permits a secret to be shared among participants in such a way that only quali ed subsets of participants can recover the secret, but any non-quali ed subset has absolutely no information on the secret. In this paper we derive new limitations on the information rate of secret sharing schemes, that measures how much information is being distributed as shares as compared to the size of the secret key, and the average information rate, that is the ratio between the secret size and the arithmetic mean of the size of the shares. By applying the substitution technique, we are able to construct many new examples of access structures where the information rate is bounded away from 1. The substitution technique is a method to obtain a new access structure by replacing a participant in a previous structure with a new access structure.

Abstract.A multi-secret sharing scheme is a protocol to share m arbitrarily related secrets s1,. . . , sm among a set of participants P. In this paper we put forward a general theory of multi-secret sharing schemes by using an information theoretical framework. We prove lower bounds on the size of information held by each participant for various access structures. Finally, we prove the optimality of the bounds by providing protocols.

Abstract. The problem we deal with in this paper is the research of upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme. We give both lower and upper bounds for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v, k, )~) designs). We prove a general result on the randomness needed to construct a scheme for the cycle C,~; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)

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