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In this paper we provide upper and lower bounds on the randomness requiredby the dealer to set up a secret sharing scheme for infinite classes ofaccess structures. Lower bounds are obtained using entropy arguments. Upperbounds derive from a decomposition construction based on combinatorialdesigns (in particular,t-(v,k,λ) designs). We prove a general result on therandomness needed to construct a scheme for the cycle Cn; whenn is odd our bound is tight. We study the access structures onat most four participants and the connected graphs on five vertices,obtaining exact values for the randomness for all them. Also, we analyze thenumber of random bits required to construct anonymous threshold schemes,giving upper bounds. (Informally, anonymous threshold schemes are schemes inwhich the secret can be reconstructed without knowledge of which participantshold which shares.)