Consider the following secret-sharing problem. Your goal is to distribute a long file s between n servers such that ( d − 1 ) -subsets cannot recover the file, ( d + 1 ) -subsets can recover the file, and d -subsets should be able to recover s if and only if they appear in some predefined list L . How small can the information ratio (i.e., the number of bits stored on a server per each bit of the secret) be?

We initiate the study of such d -uniform access structures, and view them as a useful scaled-down version of general access structures. Our main result shows that, for constant d , any d -uniform access structure admits a secret sharing scheme with a \emph{constant} asymptotic information ratio of c d that does not grow with the number of servers n . This result is based on a new construction of d -party Conditional Disclosure of Secrets (Gertner et al., JCSS '00) for arbitrary predicates over n -size domain in which each party communicates at most four bits per secret bit.

In both settings, previous results achieved non-constant information ratio which grows asymptotically with n even for the simpler (and widely studied) special case of d = 2 . Moreover, our results provide a unique example for a natural class of access structures F that can be realized with information rate smaller than its bit-representation length log F (i.e., ( d log n ) for d -uniform access structures) showing that \emph{amortization can beat the representation size barrier}.