General Short Computational Secret Sharing Schemes

A secret sharing scheme permits a secret to be shared among participants in such a way that only qualified subsets of participants can recover the secret. If any non qualified subset has absolutely no information about the secret, then the scheme is called perfect. Unfortunately, in this case the size of the shares cannot be less than the size of the secret. Krawczyk [9] showed how to improve this bound in the case of computational threshold schemes by using Rabin's information dispersal algorithms [14], [15]. We show how to extend the information dispersal algorithm for general access structure (we call access structure, the set of all qualified subsets). We give bounds on the amount of information each participant must have. Then we apply this to construct computational schemes for general access structures. The size of shares each participant must have in our schemes is nearly minimal: it is equal to the minimal bound plus a piece of information whose length does not depend on the secret size but just on the security parameter.

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