On the Power of Computational Secret Sharing

Secret sharing is a very important primitive in cryptography and distributed computing. In this work, we consider computational secret sharing (CSS) which provably allows a smaller share size (and hence greater efficiency) than its information-theoretic counterparts. Extant CSS schemes result in succinct share-size and are in a few cases, like threshold access structures, optimal. However, in general, they are not efficient (share-size not polynomial in the number of players n), since they either assume efficient perfect schemes for the given access structure (as in [10]) or make use of exponential (in n) amount of public information (like in [5]). In this paper, our goal is to explore other classes of access structures that admit of efficient CSS, without making any other assumptions. We construct efficient CSS schemes for every access structure in monotone P. As of now, most of the efficient information-theoretic schemes known are for access structures in algebraic NC2. Monotone P and algebraic NC2 are not comparable in the sense one does not include other. Thus our work leads to secret sharing schemes for a new class of access structures. In the second part of the paper, we introduce the notion of secret sharing with a semi-trusted third party, and prove that in this relaxed model efficient CSS schemes exist for a wider class of access structures, namely monotone NP.