Space-efficient Verifiable Secret Sharing Using Polynomial Interpolation

Secret sharing deals with the problem of securely distributing confidential information among a certain number of users, in such a way that only some subsets of them are able to jointly decrypt it. Several schemes and variants of secret sharing have been proposed, from the seminal schemes of Shamir and Blakley, which are based respectively on polynomial interpolation, and hyperplanes intersection, to the newest approaches closely involving number theory, such as the ones based on the Chinese Remainder Theorem (Mignotte and Asmuth-Bloom). One important issue in the design of a secret sharing protocol is its robustness against cheaters: common solutions proposed in literature rely on checking consistency of the secret information after reconstruction from more than one group of users, or on adding helpful data to the shares in order to detect and/or identify mistrustful behavior. We present new verification algorithms providing arbitrary secret sharing schemes with cheater detection capabilities, and prove their space efficiency with regard to other schemes appeared in the literature. We also introduce, in one of our schemes, the Exponentiating Polynomial Root Problem (EPRP), which is believed to be NP-Intermediate and therefore difficult.