Homomorphic Zero-Knowledge Threshold Schemes over any Finite Abelian Group

Abstract

A threshold scheme is an algorithm in which a distributor creates l shares of a secret such that a fixed minimum number $( t )$ of shares are needed to regenerate the secret. A perfect threshold scheme does not reveal anything new from an information theoretical viewpoint to $t - 1$ shareholders about the secret. When the entropy of the secret is zero all sharing schemes are perfect, so perfect sharing loses its intuitive meaning. The concept of zero-knowledge sharing scheme is introduced to prove that the distributor does not reveal anything, even from a computational viewpoint. New homomorphic perfect secret threshold schemes over any finite Abelian group for which the group operation and inverses are computable in polynomial time are developed. One of the new threshold schemes also satisfies the zero-knowledge property. A generalization toward a homomorphic zero-knowledge general sharing scheme over any finite Abelian group is discussed and it is proven that ideal homomorphic threshold schemes do not always exist.