HOMOMORPHIC ZERO-KNOWLEDGE THRESHOLD SCHEMES OVER ANY FINITE ABELIAN GROUP

A threshold scheme is an algorithm in which a distributor creates l sh ares of a secret such that a fixed minimum number (t) of shares are ne eded to regenerate the secret. A perfect threshold scheme does not rev eal anything new from an information theoretical viewpoint to t - 1 sh areholders about the secret. When the entropy of the secret is zero al l 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 comp utational viewpoint. New homomorphic perfect secret threshold schemes over any finite Abelian group for which the group operation and invers es are computable in polynomial time are developed. One of the new thr eshold schemes also satisfies the zero-knowledge property. A generaliz ation toward a homomorphic zero-knowledge general sharing scheme over any finite Abelian group is discussed and it is proven that ideal homo morphic threshold schemes do not always exist.