Hybrid Multiplicative Non-perfect Secret Sharing

A secret sharing scheme is said to be d-multiplicative if the scheme allows the players to multiply shared d secrets by locally converting their shares into an additive sharing of the product. This paper considers the most general setting where d secrets are independently shared by different adversary structures (hybrid) and a partial information can be leaked (non-perfect) and presents a necessary and sufficient condition for d-multiplication to be possible. In particular, even for the extreme case that the i-th secret is known by all but the i-th player who has partial information on the secret (e.g., all but one bit), d-multiplication is impossible.