Abstract
A multiplicative secret sharing scheme allows players to multiply two secret-shared field elements by locally converting their shares of the two secrets into an additive sharing of their product. Multiplicative secret sharing serves as a central building block in protocols for secure multiparty computation (MPC). Motivated by open problems in the area of MPC, we introduce the more general notion of d-multiplicative secret sharing, allowing to locally multiply d shared secrets, and study the type of access structures for which such secret sharing schemes exist.
While it is easy to show that d-multiplicative schemes exist if no d unauthorized sets of players cover the whole set of players, the converse direction is less obvious for d≥3. Our main result is a proof of this converse direction, namely that d-multiplicative schemes do not exist if the set of players is covered by d unauthorized sets. In particular, t-private d-multiplicative secret sharing among k players is possible if and only if k>dt.
Our negative result holds for arbitrary (possibly inefficient or even nonlinear) secret sharing schemes and implies a limitation on the usefulness of secret sharing in the context of MPC. Its proof relies on a quantitative argument inspired by communication complexity lower bounds.
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Communicated by Ronald Cramer
Research supported by grant 1310/06 from the Israel Science Foundation and grant 2004361 from the U.S.-Israel Binational Science Foundation. Work done in part while the authors were visiting the Institute for Pure & Applied Mathematics (IPAM) at UCLA.
Work done in part at the Computer Science Department, Technion.
Work done in part at the Computer Science Department, Technion, and at CWI Amsterdam.
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Barkol, O., Ishai, Y. & Weinreb, E. On d-Multiplicative Secret Sharing. J Cryptol 23, 580–593 (2010). https://doi.org/10.1007/s00145-010-9056-z
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DOI: https://doi.org/10.1007/s00145-010-9056-z