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On d-Multiplicative Secret Sharing.

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Title: On d-Multiplicative Secret Sharing.
Authors: Barkol, Omer1 omer.barkol@hp.com, Ishai, Yuval yuvali@cs.technion.ac.il, Weinreb, Enav2 weinreb@cs.technion.ac.il
Source: Journal of Cryptology. Autumn2010, Vol. 23 Issue 4, p580-593. 14p.
Subjects: Quantitative research, Complex multiplication, Linear statistical models, Mathematical variables, Statistical bias
Abstract: 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. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Cryptology is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Cryptology is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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