Abstract
Efficient secure multiparty computation (SMPC) schemes over secret shares are presented. We consider scenarios in which the secrets are elements of a finite field, \(\mathbb {F}_{p}\), and are held and shared by a single participant, the user. Evaluation of any function \(f:\mathbb {F}_{p}^n\rightarrow \mathbb {F}_{p}\) is implemented in one round of communication by representing f as a multivariate polynomial. Our schemes are based on partitioning secrets to sums or products of random elements of the field. Secrets are shared using either (multiplicative) shares whose product is the secret or (additive) shares that sum up to the secret. Sequences of additions of secrets are implemented locally by addition of local shares, requiring no communication among participants, and so does sequences of multiplications of secrets. The shift to handle a sequence of additions from the execution of multiplications or vice versa is efficiently handled as well with no need to decrypt the secrets in the course of the computation. On each shift from multiplications to additions or vice versa, the current set of participants is eliminated, and a new set of participants becomes active. Assuming no coalitions among the active participants and the previously eliminated participants are possible, our schemes are information-theoretically secure with a threshold of all active participants. Our schemes can also be used to support SMPC of boolean circuits.
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Notes
- 1.
The total degree of a multivariate polynomial is the maximal sum of exponents in a single monomial of it.
- 2.
Actually, all functions \(f:\mathbb {F}_{q}^n\rightarrow \mathbb {F}_{q}\) are p-bounded for \(p\ge q^{nq+1}\) (considering the minimal-multivariate-polynomial-representation of f). This fact is not useful for large p.
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Acknowledgments
We thank Dani Berend for being involved during the entire research providing original ideas throughout, in particular suggesting to use polynomial representation instead of circuits.
The research was partially supported by the Rita Altura Trust Chair in Computer Sciences; the Lynne and William Frankel Center for Computer Science; the Ministry of Foreign Affairs, Italy; the grant from the Ministry of Science, Technology and Space, Israel, and the National Science Council (NSC) of Taiwan; the Ministry of Science, Technology and Space, Infrastructure Research in the Field of Advanced Computing and Cyber Security; and the Israel National Cyber Bureau.
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Bitan, D., Dolev, S. (2018). One-Round Secure Multiparty Computation of Arithmetic Streams and Functions. In: Dinur, I., Dolev, S., Lodha, S. (eds) Cyber Security Cryptography and Machine Learning. CSCML 2018. Lecture Notes in Computer Science(), vol 10879. Springer, Cham. https://doi.org/10.1007/978-3-319-94147-9_20
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