Paper 2020/691

LSS Homomorphisms and Applications to Secure Signatures, Proactive Secret Sharing and Input Certification

Diego Aranha and Anders Dalskov and Daniel Escudero and Claudio Orlandi

Abstract

In this paper we present the concept of linear secret-sharing homomorphisms, which are linear transformations between different secret-sharing schemes defined over vector spaces over a field $\mathbb{F}$ and allow for efficient multiparty conversion from one secret-sharing scheme to the other. This concept generalizes the observation from (Smart and Talibi, IMACC 2019) and (Dalskov et al., EPRINT 2019) that moving from a secret-sharing scheme over $\mathbb{F}$ to a secret sharing over an elliptic curve group $\mathbb{G}$ of order $p$ can be done very efficiently with no communication by raising the generator of $\mathbb{G}$ to each share over $\mathbb{F}$. We then show how to securely evaluate arbitrary bilinear maps, which can be instantiated in particular with pairings over elliptic curves. We illustrate the benefits of being able to efficiently perform secure computation over elliptic curves by providing several applications and use-cases. First, we show methods for securely encoding and decoding field elements into elliptic curve points, which enable applications that require computation back and forth between fields and elliptic curves. Then, we show how to use use the secure pairing evaluation to sign and verify Pointcheval-Sanders signatures (D. Pointcheval and O. Sanders, CT-RSA 2016) in MPC, which enable multiple applications in which some authenticity property is required on secret-shared data. We consider some of these applications in our work, namely Dynamic Proactive Secret Sharing, on which a shared secret is intended to be transferred from one set of parties to another, and Input Certification, on which the "validity'' of the input provided by some party to some MPC protocol can be verified.