Scooby: Improved multi-party homomorphic secret sharing based on FHE

In this paper we present new constructions of multi-party homomorphic secret sharing (HSS) based on a new primitive that we call homomorphic encryption with decryption to shares (HEDS). Our first scheme, which we call $\mathrm{Scooby}$, is based on many popular fully homomorphic encryption (FHE) schemes with a linear decryption property. $\mathrm{Scooby}$ achieves an n-party HSS for general circuits with complexity $O\left(\right|F|+\log n)$, as opposed to $O(n^2\cdot |F\left|\right)$ for the prior best construction based on multi-key FHE. $\mathrm{Scooby}$ relies on a trusted setup procedure, and can be based on (ring)-LWE with a super-polynomial modulus-to-noise ratio. In our second construction, $\mathrm{Scrappy}$, assuming any generic FHE plus HSS for NC1-circuits, we obtain a HEDS scheme which does not require a super-polynomial modulus. While these schemes all require FHE, in another instantiation, $\mathrm{Shaggy}$, we show how it is also possible to obtain multi-party HSS without FHE, instead relying on the DCR assumption to obtain 4-party HSS for constant-degree polynomials.