On Homomorphic Secret Sharing from Polynomial-Modulus LWE
Homomorphic secret sharing (HSS) is a form of secret sharing that supports the local evaluation of functions on the shares, with applications to multi-server private information retrieval, secure computation, and more. Insisting on additive reconstruction, all known instantiations of HSS from "Learning with Error (LWE)"-type assumptions either have to rely on LWE with superpolynomial modulus, come with non-negligible error probability, and/or have to perform expensive ciphertext multiplications, resulting in bad concrete efficiency. In this work, we present a new 2-party local share conversion procedure, which allows to locally convert noise encoded shares to non-noise plaintext shares such that the parties can detect whenever a (potential) error occurs and in that case resort to an alternative conversion procedure. Building on this technique, we present the first HSS for branching programs from (Ring-)LWE with polynomial input share size which can make use of the efficient multiplication procedure of Boyle et al. (Eurocrypt 2019) and and has no correctness error. Our construction comes at the cost of a - on expectation - slightly increased output share size (which is insignificant compared to the input share size) and a more involved reconstruction procedure. More concretely, we show that in the setting of 2-server private information retrieval we can choose ciphertext sizes of only a quarter of the size of the scheme of Boyle et al. at essentially no extra cost.