Statistically Secure Evaluation of Quadratic Functions and 2-CNF Circuits Over a Dynamic Set of Secret Shares

One of the most interesting research topics in cryptography is nding schemes for an e cient fully-homomorphic encryption (FHE), preferably information-theoretically secure schemes, which are not based on unproven computational hardness assumptions. The greatest breakthrough in this eld was made by Gentry [11] in 2009, and since then there were some interesting developments, e.g., Boneh et al. [3] and Brakerski and Perlman [4]. All currently known FHE schemes are computationally secure and time-wise ine cient. We suggest here an information-theoretically secure secret sharing scheme of restricted values that e ciently supports one homomorphic multiplication of secrets in addition to homomorphic additions of, practically, any number of such multiplied secrets. In particular, our scheme enables sharing a dynamic set of secrets amongst N participants, using polynomials of degree N −1. Quadratic functions and 2-CNF circuits over the set of secrets can then be homomorphically evaluated, while no information is revealed to any single participant, both before and after the computation. Moreover, we prove that our scheme is statistically secure against coalitions of less than N − 1 participants. Namely, assuming the secrets are elements of Fp, under an attack of no more than N − 2 participants, the statistical di erence between two distributions induced by two pairs of secrets over the information held by a coalition of k participants is ≈ 2 pN−1−k .