Polynomials Whose Secret Shares Multiplication Preserves Degree for 2-CNF Circuits Over a Dynamic Set of Secrets

One of the most interesting research topics in cryptography is nding e cient homomorphic encryption schemes, preferably information-theoretically secure, which are not based on unproven computational hardness assumptions. The most signi cant breakthrough in this eld was made by Gentry [12] in 2009, and since then, there were various developments. We suggest here an information-theoretically secure secret sharing scheme 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. Our scheme is statistically secure against coalitions of less than N − 1 participants. One possible application of our scheme is a secure homomorphic evaluation of multi-variate quadratic functions and 2-CNF circuits.

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