Evaluating 2-DNF Formulas on Ciphertexts

Abstract

Let ψ be a 2-DNF formula on boolean variables x 1,...,x n ∈ {0,1}. We present a homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x 1,...,x n . In other words, given the encryption of the bits x 1,...,x n , anyone can create the encryption of ψ(x 1,...,x n ). More generally, we can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set. We present a number of applications of the system:

  1. In a database of size n, the total communication in the basic step of the Kushilevitz-Ostrovsky PIR protocol is reduced from \(\sqrt{n}\) to \(\sqrt[3]{n}\) .

  2. An efficient election system based on homomorphic encryption where voters do not need to include non-interactive zero knowledge proofs that their ballots are valid. The election system is proved secure without random oracles but still efficient.

  3. A protocol for universally verifiable computation.