Evaluating 2-DNF formulas on ciphertexts

Let ψ be a 2-DNF formula on boolean variables x₁,...,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 x₁,...,x_n. In other words, given the encryption of the bits x₁,...,x_n, anyone can create the encryption of ψ(x₁,...,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 √n to 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.