This paper presents a homomorphic public key encryption scheme that allows the evaluation of 2-DNF formulas on encrypted data. The scheme supports addition and one multiplication on encrypted values, enabling the evaluation of quadratic multi-variate polynomials on ciphertexts. The system is based on bilinear groups and a new hardness assumption called the subgroup decision problem. The security of the scheme is proven under this assumption. The paper describes several applications of the system, including a protocol for securely evaluating 2-DNF formulas, which improves the communication complexity of the Kushilevitz-Ostrovsky PIR protocol from √n to ∛n. It also presents an efficient election system where voters do not need to provide proofs of vote validity, and a protocol for universally verifiable computation. The homomorphic encryption scheme is constructed using bilinear groups of composite order and supports both additive and multiplicative homomorphism. The system allows for the evaluation of multi-variate polynomials of total degree 2 on encrypted values, and can be used to compute dot products on ciphertexts. The scheme is secure against malicious adversaries and has efficient decryption and homomorphic properties. The paper also discusses the use of the scheme in private information retrieval (PIR) and other applications, including the construction of a PIR scheme that reduces communication complexity. The system is shown to be secure in the random oracle model and has efficient performance for both voters and election authorities. The paper concludes with a discussion of open problems related to the encryption scheme, including the construction of n-linear maps and the expansion of the message space.This paper presents a homomorphic public key encryption scheme that allows the evaluation of 2-DNF formulas on encrypted data. The scheme supports addition and one multiplication on encrypted values, enabling the evaluation of quadratic multi-variate polynomials on ciphertexts. The system is based on bilinear groups and a new hardness assumption called the subgroup decision problem. The security of the scheme is proven under this assumption. The paper describes several applications of the system, including a protocol for securely evaluating 2-DNF formulas, which improves the communication complexity of the Kushilevitz-Ostrovsky PIR protocol from √n to ∛n. It also presents an efficient election system where voters do not need to provide proofs of vote validity, and a protocol for universally verifiable computation. The homomorphic encryption scheme is constructed using bilinear groups of composite order and supports both additive and multiplicative homomorphism. The system allows for the evaluation of multi-variate polynomials of total degree 2 on encrypted values, and can be used to compute dot products on ciphertexts. The scheme is secure against malicious adversaries and has efficient decryption and homomorphic properties. The paper also discusses the use of the scheme in private information retrieval (PIR) and other applications, including the construction of a PIR scheme that reduces communication complexity. The system is shown to be secure in the random oracle model and has efficient performance for both voters and election authorities. The paper concludes with a discussion of open problems related to the encryption scheme, including the construction of n-linear maps and the expansion of the message space.