The paper presents a homomorphic public key encryption scheme that supports addition and one multiplication, enabling the evaluation of 2-DNF formulas and quadratic multi-variate polynomials on encrypted data. The scheme is based on finite groups of composite order with a bilinear map, providing an additive homomorphism and one multiplication operation. The security of the scheme is based on a new hardness assumption called the *subgroup decision problem*. The authors demonstrate several applications of the scheme: 1. **Private Information Retrieval (PIR)**: The scheme reduces the communication complexity in the basic step of the Kushilevitz-Ostrovsky PIR protocol from $\sqrt{n}$ to $\sqrt[3]{n}$. 2. **Efficient Election System**: An election protocol where voters do not need to provide proofs of vote validity, and the system is secure without random oracles. 3. **Universally Verifiable Computation**: A protocol for verifying the correctness of a computation defined by a circuit over private inputs. The paper also discusses the construction of the scheme, its security, and the applications, including a detailed description of the 2-DNF protocol and its variants. The authors conclude with open problems related to extending the scheme to support more complex operations and larger message spaces.The paper presents a homomorphic public key encryption scheme that supports addition and one multiplication, enabling the evaluation of 2-DNF formulas and quadratic multi-variate polynomials on encrypted data. The scheme is based on finite groups of composite order with a bilinear map, providing an additive homomorphism and one multiplication operation. The security of the scheme is based on a new hardness assumption called the *subgroup decision problem*. The authors demonstrate several applications of the scheme: 1. **Private Information Retrieval (PIR)**: The scheme reduces the communication complexity in the basic step of the Kushilevitz-Ostrovsky PIR protocol from $\sqrt{n}$ to $\sqrt[3]{n}$. 2. **Efficient Election System**: An election protocol where voters do not need to provide proofs of vote validity, and the system is secure without random oracles. 3. **Universally Verifiable Computation**: A protocol for verifying the correctness of a computation defined by a circuit over private inputs. The paper also discusses the construction of the scheme, its security, and the applications, including a detailed description of the 2-DNF protocol and its variants. The authors conclude with open problems related to extending the scheme to support more complex operations and larger message spaces.