Homomorphic Encryption allows access to highly scalable, inexpensive, on-demand computing resources that can execute the code and store the data that are provided to them. This aspect, known as data outsourced computation is very attractive, as it alleviates most of the burden on IT services from the consumer. Nevertheless, the adoption of data outsourced computation by business has a major obstacle, since the data owner does not want to allow the un trusted cloud provider to have access to the data being outsourced. Merely encrypting the data prior to storing it on the cloud is not a viable solution, since encrypted data cannot be further manipulated. This means that if the data owner would like to search for particular information, then the data would need to be retrieved and decrypted a very costly operation, which limits the usability of the cloud to merely be used as a data storage centre. Homomorphic Encryption systems are used to perform operations on encrypted data without knowing the private key (without decryption), the client is the only holder of the secret key. When we decrypt the result of any operation, it is the same as if we had carried out the calculation on the raw data. Definition: An encryption is homomorphic, if: from Enc(a) and Enc(b) it is possible to compute Enc(f (a, b)), where f can be: +, ×, ⊕ and without using the private key. For plaintexts P1 and P2 and corresponding ciphertext C1 and C2, a homomorphic encryption scheme permits meaningful computation of P1 Θ P2 from C1 and C2 without revealing P1 or P2.The cryptosystem is additive or multiplicative homomorphic depending upon the operation Θ which can be addition or multiplication. A homomorphic encryption scheme consists of the followi... ... middle of paper ... ...S:  Vic (J.R.) Winkler, “Securing the Cloud, Cloud Computer Security, Techniques and Tactics”, Elsevier, 2011.  Pascal Paillier. Public-key cryptosystems based on composite degree residuosity classes. In 18th Annual Eurocrypt Conference (EUROCRYPT'99), Prague, Czech Republic, volume 1592, 1999  Julien Bringe and al. An Application of the Goldwasser-Micali Cryptosystem to Biometric Authentication, Springer-Verlag, 2007.  R. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signatures and public key cryptosystems. Communications of the ACM, 21(2):120-126, 1978. Computer Science, pages 223-238. Springer, 1999.  Taher ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on Information Theory, 469-472, 1985.  Craig Gentry, A Fully Homomorphic Encryption Scheme, 2009.