Fully Homomorphic Encryption based on Euler’s Theorem

Cloud computing enables sharing of resources and services. In the cloud computing the client (user) puts his data in the cloud, and any computations on his stored data will be implemented in the cloud. To produce the effective services, some of the cloud service providers share the information with third parties. The third party can reach to the client private data and modifies the information to make it beneficial to him. Therefore, the security becomes the main thing in cloud computing. The scheme that allows to executing several computations on the encrypted message without decrypting the message is called homomorphic encryption. This paper provides a Fully Homomorphic Encryption Scheme SA that we can apply it for security of a cloud computing. The proposed scheme is based on Euler’s theorem, which proved both addition and multiplication operations at the same time on ciphertext without decryption. We compute the time complexity of the encryption and decryption function for the Elgamal and RSA cryptosystems compared to the proposed SA scheme and also compared the execution time among the three schemes with varies sizes of messages and varies lengths of secret key, and finally the security of the three schemes is analyzed. As a result, in calculations of the execution time of ElGamal and RSA cryptosystems compared to the proposed SA scheme, it was observed that the length of the secret key effect on the execution time, in which the execution time increased when the length of the private key was increased. The proposed scheme showed a good security for the stored data on the cloud.