With the wide adoption of cloud computing paradigm, it is important to develop appropriate techniques to protect client data privacy in the cloud. Encryption is one of the major techniques that could be used to achieve this goal. However, data encryption at the rest alone is insufficient for secure cloud computation environments. There is also the need for efficient techniques to carry out computation over encrypted data. Fully homomorphic encryption (FHE) and garbled circuits are naturally used to process encrypted data without leaking any information about the data. However, existing FHE schemes are inefficient for processing large amount of data in cloud and garbled circuits are one time programs and cannot be reused. Based on quaternion/octo-nion algebra and Jordan algebra over finite rings Z_q, this paper designs efficient fully homomorphic symmetric key encryption (FHE) schemes without bootstrapping (that is, noise-free FHE schemes) that are secure in the weak ciphertext-only security model assuming the hardness of solving multivariate quadratic equation systems and solving univariate high degree polynomial equation systems in Z_q. The FHE scheme designed in this paper is sufficient for privacy preserving computation in cloud.
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