Fully Homomorphic Encryption without bootstrapping

Gentry’s bootstrapping technique is the most famous method of obtaining fully homomorphic encryption. In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the plaintext [1]. In this paper I propose the improved fully homomorphic encryption scheme on non-associative octonion ring over finite field without bootstrapping technique where the plaintext consists of a plaintext u and two random numbers v,w. The proposed fully homomorphic encryption scheme is immune from the “u and -u attack”. As the scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree while the almost all multivariate cryptosystems [2],[3],[4],[5],[6],[7] proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. Because proposed fully homomorphic encryption scheme is based on multivariate algebraic equations with high degree or too many variables, it is against the Gröbner basis [8] attack, the differential attack, rank attack and so on. The key size of this system and complexity for enciphering /deciphering become to be small enough to handle. keywords: fully homomorphic encryption, multivariate algebraic equation, Gröbner basis, octonion