SUMMARY: 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.

SUMMARY: In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the enciphering function. In this paper I propose the improved fully homomorphic encryption scheme on non-associative octonion ring over finite field without bootstrapping technique. I improve the previous scheme by (1) adopting the enciphering function such that it is difficult to express simply by using the matrices and (2) constructing the composition of the plaintext p with two sub-plaintexts u and v. The improved scheme is immune from the “p and-p attack”. The improved scheme is based on multivariate algebraic equations with high degree or too many variables while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. The improved scheme is against the Gröbner basis attack. The key size of this scheme and complexity for enciphering /deciphering become to be small enough to handle.

A framework of noise-free symmetric fully homomorphic en-cryption (FHE) is proposed in this work. Different from the frameworks that are defined over non-commutative group-s, our framework is constructed from matrices over non-commutative rings. The scheme is one-way secure against chosen plaintext attacks (OW-CPA) based on the factor-ization problem of matrices over noncommutative rings as well as the hardness of an overdefined system of multivari-ate polynomial equations over the given non-commutative algebraic structure. On the basis of this framework, a ver-ifiable FHE is proposed, where the receiver can check the validity of ciphertexts.

SUMMARY: 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][18]. In this paper I propose the improved fully homomorphic encryption scheme on non-associative octonion ring over finite ring with composite number modulus where the plaintext p consists of three numbers u,v,w. The proposed fully homomorphic encryption scheme is immune from the “p and-p 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. It is proved that if there exists the PPT algorithm that decrypts the plaintext from the ciphertexts of the proposed scheme, there exists the PPT algorithm that factors the given composite number modulus.