Paper 2015/1040

Fully Homomorphic Encryption with Composite Number Modulus

Masahiro Yagisawa

Abstract

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. 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 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 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.

Note: In previous report 2015/474 in Cryptology ePrint Archive, I proposed “fully homomorphic encryption without bootstrapping” which has the weak point in the enciphering function and is not immune from “p and -p attack”. In this report I propose the improved scheme which overcomes the weak point.