Fully Homomorphic Public-Key Encryption Against Ciphertext Square Attack with Two Ciphertexts

Abstract

A fully homomorphic public-key encryption (FHPKE) is the important cryptosystem as the basic scheme for the cloud computing. Since Gentry discovered in 2009 the first fully homomorphic encryption scheme, some fully homomorphic encryption schemes were proposed. In the systems proposed until now the bootstrapping process is the main bottleneck and the large complexity for computing the ciphertext is required. The existence of an efficient fully homomorphic cryptosystem would have great practical implications in the outsourcing of private computations, for instance, in the context of cloud computing. In recent year Yagisawa proposed fully homomorphic encryptions without bootstrapping which have the weak point in the enciphering function or not immune from “ciphertext square attack” which is the attack proposed in this article. In this article, a new FHPKE against “ciphertext square attack” is proposed which does not need the bootstrapping and does not require the large complexity for enciphering. The scheme has the following features; (a) its security bases on computational difficulty to solve the multivariate algebraic equations of high degrees; (b) it requires two ciphertexts corresponding to a plaintext. We describe concretely how to construct the proposed system over octonion ring. It is shown that proposed system is immune from “ciphertext square attack”, “m and -m attack” and the Gröbner basis attacks and the complexity to encipher and decipher is not large.