A (Leveled) fully homomorphic encryption scheme based on error-free approximate GCD

Fully homomorphic encryption has been viewed as a mechanism that enables to do arbitrary computations on encrypted data, without ever decrypting it. So fully homomorphic encryption owns great value in cloud computing, secure multiparty computation and so on. To obtain a leveled fully homomorphic encryption scheme, we present a comparatively simple and natural method, that is the approximate eigenvector method. The homomorphic addition and multiplication involved are just matrix addition and multiplication, and the error growth is slow and linear. In this essay we show an approximate-eigenvector leveled fully homomorphic encryption scheme based on Error-Free Approximate GCD, whose plaintext is a diagonal matrix. So this scheme can encrypt l bits at a time.