Fully homomorphic encryption scheme and Fermat’s little theorem

In this paper, we take an algebraic approach to the generalized DGHV scheme without using bootstrapping technique. We investigate the homomorphic evaluation algorithm in detail and provide several sufficient conditions for correct homomorphic evaluation of arbitrary polynomial circuits. Compared with the bootstrapping procedure, we show that our approach is much simpler and more efficient. Moreover, we prove that both of the key sizes are actually bounded above and that these upper-bounds depend only on the plaintext space and the security parameter. Hence we need only one pair of secret key and public key for correct homomorphic evaluation of all polynomial circuits, which implies the generalized DGHV scheme is a fully homomorphic encryption in itself. Thus our approach shows that the generalized DGHV scheme provides extremely simple and efficient algorithms.