We propose an unbounded fully homomorphic encryption scheme, i.e. a scheme that allows one to compute on encrypted data for any desired functions without needing to decrypt the data or knowing the decryption keys. This is a rational solution to an old problem proposed by Rivest, Adleman, and Dertouzos \cite{32} in 1978, and to some new problems appeared in Peikert \cite{28} as open questions 10 and open questions 11 a few years ago.

Our scheme is completely different from the breakthrough work \cite{14,15} of Gentry in 2009. Gentry’s bootstrapping technique constructs a fully homomorphic encryption (FHE) scheme from a somewhat homomorphic one that is powerful enough to evaluate its own decryption function. To date, it remains the only known way of obtaining unbounded FHE. Our construction of unbounded FHE scheme is straightforward and noise-free that can handle unbounded homomorphic computation on any refreshed ciphertexts without bootstrapping transformation technique.