  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.
