We present a somewhat homomorphic encryption scheme that is both very simple to describe and analyze,
and whose security (quantumly) reduces to the worst-case hardness of problems on ideal lat- tices. We
then transform it into a fully homomorphic encryption scheme using standard “squashing” and
“bootstrapping” techniques introduced by Gentry (STOC 2009).
One of the obstacles in going from “somewhat” to full homomorphism is the requirement that the somewhat
homomorphic scheme be circular secure, namely, the scheme can be used to securely encrypt its own se-
cret key. For all known somewhat homomorphic encryption schemes, this requirement was not known to be
achievable under any cryptographic assumption, and had to be explicitly assumed. We take a step forward
towards removing this additional assumption by proving that our scheme is in fact secure when encrypting
polynomial functions of the secret key. Our scheme is based on the ring learning with errors (RLWE)
assumption that was recently introduced by Lyubashevsky, Peikert and Regev (Euro- crypt 2010). The RLWE
assumption is reducible to worst-case problems on ideal lattices, and allows us to completely abstract
out the lattice in- terpretation, resulting in an extremely simple scheme. For example, our secret key
is s, and our public key is (a,b = as+2e), where s,a,e are all degree (n − 1) integer polynomials whose
coefficients are independently drawn from easy to sample distributions.