REFHE: Fully Homomorphic ALU

We present a fully homomorphic encryption scheme which natively supports arithmetic and logical operations over large machine words'', namely plaintexts of the form $\mathbb{Z}_{2^n}$ (e.g. $n=64$). Our scheme builds on the well-known BGV framework, but deviates in the selection of number field and in the encoding of messages. This allows us to support large message spaces with only modest effect on the noise growth. Arithmetic operations (modulo $2^n$) are supported natively similarly to BGV-style FHE schemes, and we present an efficient bootstrapping procedure for our scheme. Our bootstrapping algorithm has the feature that along the way it decomposes our machine word into bits, so that during bootstrapping it is possible to perform logical operations (essentially addressing each bit in the message independently). This means that during a single bootstrapping cycle we can perform logical operations on $n$ bits. For example, a greater than'' operation (if $x> y$ output $1$, otherwise $0$), only requires a single subtraction and a single bootstrapping cycle.

Along the way we present a number of new tools and techniques, such as a generalization of the BGV modulus switching technique to a setting where the plaintext and ciphertext moduli are ideals (and not numbers).