International Association
for Cryptologic Research

We present fully homomorphic encryption schemes for fixed-point arithmetic with fixed precision. Our scheme achieves $\mathsf{IND}$-$\mathsf{CPA^D}$ security and uses $\mathsf{RLWE}$ ring with dimension ${2^{13}}$ or less. Our techniques could also be extended to construct fully homomorphic encryption schemes for approximate numbers with $\mathsf{IND}$-$\mathsf{CPA}$ security. The bootstrapping process of our $\mathsf{IND}$-$\mathsf{CPA}$ scheme preserves about 39-bit precision with ring dimension $2^{13}$, which is the first construction that preserves high precision while keeping the parameters small.

The core technique in this paper is a new and efficient functional bootstrapping algorithm that avoids the negacyclicity constraint of the evaluated functions, which enables us to extract bits blocks homomorphically. This new functional bootstrapping algorithm could be applied to BFV and TFHE schemes as well, and is of independent interest.