Ring-LWE based encrypted controller with unlimited number of recursive multiplications and effect of error growth

TL;DR

This paper introduces a Ring-LWE based encrypted dynamic controller capable of unlimited recursive homomorphic multiplications without bootstrapping, analyzing error growth effects, and demonstrating improved efficiency and stability through simulations.

Contribution

It presents a novel encrypted controller using Ring-LWE that supports unlimited recursive multiplications and introduces a vector packing method for efficiency.

Findings

The controller performs recursive homomorphic multiplications without bootstrapping.

Error effects can be bounded to ensure control performance.

Numerical simulations validate the proposed approach.

Abstract

In this paper, we propose an encrypted dynamic controller that executes an unlimited number of recursive homomorphic multiplications on a Ring Learning With Errors (Ring-LWE) based cryptosystem without bootstrapping. The proposed controller exhibits lower computational complexity compared to existing encrypted controllers implemented on LWE based schemes due to the polynomial structure of Ring-LWE. However, the structural difference introduces additional difficulties in analyzing the effect of error growth; Ring-LWE based schemes inject multiple error coefficients when encrypting a single message, which accumulate under recursive homomorphic multiplications. We show that their effect on the control performance can be arbitrarily bounded by the closed-loop stability, thus recovering the performance of the unencrypted controller. Furthermore, a novel method to ``pack'' a vector into a polynomial is presented, which enhances computational and memory efficiency when applied to the proposed encrypted controller. The effectiveness of the proposed design is demonstrated through numerical simulations.