Taking Advantage of Rational Canonical Form for Faster Ring-LWE based Encrypted Controller with Recursive Multiplication

TL;DR

This paper introduces a system-theoretical approach using rational canonical form to optimize encrypted linear controllers based on Ring-LWE, significantly reducing computational complexity and enabling faster encrypted control implementations.

Contribution

It presents a novel method to transform state matrices into rational canonical form and pack matrices into polynomials, reducing homomorphic operations in encrypted control systems.

Findings

Reduced homomorphic operations for recursive multiplication

Faster encrypted controller implementation in simulations

Efficient encryption of sparse, circulant matrices

Abstract

This paper aims to provide an efficient implementation of encrypted linear dynamic controllers that perform recursive multiplications on a Ring-Learning With Errors (Ring-LWE) based cryptosystem. By adopting a system-theoretical approach, we significantly reduce both time and space complexities, particularly the number of homomorphic operations required for recursive multiplications. Rather than encrypting the entire state matrix of a given controller, the state matrix is transformed into its rational canonical form, whose sparse and circulant structure enables that encryption and computation are required only on its nontrivial columns. Furthermore, we propose a novel method to ``pack'' each of the input and the output matrices into a single polynomial, thereby reducing the number of homomorphic operations. Simulation results demonstrate that the proposed design enables a remarkably fast implementation of encrypted controllers.