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

arXiv: https://arxiv.org/abs/2406.14372 · HTML: https://mlsanity.com/paper/2406.14372/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2024.

## Built on by (most cited first, 1 of 1)

- [Relative Entropy Regularized Reinforcement Learning for Efficient Encrypted Policy Synthesis](https://mlsanity.com/paper/2506.12358/) (2025)

## Abstract (arXiv)

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.