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Client-Aided Secure Two-Party Computation of Dynamic Controllers

Kaoru Teranishi, Takashi Tanaka

TL;DR

The paper addresses privately computing discrete-time dynamic controllers by introducing a secret-sharing based two-party computation that outsources controller evaluation to two servers while keeping parameters, states, and signals secret. It keyly combines a truncation-based overflow prevention scheme with Beaver-triple based arithmetic to realize controller updates without state decryption or input re-encryption, achieving performance arbitrarily close to the unencrypted controller given sufficient fixed-point precision. The authors analyze and compare the computational and communication costs against single-server HE-based schemes, and propose improvements via Brunovsky normal form and cryptographic primitives to reduce communication. Numerical demonstrations with PID and observer-based controllers corroborate feasibility and show controllability within the prescribed error bounds, while outlining future work toward MPC and active adversary resistance.

Abstract

In this paper, we propose a secure two-party computation protocol for dynamic controllers using a secret sharing scheme. The proposed protocol realizes outsourcing of controller computation to two servers, while controller parameters, states, inputs, and outputs are kept secret against the servers. Unlike previous encrypted controls in a single-server setting, the proposed method can operate a dynamic controller for an infinite time horizon without controller state decryption or input re-encryption. We show that the control performance achievable by the proposed protocol can be made arbitrarily close to that attained by the unencrypted controller. Furthermore, system-theoretic and cryptographic modifications of the protocol are presented to improve the communication complexity. The feasibility of the protocol is demonstrated through numerical examples of PID and observer-based controls.

Client-Aided Secure Two-Party Computation of Dynamic Controllers

TL;DR

The paper addresses privately computing discrete-time dynamic controllers by introducing a secret-sharing based two-party computation that outsources controller evaluation to two servers while keeping parameters, states, and signals secret. It keyly combines a truncation-based overflow prevention scheme with Beaver-triple based arithmetic to realize controller updates without state decryption or input re-encryption, achieving performance arbitrarily close to the unencrypted controller given sufficient fixed-point precision. The authors analyze and compare the computational and communication costs against single-server HE-based schemes, and propose improvements via Brunovsky normal form and cryptographic primitives to reduce communication. Numerical demonstrations with PID and observer-based controllers corroborate feasibility and show controllability within the prescribed error bounds, while outlining future work toward MPC and active adversary resistance.

Abstract

In this paper, we propose a secure two-party computation protocol for dynamic controllers using a secret sharing scheme. The proposed protocol realizes outsourcing of controller computation to two servers, while controller parameters, states, inputs, and outputs are kept secret against the servers. Unlike previous encrypted controls in a single-server setting, the proposed method can operate a dynamic controller for an infinite time horizon without controller state decryption or input re-encryption. We show that the control performance achievable by the proposed protocol can be made arbitrarily close to that attained by the unencrypted controller. Furthermore, system-theoretic and cryptographic modifications of the protocol are presented to improve the communication complexity. The feasibility of the protocol is demonstrated through numerical examples of PID and observer-based controls.

Paper Structure

This paper contains 24 sections, 4 theorems, 53 equations, 7 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Dynamic Controllers Over Integers
  4. Secret Sharing and Beaver Triple
  5. Secret Sharing
  6. Beaver Triple
  7. Two-Party Computation of Dynamic Controllers
  8. Truncation Protocol
  9. Controller Computation Protocol
  10. Comparison to Previous Works
  11. LWE-based encrypted controller in Kim2023-pk
  12. RLWE-based encrypted controller in Teranishi2024-ha
  13. RLWE-based encrypted controller in Lee2025-jo
  14. Proposed protocol (Protocol \ref{['pro:controller']})
  15. Improvement of Communication Complexities
  16. ...and 9 more sections

Key Result

Lemma 1

Suppose $q$ is a prime. Let $\kappa, \ell, \lambda \in \mathbb{N}$ such that $\kappa = \lfloor \log_2 q \rfloor - \lambda - 1 > \ell$. It holds that, for all $m \in \mathbb{Z}_{\langle \kappa \rangle}$, where $\llbracket m\rrbracket \gets \mathsf{Share}(m)$ and $w \in \{-1, 0, 1\}$.

Figures (7)

  • Figure 1: System architectures of encrypted controls in single-server and two-party computation settings.
  • Figure 2: Client-aided two-party dynamic controller computation.
  • Figure 3: Input errors between the original and two-party PID controls with $\ell = 32, 40, 48, 56$ bit.
  • Figure 4: Input errors between the original and two-party observer-based controls with $\ell = 32, 40, 48, 56$ bit.
  • Figure : Multiplication ($\mathsf{Mult}$)
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1: $2$-out-of-$2$ secret sharing
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 5 more