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Variational Encrypted Model Predictive Control

Jihoon Suh, Yeongjun Jang, Junsoo Kim, Takashi Tanaka

Abstract

We develop a variational encrypted model predictive control (VEMPC) protocol whose online execution relies only on encrypted polynomial operations. The proposed approach reformulates the MPC problem into a sampling-based estimator, in which the computation of the quadratic cost is naturally handled by tilting the sampling distribution, thus reducing online encrypted computation. The resulting protocol requires no additional communication rounds or intermediate decryption, and scales efficiently through two complementary levels of parallelism. We analyze the effect of encryption-induced errors on optimality, and simulation results demonstrate the practical applicability of the proposed method.

Variational Encrypted Model Predictive Control

Abstract

We develop a variational encrypted model predictive control (VEMPC) protocol whose online execution relies only on encrypted polynomial operations. The proposed approach reformulates the MPC problem into a sampling-based estimator, in which the computation of the quadratic cost is naturally handled by tilting the sampling distribution, thus reducing online encrypted computation. The resulting protocol requires no additional communication rounds or intermediate decryption, and scales efficiently through two complementary levels of parallelism. We analyze the effect of encryption-induced errors on optimality, and simulation results demonstrate the practical applicability of the proposed method.
Paper Structure (21 sections, 7 theorems, 41 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 21 sections, 7 theorems, 41 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Main Contributions
  3. Notation
  4. Preliminaries: CKKS Cryptosystem
  5. Problem Formulation
  6. A Variational Approach to Encrypted MPC
  7. Variational Reformulation
  8. Exponential Tilting and Efficient Encrypted Sampling
  9. Polynomial Surrogate of Feasibility
  10. Aggregate violation score
  11. Chebyshev polynomial approximation of $[\,\cdot\,]_+$
  12. Threshold correction and final estimator
  13. Variational Encrypted MPC Protocol
  14. Offline preprocessing
  15. Online execution
  16. ...and 6 more sections

Key Result

Lemma 1

There exist $\mathsf{B}^{\mathsf{Enc}}, \mathsf{B}^{\mathsf{Mult}},\mathsf{B}^{\mathsf{Rot}} \in \mathcal{O}(N_\mathsf{ct}/\Delta)$ for any $\mathsf{pt} \in \mathbb{C}^{N_\mathsf{ct}/2}$, $\mathbf{c}, \mathbf{c}' \in \mathcal{C}$ and $\rho\in\mathbb{Z}$, such that

Figures (1)

  • Figure 1: Comparison of unencrypted variational MPC (orange dashed) and VEMPC (blue solid) subject to constraints (red dotted).

Theorems & Definitions (9)

  • Lemma 1
  • Corollary 1
  • Lemma 2: Variational formula boue1998variational
  • Theorem 1
  • proof
  • Lemma 3
  • Corollary 2
  • Theorem 2
  • proof