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Bootstrapping Guarantees: Stability and Performance Analysis for Dynamic Encrypted Control

Sebastian Schlor, Frank Allgöwer

TL;DR

The paper addresses stability and performance of dynamic encrypted control when bootstrapping introduces a controllable but uncertain error. It recasts bootstrapping as a sector-bounded static uncertainty and derives robust-quadratic-performance tests, including a lifted-dynamics formulation that reduces conservatism by accounting for bootstrapping only at refresh times $t = k T_{BS}$. The contribution includes a practical framework that unifies bootstrapping with periodic resets and FIR approaches, and demonstrates how tailored bootstrapping polynomials can balance accuracy and computation. A numerical example illustrates the tighter bound achievable by lifting and the feasibility of the approach, highlighting its potential impact for privacy-preserving control in real-time, resource-constrained settings. The work establishes a foundation for integrating cryptographic bootstrapping with robust control theory to enable stable, private, dynamic control over long horizons, with clear directions for improving efficiency and cryptographic practicality.

Abstract

Encrypted dynamic controllers that operate for an unlimited time have been a challenging subject of research. The fundamental difficulty is the accumulation of errors and scaling factors in the internal state during operation. Bootstrapping, a technique commonly employed in fully homomorphic cryptosystems, can be used to avoid overflows in the controller state but can potentially introduce significant numerical errors. In this paper, we analyze dynamic encrypted control with explicit consideration of bootstrapping. By recognizing the bootstrapping errors occurring in the controller's state as an uncertainty in the robust control framework, we can provide stability and performance guarantees for the whole encrypted control system. Further, the conservatism of the stability and performance test is reduced by using a lifted version of the control system.

Bootstrapping Guarantees: Stability and Performance Analysis for Dynamic Encrypted Control

TL;DR

The paper addresses stability and performance of dynamic encrypted control when bootstrapping introduces a controllable but uncertain error. It recasts bootstrapping as a sector-bounded static uncertainty and derives robust-quadratic-performance tests, including a lifted-dynamics formulation that reduces conservatism by accounting for bootstrapping only at refresh times . The contribution includes a practical framework that unifies bootstrapping with periodic resets and FIR approaches, and demonstrates how tailored bootstrapping polynomials can balance accuracy and computation. A numerical example illustrates the tighter bound achievable by lifting and the feasibility of the approach, highlighting its potential impact for privacy-preserving control in real-time, resource-constrained settings. The work establishes a foundation for integrating cryptographic bootstrapping with robust control theory to enable stable, private, dynamic control over long horizons, with clear directions for improving efficiency and cryptographic practicality.

Abstract

Encrypted dynamic controllers that operate for an unlimited time have been a challenging subject of research. The fundamental difficulty is the accumulation of errors and scaling factors in the internal state during operation. Bootstrapping, a technique commonly employed in fully homomorphic cryptosystems, can be used to avoid overflows in the controller state but can potentially introduce significant numerical errors. In this paper, we analyze dynamic encrypted control with explicit consideration of bootstrapping. By recognizing the bootstrapping errors occurring in the controller's state as an uncertainty in the robust control framework, we can provide stability and performance guarantees for the whole encrypted control system. Further, the conservatism of the stability and performance test is reduced by using a lifted version of the control system.
Paper Structure (19 sections, 3 theorems, 23 equations, 3 figures)

This paper contains 19 sections, 3 theorems, 23 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Related work
  3. Contribution
  4. Preliminaries
  5. Notation
  6. CKKS
  7. Integer representation and rescaling
  8. Bootstrapping
  9. Bootstrapping polynomial
  10. Problem description and formulation as robust control problem
  11. System description
  12. Bootstrapping error
  13. Problem description
  14. Dynamic control with bootstrapping
  15. Stability and performance analysis
  16. ...and 4 more sections

Key Result

Theorem 1

The encrypted closed-loop system eq:clsys with the bootstrapping uncertainty eq:sectorP satisfies robust quadratic performance with performance index $P_p = $ with $R_p \succeq 0$, if there exist $X \succ 0$ and $\tau>0$ such that

Figures (3)

  • Figure 1: The modulo function and its polynomial approximation for bootstrapping.
  • Figure 2: Relative error of the polynomial approximation to the modulo function for bootstrapping. The figure shows multiple error functions since the bootstrapping polynomial in Fig. is evaluated at different intervals depending on the offset $rq$.
  • Figure 3: Block diagram of the encrypted control system with bootstrapping. The bootstrapping is is interpreted as static nonlinearity acting on the controller state.

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof