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Encrypted system identification as-a-service via reliable encrypted matrix inversion

Janis Adamek, Philipp Binfet, Nils Schlüter, Moritz Schulze Darup

TL;DR

An iterative algorithm for matrix inversion is devised and reliable initializations as well as certificates for the achieved accuracy without compromising the privacy of provided I/O-data are presented.

Abstract

Encrypted computation opens up promising avenues across a plethora of application domains, including machine learning, health-care, finance, and control. Arithmetic homomorphic encryption, in particular, is a natural fit for cloud-based computational services. However, computations are essentially limited to polynomial circuits, while comparisons, transcendental functions, and iterative algorithms are notoriously hard to realize. Against this background, the paper presents an encrypted system identification service enabled by a reliable encrypted solution to least squares problems. More precisely, we devise an iterative algorithm for matrix inversion and present reliable initializations as well as certificates for the achieved accuracy without compromising the privacy of provided I/O-data. The effectiveness of the approach is illustrated with three popular identification tasks.

Encrypted system identification as-a-service via reliable encrypted matrix inversion

TL;DR

An iterative algorithm for matrix inversion is devised and reliable initializations as well as certificates for the achieved accuracy without compromising the privacy of provided I/O-data are presented.

Abstract

Encrypted computation opens up promising avenues across a plethora of application domains, including machine learning, health-care, finance, and control. Arithmetic homomorphic encryption, in particular, is a natural fit for cloud-based computational services. However, computations are essentially limited to polynomial circuits, while comparisons, transcendental functions, and iterative algorithms are notoriously hard to realize. Against this background, the paper presents an encrypted system identification service enabled by a reliable encrypted solution to least squares problems. More precisely, we devise an iterative algorithm for matrix inversion and present reliable initializations as well as certificates for the achieved accuracy without compromising the privacy of provided I/O-data. The effectiveness of the approach is illustrated with three popular identification tasks.

Paper Structure

This paper contains 17 sections, 4 theorems, 40 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics on linear system identification
  4. Basics on public-key homomorphic encryption
  5. Basics on iterative matrix inversions and scalar divisions
  6. Reliable encrypted system identification
  7. Reliable initialization and division
  8. Reliable matrix inversion and system identification
  9. Validation via encrypted certificates
  10. Implementation Details
  11. Overview
  12. Iteration variants with smaller multiplicative depth
  13. Specifications of the cryptostem
  14. Numerical case study
  15. Setup
  16. ...and 2 more sections

Key Result

Lemma 1

Let $M \in \mathbb{R}^{l \times \nu}$ and let $\beta$ be as in eq:boundBeta. Assume the entries of $M$ reflect entries of $U_L$ or $Y_L$. Then,

Figures (2)

  • Figure 1: Overview of the proposed encrypted system identification.
  • Figure 2: Evolution of the approximation errors for the three identification tasks depending on the inversion iteration $k$. For each task, intermediate results of the encrypted implementation (solid) are compared with the plaintext counterpart (dashed).

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • proof