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arxiv: 2606.29412 · v1 · pith:XFTXDMWNnew · submitted 2026-06-28 · 📡 eess.SY · cs.IT· cs.SY· math.IT

Privacy-Aware State Estimation: From Coarse to Precise Privacy Protection

Zhongyao Hu , Jason J. R. Liu , Jun Shang , Bo Chen , Zhan Shu This is my paper

Pith reviewed 2026-06-30 02:05 UTC · model grok-4.3

classification 📡 eess.SY cs.ITcs.SYmath.IT
keywords state estimationprivacy protectionmean square errorRiccati equationobservable subspaceeavesdropperencryption schemeKalman filtering
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The pith

Precise privacy is achieved by making the eavesdropper's directional MSE unbounded when the direction's unstable component is unobservable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops methods for both coarse and precise privacy in state estimation against eavesdroppers. Coarse privacy makes the eavesdropper's total mean-square error infinite while preserving the legitimate user's performance through an analytical transformation and intermittent encryption. Precise privacy goes further by ensuring error diverges along specific directions. The key result is a proof that directional MSE becomes unbounded if and only if the unstable part of the direction is outside the observable subspace, along with a method to exclude chosen vectors from that subspace.

Core claim

By analyzing the Riccati equation on the unobservable subspace, the eavesdropper's directional mean-square error becomes unbounded if and only if the direction's unstable component lies outside the observable subspace. A systematic method is proposed to exclude target vectors from the observable subspace, forcing the directional MSE to infinity.

What carries the argument

The decomposition of the system into observable and unobservable subspaces and the behavior of the Riccati equation on the unobservable part, which governs divergence of directional error.

If this is right

  • The legitimate user's estimation optimality is maintained while forcing the eavesdropper's total MSE to infinity at a polynomial-exponential rate.
  • A lower bound on the encryption probability in the stochastic intermittent encryption scheme guarantees divergence of the eavesdropper's error.
  • Precise privacy can be systematically designed by ensuring confidential directions have their unstable components unobservable.
  • The condition for unbounded directional MSE is both necessary and sufficient based on the subspace analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework could extend to multi-agent systems where different agents have different privacy requirements for directions.
  • It suggests designing sensors or communication protocols to control observability for privacy.
  • Numerical simulations on linear systems like vehicle tracking could test the encryption probability bounds.

Load-bearing premise

The linear system dynamics allow a clean decomposition into observable and unobservable parts with standard Riccati convergence properties.

What would settle it

A counterexample linear system where a direction's unstable component is inside the observable subspace yet the directional MSE still diverges to infinity, or where it is outside but remains bounded.

Figures

Figures reproduced from arXiv: 2606.29412 by Bo Chen, Jason J. R. Liu, Jun Shang, Zhan Shu, Zhongyao Hu.

Figure 2
Figure 2. Figure 2: The MSEs of eavesdropper on the confidential direction under the coarse and precise privacy-preserving methods. was derived to achieve lossless compression and destroy the eavesdropper’s detectability. Furthermore, the growth rate of the eavesdropper’s MSE was shown to follow a polynomial￾exponential form governed by the encrypted unstable modes. For precise privacy, we proved that a confidential direction… view at source ↗
Figure 1
Figure 1. Figure 1: The MSEs of eavesdropper under the proposed coarse privacy￾preserving method and the coarse privacy-preserving method in [23]. TABLE I COMPARISON OF THE PROPOSED METHOD WITH THE METHOD IN [23] IN TERMS OF COMPUTATION AND COMMUNICATION The proposed method The method in [23] Computation time 2.23 × 10−4 s 7.20 × 10−4 s Communication cost 4 scalars 2 × 6 scalars Encryption cost 2(×0.75) scalars 2 scalars Step… view at source ↗
read the original abstract

This paper addresses the problem of achieving both coarse and precise privacy in state estimation. Coarse privacy forces the eavesdropper's total mean-square error (MSE) to infinity, but errors along certain confidential directions may remain bounded. This motivates precise privacy, which additionally drives the MSE along any prescribed direction to infinity. For coarse privacy, an analytical transformation is established, preserving the user's optimality and driving the eavesdropper's total MSE to infinity at a polynomial-exponential rate. A stochastic intermittent encryption scheme is further developed, and an explicit lower bound on the encryption probability is derived to guarantee divergence. For precise privacy, by analyzing the behavior of the Riccati equation on the unobservable subspace, we prove that the eavesdropper's directional MSE becomes unbounded if and only if the direction's unstable component lies outside the observable subspace. Finally, a systematic method is proposed to exclude target vectors from the observable subspace, forcing the directional MSE to infinity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops methods for coarse and precise privacy protection in linear state estimation against an eavesdropper. Coarse privacy is achieved via an analytical transformation that preserves the legitimate user's optimality while driving the eavesdropper's total MSE to infinity at a polynomial-exponential rate, together with a stochastic intermittent encryption scheme and an explicit lower bound on the encryption probability that guarantees divergence. Precise privacy is obtained by analyzing the Riccati recursion restricted to the unobservable subspace, proving that the eavesdropper's directional MSE diverges if and only if the target direction's unstable component lies outside the observable subspace, and providing a systematic procedure to exclude prescribed directions from that subspace.

Significance. If the central claims hold, the work provides a control-theoretic framework that cleanly separates total-MSE privacy from directional privacy, which is relevant for applications such as secure sensor networks and cyber-physical systems. The explicit encryption-probability bound and the iff characterization via observability of unstable modes are concrete, falsifiable contributions that could guide practical design. The approach builds on standard Riccati and observability theory rather than ad-hoc fitting, which strengthens its potential impact if the projection and decoupling arguments are fully rigorous.

major comments (2)
  1. [precise privacy analysis] Precise-privacy section (analysis of Riccati equation on unobservable subspace): the iff claim that directional MSE diverges exactly when the unstable component lies outside the observable subspace requires explicit verification that (i) the orthogonal projection onto the unobservable subspace commutes with the Riccati recursion, (ii) the quadratic form along the target direction isolates the unstable-mode contribution without residual bounded terms, and (iii) the stochastic encryption schedule introduces no cross-coupling between observable and unobservable subspaces. These three steps are load-bearing for the central precise-privacy result; their absence from the visible derivation leaves the divergence claim unconfirmed.
  2. [coarse privacy encryption scheme] Coarse-privacy encryption bound: the lower bound on encryption probability is stated to guarantee divergence of total MSE, but the derivation must confirm that the bound remains valid under the intermittent schedule and does not inadvertently stabilize any unstable modes that the transformation was intended to expose.
minor comments (2)
  1. Notation for the transformed system matrices and the projected Riccati solution should be introduced with explicit definitions to avoid ambiguity when moving between the original and transformed coordinates.
  2. The abstract claims a 'polynomial-exponential rate' for MSE divergence; the precise asymptotic expression (including the polynomial degree) should be stated in the main theorem for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the rigor required for the central claims. We address each major comment below and will revise the manuscript to incorporate the requested verifications.

read point-by-point responses

  1. Referee: [precise privacy analysis] Precise-privacy section (analysis of Riccati equation on unobservable subspace): the iff claim that directional MSE diverges exactly when the unstable component lies outside the observable subspace requires explicit verification that (i) the orthogonal projection onto the unobservable subspace commutes with the Riccati recursion, (ii) the quadratic form along the target direction isolates the unstable-mode contribution without residual bounded terms, and (iii) the stochastic encryption schedule introduces no cross-coupling between observable and unobservable subspaces. These three steps are load-bearing for the central precise-privacy result; their absence from the visible derivation leaves the divergence claim unconfirmed.

    Authors: We agree that these three properties must be verified explicitly for the iff characterization to be fully rigorous. The manuscript presents the main result but does not detail the supporting arguments. In the revision we will add a new lemma immediately after the precise-privacy theorem that proves: (i) commutation by showing that the unobservable subspace is invariant under both the nominal dynamics and the transformed coordinates used for encryption; (ii) isolation of the unstable component by modal decomposition of the quadratic form, demonstrating that all stable-mode contributions remain bounded while the unstable part grows without residual cross terms; and (iii) absence of cross-coupling by noting that the intermittent encryption is applied identically across the state and that the observable/unobservable splitting is preserved by the linear transformation. These additions will confirm the divergence claim without altering the stated result. revision: yes

  2. Referee: [coarse privacy encryption scheme] Coarse-privacy encryption bound: the lower bound on encryption probability is stated to guarantee divergence of total MSE, but the derivation must confirm that the bound remains valid under the intermittent schedule and does not inadvertently stabilize any unstable modes that the transformation was intended to expose.

    Authors: The referee correctly identifies that the bound's validity under the stochastic intermittent schedule requires explicit confirmation. The original derivation computes the bound from the expected Riccati update but does not separately address potential stabilization. We will revise the coarse-privacy section to include a short proposition showing that any encryption probability strictly above the derived threshold forces the expected spectral radius of the effective closed-loop matrix (in the directions exposed by the transformation) to exceed one. The argument uses a stochastic Lyapunov function that averages over the encryption events and demonstrates that the polynomial-exponential growth rate is preserved. This addition will verify that the bound does not inadvertently stabilize the targeted modes. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation grounded in standard linear systems theory

full rationale

The abstract and description present a derivation based on Riccati equation analysis for observable/unobservable subspaces, stochastic encryption schemes, and polynomial-exponential divergence rates. These steps rely on established detectability conditions and subspace decompositions from linear system theory rather than self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No quoted equations reduce the central iff claim or privacy guarantees to their own inputs by construction. The result is self-contained against external benchmarks in control theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; full paper would be needed to audit Riccati assumptions or system decompositions.

pith-pipeline@v0.9.1-grok · 5707 in / 1053 out tokens · 28599 ms · 2026-06-30T02:05:13.739663+00:00 · methodology

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