How Sensor Attacks Transfer Across Lie Groups

Rijad Alisic; Saurabh Amin

arxiv: 2604.03461 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

How Sensor Attacks Transfer Across Lie Groups

Rijad Alisic , Saurabh Amin This is my paper

Pith reviewed 2026-05-13 18:36 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords sensor attacksLie groupsattack transferabilityinvariant subspacesAdjoint actioncyber-physical systemsDubins unicyclestealthy attacks

0 comments

The pith

Sensor attacks transfer successfully across Lie groups only if they commute with the nominal dynamics.

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

This paper develops a geometric framework for determining when sensor spoofing attacks can transfer between operating conditions on systems that evolve on Lie groups. The central result is that an attack preserves both its physical effect and its stealth only when it commutes with the system dynamics, a condition expressed by a vanishing Lie bracket. This requirement confines transferable attacks to a specific invariant subspace while making attacks outside that subspace produce detectable changes in the residuals. The framework further shows that the Adjoint action of the flow amplifies any small deviation from perfect commutation, distorting the accumulated error even when the initial innovation perturbation remains linear. The analysis is illustrated on a Dubins unicycle, where turning maneuvers shrink the transferable subspace to a single direction.

Core claim

A sensor attack transfers across operating conditions on a Lie group while preserving both physical impact and stealthiness only when it commutes with the nominal dynamics, as measured by a vanishing Lie bracket. This isolates transferable attacks to an invariant subspace; attacks outside it identifiably alter the residuals. For small deviations from ideal commutation, the Adjoint action amplifies the physical impact of the bracket-violating component. Although the attack perturbs the innovation linearly, the accumulated error drift undergoes distortion via the Adjoint action. Turning maneuvers on a Dubins unicycle collapse the transferable subspace to a single direction, keeping imperfectly

What carries the argument

The Lie bracket commutation condition between the sensor attack and the nominal dynamics, which isolates transferable stealthy attacks to an invariant subspace via the Adjoint action.

Load-bearing premise

The analysis assumes exact continuous Lie group dynamics and sensor attack models that allow clean decomposition via the Adjoint action, without noise, model mismatch, or discretization.

What would settle it

Apply a deliberately non-commuting sensor attack to a physical vehicle executing turning maneuvers and check whether the residuals remain below detection thresholds or exhibit the predicted alteration.

Figures

Figures reproduced from arXiv: 2604.03461 by Rijad Alisic, Saurabh Amin.

**Figure 1.** Figure 1: Visualization of the commuting subspace ∆fe . (Top) A constant lateral attack perfectly commutes during straight-line motion. (Bottom) During turns, only spoofing along the nominal trajectory (purple) remains in ∆fe . Inconsistent attacks, such as purely body-frame (red) or global (green) offsets, fail to commute and cause observable distortions. C. Observational Transfer To reliably anticipate the detecto… view at source ↗

**Figure 2.** Figure 2: Estimator drift under the spoofed attack over a curved trajectory. The true path (blue) diverges from the spoofed measurement (red dashed) by the injected attack vectors (arrows), while the detector prediction (brown dots) is dragged along the spoofed signal. Realized spoofing deviations (orange) remain within the total theoretical bound (dashed blue) at all times. The steady-state tracking error stays be… view at source ↗

**Figure 3.** Figure 3: Nominal dynamical impact and detector innovation (dotted, training phase) stay near zero. Upon erroneous transfer with ϵ=0.44, both quantities (solid lines) grow with translational displacement but remain within the total bound ∥ξideal∥+ϵ∥AdR gk ∥2 (dashed blue), which peaks at the detection threshold τ =5.0 m, confirming that the margin established during training is sufficient to maintain τ -stealthiness… view at source ↗

read the original abstract

Sensor spoofing analysis in cyber-physical systems is predominantly confined to linear state spaces, where attack transferability is trivial. On Lie groups, however, the noncommutativity of the dynamics can distort certain sensor attacks, exposing nominally stealthy attacks during complex maneuvers. We present a geometric framework characterizing when a sensor attack can transfer across operating conditions, preserving both its physical impact and stealthiness. We prove that successful transfer requires the attack to commute with the nominal dynamics (a Lie bracket condition), which isolates transferable attacks to an invariant subspace, while attacks outside this subspace identifiably alter residuals. For small deviations from ideal transferable attacks, our decomposition theorem reveals a fundamental asymmetry: the flow's Adjoint action amplifies the physical impact of the bracket-violating component. Furthermore, although the attack perturbs the innovation linearly, the accumulated error drift undergoes distortion via the Adjoint action. Finally, we demonstrate how turning maneuvers on a Dubins unicycle collapse the transferable subspace to a single direction, verifying that imperfect attacks remain within theoretical detection bounds.

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.

Desk Editor's Note private letter to a colleague

The paper gives a clean Lie-bracket condition for when sensor attacks stay stealthy across maneuvers on Lie groups, plus an Adjoint-action asymmetry that shows how small deviations get amplified.

read the letter

The core result is that transferable attacks must commute with the nominal dynamics, pinning them to an invariant subspace while non-commuting parts show up in the residuals. The unicycle example then shows how a turning maneuver shrinks that subspace to one direction. This is the first explicit geometric treatment I have seen for attack transfer on non-commutative systems, and it moves past the linear state-space cases that dominate the literature. The decomposition theorem and the asymmetry claim are the parts that feel genuinely new. They give a way to reason about attack persistence without having to simulate every possible trajectory. The math is standard Lie-group machinery, but applied in a way that isolates the effect of non-commutativity on stealth. The unicycle check is useful because it makes the abstract condition concrete and shows the subspace collapse as predicted. That said, the work rests on exact dynamics and perfect sensor models. Real noise, discretization, or small model errors could easily violate the commutation condition and blur the residual signature, yet the paper does not appear to quantify those effects. The demonstration is also limited to one simple vehicle; broader validation on higher-dimensional groups or with hardware would strengthen the case. This is for people already working in geometric control or CPS security who want to think about attacks beyond linear approximations. It is worth sending to referees because the central claim is formally stated and the example supports it, even if the robustness questions need attention in revision.

Referee Report

2 major / 2 minor

Summary. The paper develops a geometric framework for characterizing sensor attack transferability on Lie groups. It proves that successful transfer requires the attack to commute with the nominal dynamics (a Lie bracket condition), confining transferable attacks to an invariant subspace while attacks outside it identifiably alter residuals. A decomposition theorem shows that the flow's Adjoint action amplifies the physical impact of bracket-violating components for small deviations, with accumulated error drift distorted accordingly. The claims are illustrated via a Dubins unicycle demonstration in which turning maneuvers collapse the transferable subspace to a single direction.

Significance. If the central claims hold, the work meaningfully extends sensor spoofing analysis beyond linear systems to noncommutative Lie group dynamics typical in robotics and autonomous vehicles. The isolation of an invariant subspace for stealthy attacks and the identified asymmetry under the Adjoint action supply a principled, parameter-free basis for predicting when nominally stealthy attacks become detectable during maneuvers, which could inform detection filter design and robust control.

major comments (2)

[Abstract and theoretical development] The abstract states proofs of the bracket condition and decomposition theorem, yet the manuscript provides neither the full derivations nor an explicit list of assumptions (e.g., exact continuous-time dynamics, absence of sensor noise or discretization). Because these elements are load-bearing for the commutation requirement and the claimed invariant subspace, their omission prevents verification of the central claims.
[Unicycle demonstration] The unicycle demonstration asserts that turning maneuvers collapse the transferable subspace, but does not report the concrete matrix representation of the Adjoint action or the explicit Lie bracket computation used to obtain the one-dimensional result. Without these steps, it is impossible to confirm that the observed collapse follows from the stated theorem rather than from the specific numerical trajectory chosen.

minor comments (2)

[Notation and preliminaries] Define the innovation and residual explicitly in the Lie-group setting (e.g., via left- or right-invariant error) before invoking them in the decomposition theorem.
[Decomposition theorem] The phrase 'for small deviations from ideal transferable attacks' should be accompanied by a precise norm or distance measure on the Lie algebra to make the asymmetry statement quantitative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our geometric framework. We address each major comment below and will incorporate the requested details into the revised manuscript.

read point-by-point responses

Referee: [Abstract and theoretical development] The abstract states proofs of the bracket condition and decomposition theorem, yet the manuscript provides neither the full derivations nor an explicit list of assumptions (e.g., exact continuous-time dynamics, absence of sensor noise or discretization). Because these elements are load-bearing for the commutation requirement and the claimed invariant subspace, their omission prevents verification of the central claims.

Authors: We agree that the full derivations and explicit assumptions were insufficiently detailed. In the revised manuscript we will add a dedicated section (or appendix) containing the complete proofs of the Lie bracket commutation condition and the decomposition theorem. We will also insert an explicit list of assumptions at the start of the theoretical development, stating the exact continuous-time Lie group dynamics, the absence of sensor noise, and the lack of discretization effects. revision: yes
Referee: [Unicycle demonstration] The unicycle demonstration asserts that turning maneuvers collapse the transferable subspace, but does not report the concrete matrix representation of the Adjoint action or the explicit Lie bracket computation used to obtain the one-dimensional result. Without these steps, it is impossible to confirm that the observed collapse follows from the stated theorem rather than from the specific numerical trajectory chosen.

Authors: We accept that the demonstration lacks the required explicit computations. In the revision we will include the concrete matrix form of the Adjoint action for the Dubins unicycle and the step-by-step Lie bracket calculations that reduce the transferable subspace to one dimension under turning maneuvers, thereby confirming that the collapse follows directly from the theorem. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained on standard Lie group theory

full rationale

The paper establishes its Lie bracket commutation condition and invariant subspace decomposition directly from the Adjoint action on Lie groups and the given sensor attack model. These steps follow from the standard geometric properties of the dynamics without any reduction to fitted parameters, data-dependent predictions, or self-citation chains that carry the uniqueness of the result. The unicycle example functions as a consistency check rather than an input that forces the general claim. No load-bearing step collapses by construction to the paper's own definitions or prior outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard Lie group structure and attack models in CPS; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract description.

axioms (2)

domain assumption System dynamics evolve exactly on a Lie group with non-commutative operations.
Explicitly stated as the setting where linear analysis fails and the bracket condition applies.
domain assumption Sensor attacks affect the innovation and residuals in a manner amenable to Adjoint action analysis.
Implicit in the claims about stealthiness preservation and error drift distortion.

pith-pipeline@v0.9.0 · 5472 in / 1321 out tokens · 37109 ms · 2026-05-13T18:36:54.528722+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

successful transfer requires the attack to commute with the nominal dynamics (a Lie bracket condition), which isolates transferable attacks to an invariant subspace Δ_fe = ker([f_e,·])

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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[1] [1]

On the requirements for successful GPS spoofing attacks

Nils Ole Tippenhauer, Christina Pöpper, Kasper Bonne Rasmussen, and Srdjan Capkun. On the requirements for successful GPS spoofing attacks. InProceedings of the 18th ACM Conference on Computer and Communications Security, CCS ’11, pages 75–86, New York, NY , USA, 2011. Association for Computing Machinery

work page 2011

[2] [2]

Kerns, Daniel P

Andrew J. Kerns, Daniel P. Shepard, Jahshan A. Bhatti, and Todd E. Humphreys. Unmanned aircraft capture and control via GPS spoofing. J. Field Robot., 31(4):617–636, July 2014

work page 2014

[3] [3]

Spoofing- resilient LiDAR-GPS factor graph localization with chimera authen- tication

Adam Dai, Tara Mina, Ashwin Kanhere, and Grace Gao. Spoofing- resilient LiDAR-GPS factor graph localization with chimera authen- tication. In2023 IEEE/ION Position, Location and Navigation Symposium (PLANS), pages 470–480, 2023

work page 2023

[4] [4]

Mohammed Aftatah, Abdelhak Khalil, and Khalid Zebbara. Secure navigation through GPS/INS integration: Comparative analysis of su- pervised deep learning and kalman filtering for precision and spoofing detection.IEEE Access, 14:18581–18594, 2026

work page 2026

[5] [5]

Distributed attack-resilient platooning against false data injection.IEEE Trans- actions on Vehicular Technology, 75(3):3888–3903, 2026

Lorenzo Lyons, Manuel Boldrer, and Laura Ferranti. Distributed attack-resilient platooning against false data injection.IEEE Trans- actions on Vehicular Technology, 75(3):3888–3903, 2026

work page 2026

[6] [6]

Saurabh Amin, Xavier Litrico, Shankar Sastry, and Alexandre M. Bayen. Cyber security of water SCADA systems—part i: Analysis and experimentation of stealthy deception attacks.IEEE Transactions on Control Systems Technology, 21(5):1963–1970, 2013

work page 1963

[7] [7]

Cárdenas, Saurabh Amin, and Shankar Sastry

Alvaro A. Cárdenas, Saurabh Amin, and Shankar Sastry. Research challenges for the security of control systems. InProceedings of the 3rd Conference on Hot Topics in Security, HOTSEC’08, USA, 2008. USENIX Association

work page 2008

[8] [8]

Secure control against replay attacks

Yilin Mo and Bruno Sinopoli. Secure control against replay attacks. In2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 911–918, 2009

work page 2009

[9] [9]

Attack detec- tion and identification in cyber-physical systems.IEEE Transactions on Automatic Control, 58(11):2715–2729, 2013

Fabio Pasqualetti, Florian Dörfler, and Francesco Bullo. Attack detec- tion and identification in cyber-physical systems.IEEE Transactions on Automatic Control, 58(11):2715–2729, 2013

work page 2013

[10] [10]

Secure control systems: A quantitative risk management approach.IEEE Control Systems Magazine, 35(1):24–45, 2015

André Teixeira, Kin Cheong Sou, Henrik Sandberg, and Karl Henrik Johansson. Secure control systems: A quantitative risk management approach.IEEE Control Systems Magazine, 35(1):24–45, 2015

work page 2015

[11] [11]

Chong, Henrik Sandberg, and André Teixeira

Michelle S. Chong, Henrik Sandberg, and André Teixeira. A tutorial introduction to security and privacy for cyber-physical systems. In 2019 18th European Control Conference (ECC), pages 968–978, 2019

work page 2019

[12] [12]

Bloch.Nonholonomic Mechanics and Control

Anthony M. Bloch.Nonholonomic Mechanics and Control. Springer New York, New York, NY , 2nd edition, 2015

work page 2015

[13] [13]

Polycarpou

Kangkang Zhang, Christodoulos Keliris, Thomas Parisini, and Mar- ios M. Polycarpou. Stealthy integrity attacks for a class of nonlinear cyber-physical systems.IEEE Transactions on Automatic Control, 67(12):6723–6730, 2022

work page 2022

[14] [14]

A micro Lie theory for state estimation in robotics, 2021

Joan Solà, Jeremie Deray, and Dinesh Atchuthan. A micro Lie theory for state estimation in robotics, 2021

work page 2021

[15] [15]

Murray, Zexiang Li, and S

Richard M. Murray, Zexiang Li, and S. Shankar Sastry.A Mathemat- ical Introduction to Robotic Manipulation. CRC Press, 1st edition, 1994

work page 1994

[16] [16]

The invariant extended Kalman filter as a stable observer.IEEE Transactions on Automatic Control, 62(4):1797–1812, 2017

Axel Barrau and Silvère Bonnabel. The invariant extended Kalman filter as a stable observer.IEEE Transactions on Automatic Control, 62(4):1797–1812, 2017

work page 2017

[17] [17]

Observer design for nonlinear systems with equivariance.Annual Review of Control, Robotics, and Autonomous Systems, 5(V olume 5, 2022):221–252, 2022

Robert Mahony, Pieter van Goor, and Tarek Hamel. Observer design for nonlinear systems with equivariance.Annual Review of Control, Robotics, and Autonomous Systems, 5(V olume 5, 2022):221–252, 2022

work page 2022

[18] [18]

Data-enabled predictive control: In the shallows of the DeePC

Jeremy Coulson, John Lygeros, and Florian Dörfler. Data-enabled predictive control: In the shallows of the DeePC. In2019 18th European Control Conference (ECC), pages 307–312, 2019

work page 2019

[19] [19]

Müller, and Frank Allgöwer

Julian Berberich, Johannes Köhler, Matthias A. Müller, and Frank Allgöwer. Linear tracking MPC for nonlinear systems—part II: The data-driven case.IEEE Transactions on Automatic Control, 67(9):4406–4421, 2022

work page 2022

[20] [20]

Data-injection attacks using historical inputs and outputs

Rijad Alisic and Henrik Sandberg. Data-injection attacks using historical inputs and outputs. In2021 European Control Conference (ECC), pages 1399–1405, 2021

work page 2021

[21] [21]

Data-driven covert-attack strategies and countermeasures for cyber- physical systems

Mahdi Taheri, Khashayar Khorasani, Iman Shames, and Nader Meskin. Data-driven covert-attack strategies and countermeasures for cyber- physical systems. In2021 60th IEEE Conference on Decision and Control (CDC), pages 4170–4175, 2021

work page 2021

[22] [22]

Data-driven attack detection for linear systems.IEEE Control Systems Letters, 5(2):671–676, 2021

Vishaal Krishnan and Fabio Pasqualetti. Data-driven attack detection for linear systems.IEEE Control Systems Letters, 5(2):671–676, 2021

work page 2021

[23] [23]

Hall.Lie Groups, Lie Algebras, and Representations: An El- ementary Introduction, volume 222 ofGraduate Texts in Mathematics

Brian C. Hall.Lie Groups, Lie Algebras, and Representations: An El- ementary Introduction, volume 222 ofGraduate Texts in Mathematics. Springer International Publishing, 2nd edition, 2015

work page 2015