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Secure Consensus via Objective Coding: Robustness Analysis to Channel Tampering

Marco Fabris, Daniel Zelazo

TL;DR

This paper tackles robust consensus in continuous- and discrete-time multiagent networks subject to channel tampering by introducing a secure-by-design framework where a network manager broadcasts encoded objectives. It develops objective coding and information localization to hide edge weights and ensure robust convergence, even under single-edge perturbations, via a small-gain analysis that links perturbation magnitudes to effective resistances. A key contribution is deriving tight robustness margins, showing that a single-edge codeword perturbation must satisfy bounds tied to the edge's Lipschitz decoding constant and the network's effective resistance; it also explores a tradeoff between encryption capability and robustness. The framework is illustrated through a decentralized estimation application (DPIA/PI-ACE) and extended to discrete-time dynamics, with numerical simulations validating the theoretical guarantees and highlighting practical implications for secure distributed estimation and opinion dynamics. Overall, the work offers a scalable, distributed approach to secure consensus with rigorous robustness guarantees and practical relevance for cyber-physical systems.

Abstract

This work mainly addresses continuous-time multiagent consensus networks where an adverse attacker affects the convergence performances of said protocol. In particular, we develop a novel secure-by-design approach in which the presence of a network manager monitors the system and broadcasts encrypted tasks (i.e., hidden edge weight assignments) to the agents involved. Each agent is then expected to decode the received codeword containing data on the task through appropriate decoding functions by leveraging advanced security principles, such as objective coding and information localization. Within this framework, a stability analysis is conducted for showing the robustness to channel tampering in the scenario where part of the codeword corresponding to a single link in the system is corrupted. A tradeoff between objective coding capability and network robustness is also pointed out. To support these novelties, an application example on decentralized estimation is provided. Moreover, an investigation of the robust agreement is as well extended in the discrete-time domain. Further numerical simulations are given to validate the theoretical results in both the time domains.

Secure Consensus via Objective Coding: Robustness Analysis to Channel Tampering

TL;DR

This paper tackles robust consensus in continuous- and discrete-time multiagent networks subject to channel tampering by introducing a secure-by-design framework where a network manager broadcasts encoded objectives. It develops objective coding and information localization to hide edge weights and ensure robust convergence, even under single-edge perturbations, via a small-gain analysis that links perturbation magnitudes to effective resistances. A key contribution is deriving tight robustness margins, showing that a single-edge codeword perturbation must satisfy bounds tied to the edge's Lipschitz decoding constant and the network's effective resistance; it also explores a tradeoff between encryption capability and robustness. The framework is illustrated through a decentralized estimation application (DPIA/PI-ACE) and extended to discrete-time dynamics, with numerical simulations validating the theoretical guarantees and highlighting practical implications for secure distributed estimation and opinion dynamics. Overall, the work offers a scalable, distributed approach to secure consensus with rigorous robustness guarantees and practical relevance for cyber-physical systems.

Abstract

This work mainly addresses continuous-time multiagent consensus networks where an adverse attacker affects the convergence performances of said protocol. In particular, we develop a novel secure-by-design approach in which the presence of a network manager monitors the system and broadcasts encrypted tasks (i.e., hidden edge weight assignments) to the agents involved. Each agent is then expected to decode the received codeword containing data on the task through appropriate decoding functions by leveraging advanced security principles, such as objective coding and information localization. Within this framework, a stability analysis is conducted for showing the robustness to channel tampering in the scenario where part of the codeword corresponding to a single link in the system is corrupted. A tradeoff between objective coding capability and network robustness is also pointed out. To support these novelties, an application example on decentralized estimation is provided. Moreover, an investigation of the robust agreement is as well extended in the discrete-time domain. Further numerical simulations are given to validate the theoretical results in both the time domains.

Paper Structure

This paper contains 19 sections, 9 theorems, 42 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries and models
  3. The secure-by-design consensus protocol
  4. Objective coding and information localization
  5. Secure-by-design consensus dynamics
  6. Robustness to channel tampering
  7. Model for the channel tampering
  8. Robustness of the SBDC
  9. An application to decentralized estimation
  10. On the spectral properties of the PI-ACE
  11. Application scenario
  12. Numerical examples on the DPIA criticalities
  13. Extension to the discrete-time domain
  14. Secure-by-design consensus in discrete time
  15. Robustness to channel tampering in discrete time
  16. ...and 4 more sections

Key Result

Lemma 2.1

Consider the nominal weighted consensus protocol eq:LAP. Then, for a single edge attack $\Delta^W = \delta_{uv}^{w} \mathbf{e}_z \mathbf{e}_z^{\top} \in \bm{\Delta}^{W}$ on the edge $z=(u,v)\in\mathcal{E}$, such that $\delta_{uv}^{w}$ is a scalar function of $t$, the perturbed consensus protocol is stable for all $\delta_{uv}^{w}$ satisfying where $\mathcal{R}_{uv}(\mathcal{G}) = [L^{\dagger}(\m

Figures (5)

  • Figure 1: Block diagram depicting relation \ref{['eq:SBDC']} and the presence of a cyber-physical attack $\delta^{\theta}$ deviating a sent codeword $\theta$.
  • Figure 2: Numerical results obtained from the application of the SBDC approach to the DPIA.
  • Figure 3: Computation of the convergence rate for several different random topologies, depicted via diverse colored markers. Dots and diamonds represent respectively the results for the nominal DPIA and the perturbed DPIA through $\delta_{\alpha}^{\theta} = -0.99 \alpha$ according to setup $S2$. Items marked in black are acceptable while those marked in red are not, as $r(10,100)\leq0$ for their associated nominal simulation.
  • Figure 4: (a) Considered network topology and attack on edge $(u,v)=(3,4)$; (b) Decoding function in \ref{['eq:p_rs']}, Lipschitz constants $K^{(\beta)}_{uv} = 1/\ln(\beta)$, $\beta = 2,3$, are highlighted (dashed lines); (c-d) Agent dynamics as objective coding and perturbation vary.
  • Figure 5: Results obtained simulating system \ref{['eq:opdyn_sys']} subject to different perturbations on edge $(3,4)$ in $\mathcal{G}_{\alpha}$, with $\alpha = 3/13$.

Theorems & Definitions (20)

  • Definition 2.1: Weighted Consensus,MesbahiEgerstedt2010
  • Lemma 2.1: ZelazoBurger2017
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.1
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • ...and 10 more