An End-to-End Encrypted Control Pipeline for Multi-Agent Coordination via CKKS Homomorphic Encryption

Technical Summary: End-to-End Encrypted Control Pipeline for Multi-Agent Coordination

Problem Statement
Cloud-based coordination of multi-agent systems offers significant computational advantages, such as centralized sensor fusion and global optimization. However, this architecture necessitates that agents transmit raw state data (positions, velocities, sensor readings) to a central server in plaintext. This creates a fundamental conflict between coordination efficiency and privacy, particularly in adversarial environments, coalition military operations, or settings involving competing organizations. While Fully Homomorphic Encryption (FHE) theoretically resolves this by allowing computation on ciphertexts without decryption, its severe arithmetic constraints—specifically the lack of native support for division, comparisons, and complex matrix operations—have historically prevented the implementation of a complete control loop. Existing works have addressed individual components (controllers, estimators, or optimization) in isolation, but a unified, end-to-end encrypted framework for multi-agent formation control has remained unavailable.

Methodology
The authors propose an end-to-end encrypted control pipeline for Linear Time-Invariant (LTI) multi-agent systems that operates entirely on CKKS-encrypted data. The design adheres strictly to the three primitive operations native to the CKKS scheme: element-wise addition, element-wise (Hadamard) multiplication, and cyclic rotation.

Key methodological adaptations include:

1. Pre-computation of Gains: To avoid non-CKKS operations like matrix inversion and online Riccati equation solving, the system employs pre-computed steady-state Kalman gains ($K_{ss}$) and feedback gains ($K, K_r$). These are applied as plaintext operands to ciphertexts, leveraging the significant cost asymmetry where ciphertext-plaintext operations are far cheaper and introduce less noise than ciphertext-ciphertext operations.
2. The Diagonal Method for Laplacians: The consensus control law requires applying the graph Laplacian ($L \otimes I_n$) to the encrypted state vector. The authors utilize the "diagonal method," decomposing the Laplacian into a sum of cyclic diagonals. Each diagonal is applied via a single CKKS rotation followed by a plaintext mask. This allows the cost of the consensus stage to scale with the number of nonzero cyclic diagonals rather than the matrix dimension, accommodating ring, torus, and complete-graph topologies within a unified framework.
3. Pipeline Architecture: The control loop is realized in five stages:
   • Sensing: Agents encrypt measurements and transmit them to the coordinator.
   • State Estimation: The coordinator updates encrypted state estimates using a steady-state Kalman filter (matrix-vector products via the diagonal method).
   • State Propagation: The system matrix $A$ is applied to the encrypted state.
   • Consensus Control: The graph Laplacian is applied to compute the consensus coupling term.
   • Actuation: The coordinator returns encrypted control inputs, which agents decrypt and execute.
4. Bootstrapping Management: To manage the finite multiplicative depth of CKKS, the system performs periodic bootstrapping (refreshing the ciphertext). The authors model this periodic refresh as an impulsive disturbance in the error dynamics.

Key Contributions
The paper makes two primary contributions:

1. Unified Encrypted Pipeline: It presents the first complete control chain for multi-agent formation control that integrates sensing, estimation, propagation, and consensus into a single CKKS-compatible pipeline. It demonstrates that the diagonal method can handle non-circulant topologies (like torus graphs) efficiently, not just circulant ones (like rings).
2. End-to-End Error Analysis: The authors derive a formal stability analysis and a closed-form steady-state error bound. By applying the separation principle to decouple controller and observer dynamics, they model periodic bootstrapping as a discrete-time impulsive disturbance. They derive the steady-state error ball $\varepsilon_{ss} = \bar{\delta}_{eff} / (1 - \rho_{cl}^{T_{boot}})$, where $\rho_{cl}$ is the closed-loop spectral radius and $T_{boot}$ is the bootstrapping period. This provides a direct design equation linking CKKS parameters, communication topology, and tracking accuracy.

Results
The framework was validated using OpenFHE.jl on a formation control scenario with $M=9$ double-integrator agents across three topologies: a ring, a $3 \times 3$ torus, and a complete graph.

• Stability: The system maintained stable closed-loop operation under encryption for all topologies.
• Accuracy: The encrypted trajectories were visually indistinguishable from plaintext trajectories. The tracking error remained bounded, with the encryption-induced gap stabilizing at approximately $10^{-6}$, dominated by bootstrapping noise rather than arithmetic accumulation.
• Computational Cost: The cost of the consensus stage scaled linearly with the number of nonzero cyclic diagonals (3 for the ring, 7 for the torus, 9 for the complete graph). The total pipeline step time was approximately 5.5 seconds, yielding an update rate of ~0.18 Hz. The authors note that while this rate is slow, it is viable for supervisory coordination applications with dynamics on the order of seconds.
• Noise Behavior: The closed-loop feedback successfully suppressed the accumulation of CKKS noise, preventing the error from diverging, whereas open-loop consensus showed error growth without corrective measurements.

Significance
The paper claims to close the loop on encrypted control by connecting all stages of the control cycle into a single pipeline with an end-to-end error guarantee. It establishes that FHE is practically applicable to multi-agent coordination, provided that the control design accounts for the specific arithmetic constraints and noise characteristics of CKKS. The derived error bound offers practitioners a tool to quantify the privacy-accuracy tradeoff, allowing them to tune CKKS parameters (ring dimension) and control gains (spectral radius) to meet specific performance requirements. The work suggests that while current FHE limitations restrict the framework to slow-dynamics applications, ongoing acceleration efforts may make faster encrypted control practical in the future.