[Paper Review] Privacy Preserving Set-Based Estimation Using Partially Homomorphic Encryption

Amr Alanwar, Victor Gassmann+5 more|arXiv (Cornell University)|Oct 19, 2020

Security in Wireless Sensor Networks4 citations

TL;DR

This paper proposes a privacy-preserving set-based estimation protocol using partially homomorphic encryption to protect sensor measurements and state estimates in distributed systems. By encrypting zonotopic and constrained zonotopic set representations, the method enables secure aggregation and intersection of sets in the encrypted domain, ensuring computational privacy through indistinguishability while maintaining accurate, bounded state estimation for safety-critical applications.

ABSTRACT

The set-based estimation has gained a lot of attention due to its ability to guarantee state enclosures for safety-critical systems. However, collecting measurements from distributed sensors often requires outsourcing the set-based operations to an aggregator node, raising many privacy concerns. To address this problem, we present set-based estimation protocols using partially homomorphic encryption that preserve the privacy of the measurements and sets bounding the estimates. We consider a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Sets are represented by zonotopes and constrained zonotopes as they can compactly represent high-dimensional sets and are closed under linear maps and Minkowski addition. By selectively encrypting parameters of the set representations, we establish the notion of encrypted sets and intersect sets in the encrypted domain, which enables guaranteed state estimation while ensuring privacy. In particular, we show that our protocols achieve computational privacy using the cryptographic notion of computational indistinguishability. We demonstrate the efficiency of our approach by localizing a real mobile quadcopter using ultra-wideband wireless devices.

Motivation & Objective

• To address privacy leaks in distributed set-based estimation where sensors outsource data to untrusted cloud aggregators.
• To enable secure, private state estimation in safety-critical systems without compromising estimation accuracy or set enclosure guarantees.
• To develop a protocol that supports encrypted set operations—particularly intersection—on zonotopic and constrained zonotopic representations.
• To achieve computational privacy using cryptographic notions such as computational indistinguishability under the decisional Diffie-Hellman assumption.
• To demonstrate practical feasibility through real-world localization of a mobile quadcopter using ultra-wideband devices.

Proposed method

• The method uses partially homomorphic encryption (PHE) to encrypt key parameters of zonotopes and constrained zonotopes, including centers, generators, and bias vectors.
• Zonotopes and constrained zonotopes are employed due to their compact representation and closure under linear transformations and Minkowski addition.
• Set intersection is performed in the encrypted domain by homomorphically combining encrypted set representations from multiple sensor groups.
• The protocol ensures privacy by simulating the view of any coalition of honest-but-curious parties using a simulator that produces indistinguishable outputs.
• The security model relies on computational indistinguishability, ensuring that no coalition can learn the actual values of encrypted sets beyond what is publicly available.
• The system is implemented using the Paillier cryptosystem, enabling additive homomorphism for secure aggregation of encrypted measurements and set parameters.

Experimental results

Research questions

• RQ1 Can set-based estimation be performed securely in the cloud while preserving the privacy of sensor measurements and state estimates?
• RQ2 How can zonotopic and constrained zonotopic set representations be securely homomorphically processed in the encrypted domain without decryption?
• RQ3 What level of privacy leakage is possible under a coalition of honest-but-curious sensor groups, and can it be cryptographically bounded?
• RQ4 Can the proposed protocol maintain tight state estimation bounds while ensuring computational privacy?
• RQ5 How efficient is the protocol in real-world applications such as mobile robot localization?

Key findings

• The proposed protocol achieves computational privacy by ensuring that the view of any coalition of sensor groups is computationally indistinguishable from a simulated view, even when they collude.
• The method enables secure intersection of encrypted sets using homomorphic operations, preserving the accuracy of state estimation under bounded uncertainties.
• The system supports efficient computation of linear maps and Minkowski sums on encrypted zonotopes, which are essential for set-based estimation.
• The protocol was successfully evaluated in a real-world quadcopter localization task using ultra-wideband (UWB) ranging, demonstrating practical feasibility.
• The security model guarantees that even a coalition of sensor groups cannot infer the private set parameters of non-colluding groups beyond the public information.
• The use of constrained zonotopes allows for more compact and tighter set representations compared to standard zonotopes, improving estimation performance.