Private Computation of Polynomials over Networks

This study concentrates on preserving privacy in a network of agents where each agent seeks to evaluate a general polynomial function over the private values of her immediate neighbors. We provide an algorithm for the exact evaluation of such functions while preserving privacy of the involved agents. The solution is based on a reformulation of polynomials and adoption of two cryptographic primitives: Paillier as a Partially Homomorphic Encryption scheme and multiplicative-additive secret sharing. The provided algorithm is fully distributed, lightweight in communication, robust to dropout of agents, and can accommodate a wide class of functions. Moreover, system theoretic and secure multi-party conditions guaranteeing the privacy preservation of an agent's private values against a set of colluding agents are established. The theoretical developments are complemented by numerical investigations illustrating the accuracy of the algorithm and the resulting computational cost.Comment: 11 pages, 2 figure

Private Computation of Polynomials over Networks

This study concentrates on preserving privacy in a network of agents where each agent seeks to evaluate a general polynomial function over the private values of her immediate neighbors. We provide an algorithm for the exact evaluation of such functions while preserving privacy of the involved agents. The solution is based on a reformulation of polynomials and adoption of two cryptographic primitives: Paillier as a Partially Homomorphic Encryption scheme and multiplicative-additive secret sharing. The provided algorithm is fully distributed, lightweight in communication, robust to dropout of agents, and can accommodate a wide class of functions. Moreover, system theoretic and secure multi-party conditions guaranteeing the privacy preservation of an agent's private values against a set of colluding agents are established. The theoretical developments are complemented by numerical investigations illustrating the accuracy of the algorithm and the resulting computational cost.Comment: 11 pages, 2 figure

Secure and Private Implementation of Dynamic Controllers Using Semi-Homomorphic Encryption

This paper presents a secure and private implementation of linear time-invariant dynamic controllers using Paillier's encryption, a semi-homomorphic encryption method. To avoid overflow or underflow within the encryption domain, the state of the controller is reset periodically. A control design approach is presented to ensure stability and optimize performance of the closed-loop system with encrypted controller.Comment: Improved numerical exampl

Homomorphically encrypted gradient descent algorithms for quadratic programming

In this paper, we evaluate the different fully homomorphic encryption schemes, propose an implementation, and numerically analyze the applicability of gradient descent algorithms to solve quadratic programming in a homomorphic encryption setup. The limit on the multiplication depth of homomorphic encryption circuits is a major challenge for iterative procedures such as gradient descent algorithms. Our analysis not only quantifies these limitations on prototype examples, thus serving as a benchmark for future investigations, but also highlights additional trade-offs like the ones pertaining the choice of gradient descent or accelerated gradient descent methods, opening the road for the use of homomorphic encryption techniques in iterative procedures widely used in optimization based control. In addition, we argue that, among the available homomorphic encryption schemes, the one adopted in this work, namely CKKS, is the only suitable scheme for implementing gradient descent algorithms. The choice of the appropriate step size is crucial to the convergence of the procedure. The paper shows firsthand the feasibility of homomorphically encrypted gradient descent algorithms

Homomorphically encrypted gradient descent algorithms for quadratic programming

In this paper, we evaluate the different fully homomorphic encryption schemes, propose an implementation, and numerically analyze the applicability of gradient descent algorithms to solve quadratic programming in a homomorphic encryption setup. The limit on the multiplication depth of homomorphic encryption circuits is a major challenge for iterative procedures such as gradient descent algorithms. Our analysis not only quantifies these limitations on prototype examples, thus serving as a benchmark for future investigations, but also highlights additional trade-offs like the ones pertaining the choice of gradient descent or accelerated gradient descent methods, opening the road for the use of homomorphic encryption techniques in iterative procedures widely used in optimization based control. In addition, we argue that, among the available homomorphic encryption schemes, the one adopted in this work, namely CKKS, is the only suitable scheme for implementing gradient descent algorithms. The choice of the appropriate step size is crucial to the convergence of the procedure. The paper shows firsthand the feasibility of homomorphically encrypted gradient descent algorithms

Encrypted control for networked systems -- An illustrative introduction and current challenges

Cloud computing and distributed computing are becoming ubiquitous in many modern control systems such as smart grids, building automation, robot swarms or intelligent transportation systems. Compared to "isolated" control systems, the advantages of cloud-based and distributed control systems are, in particular, resource pooling and outsourcing, rapid scalability, and high performance. However, these capabilities do not come without risks. In fact, the involved communication and processing of sensitive data via public networks and on third-party platforms promote, among other cyberthreats, eavesdropping and manipulation of data. Encrypted control addresses this security gap and provides confidentiality of the processed data in the entire control loop. This paper presents a tutorial-style introduction to this young but emerging field in the framework of secure control for networked dynamical systems.Comment: The paper is a preprint of an accepted paper in the IEEE Control Systems Magazin

Secure Teleoperation Control Using Somewhat Homomorphic Encryption

Presented at 2022 Modeling, Estimation, and Control Conference (MECC) , October 2022The goal of this research is to establish control theoretic methods to enhance cyber security of networked motion control systems by utilizing somewhat homomorphic encryption. The proposed approach will encrypt the entire motion control schemes including: sensor signals, model parameters, feedback gains, and performs computation in the ciphertext space to generate motion commands to servo systems without a security hole. The paper will discuss implementation of encrypted bilateral teleoperation control schemes with nonlinear friction compensation. The paper will present (1) encrypted teleoperation control realization with somewhat homomorphic encryption and (2) simulation results.This work was supported in part by the National Science Foundation under Grant No. 2112793 and the Japan Society for the Promotion of Science KAKENHI Grant No. JP22H01509

Fully Homomorphic Encryption-enabled Distance-based Distributed Formation Control with Distance Mismatch Estimators

This paper considers the use of fully homomorphic encryption for the realisation of distributed formation control of multi-agent systems via edge computer. In our proposed framework, the distributed control computation in the edge computer uses only the encrypted data without the need for a reset mechanism that is commonly required to avoid error accumulation. Simulation results show that, despite the use of encrypted data on the controller and errors introduced by the quantization process prior to the encryption, the formation is able to converge to the desired shape. The proposed architecture offers insight on the mechanism for realising distributed control computation in an edge/cloud computer while preserving the privacy of local information coming from each agent