In this paper, we present dynamic systems over encrypted data that compute the next state and the output using homomorphic properties of the cryptosystem, which has equivalent performance to the linear dynamic controllers over real-valued signals. Assuming that the input as well as the output of the plant is encrypted and transmitted to the system, the state matrix of the system is designed to consist of integers. This allows the proposed dynamic system to operate for infinite time horizon, without decryption or reset of the state. For implementation in practice, the use of cryptosystems based on Learning With Errors problem is considered, which allows both multiplication and addition over encrypted data. The effect of injecting errors during encryption is in turn controlled under closed-loop stability.

Among many advances given by the diffusive coupling on the synchronization problem, recently a design method for a continuous-time heterogeneous network with the desired collective behavior has been developed. Meanwhile, by its use of high-gain, implementation issues arise in discrete -time frameworks. The goal of this paper is thus to introduce a simple scheme that follows the philosophy of Carver Mead, a pioneer of VLSI (Very Large Scale Integration) technology, that removes those implementation issues. In the end, we utilize what corresponds to derivative consensus in continuous-time, which can be realized by, for instance, an analog electric circuit. This provides a mixed analog-digital method of designing a heterogeneous network with the desired collective behavior and rapid convergence.

This paper considers the use of 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.

In this paper, we propose a method for dynamic systems to operate over homomorphically encrypted data for an infinite time horizon, where we do not make use of reset, re-encryption, or bootstrapping for the encrypted messages. The given system is first decomposed into the stable part and the anti-stable part. Then, the stable part is approximated to have finite impulse response, and by a novel conversion scheme, the eigenvalues of the state matrix of the anti-stable part are approximated to algebraic integers. This allows that the given system can be implemented to operate for an infinite time horizon using only addition and multiplication over encrypted data, without re-encrypting any portion of data. The performance error caused by the approximation and quantization can be made arbitrarily small, with appropriate choice of parameters.