A Secure Resilient Homomorphic Encryption Scheme for Control Systems

Abstract

Recently, the so-called resilient homomorphic encryption (RHE) scheme has been proposed by Fauser and Zhang (2021). Different from the existing homomorphic encryption schemes, the RHE scheme is able to neutralize the effect of an attack injected into the ciphertexts. Thus, a control system can still operate even if an attack takes place. In this article, the RHE scheme is further developed so that it can not only neutralize additive attacks but also satisfy the security requirement of indistinguishability under chosen plaintext attack (IND-CPA). This is achieved by choosing the modulus and the random values in the RHE scheme suitably so that the ciphertexts obtained by the secure resilient homomorphic encryption (SRHE) scheme follow the approximate greatest common divisor (AGCD) distribution. Since it can be shown that the AGCD distribution is computationally indistinguishable from the uniform distribution, the distribution of the ciphertexts obtained by the SRHE scheme is also computationally indistinguishable from the uniform distribution. Therefore, the SRHE scheme satisfies IND-CPA. An approach is given to select the parameters of the SRHE scheme systematically to guarantee the desired security level. Moreover, considering the quantization error caused by the transformation between real values and integer values, a condition for the stability of the closed-loop encrypted control system with a dynamic output feedback controller is provided. Finally, the SRHE scheme is illustrated through the well-established quadruple-tank system and analyzed with respect to the execution time of the encrypted controller, the requirement on network capacity and the usage of storage.