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Recent advances in cloud services provide greater computing ability to edge devices on cyber-physical systems (CPS) and internet of things (IoT) but cause security issues in cloud servers and networks. This paper applies homomorphic encryption (HE) to background subtraction (BGS) in CPS/IoT. Cheon et al.'s numerical methods are adopted to implement the non-linear functions of BGS in the HE domain. In particular, square- and square root-based HE-based BGS (HEBGS) designs are proposed for the input condition of the numerical comparison operation. In addition, a fast-converging method is proposed so that the numerical comparison operation outputs more accurate results with lower iterations. Although the outer loop of the numerical comparison operation is removed, the proposed square-based HEBGS with the fast-converging method shows an average peak signal-to-noise ratio value of 20dB and an average structural similarity index measure value of 0.89 compared to the non-HE-based conventional BGS. On a PC, the execution time of the proposed design for each 128×128-sized frame is 0.34 seconds. 

One of the image segmentation techniques, multilevel thresholding, is widely used in many computer vision applications because of its low computational complexity and efficient data representation. When it is used in cyber-physical systems and internet-of-things, a special technique is required to protect the sensitive information in an image. This paper proposes a novel homomorphic encryption (HE)-based multilevel thresholding method. To implement a comparison operation in the HE domain, which is not a basic homomorphic operation, a numerical method is adopted. Our proposed method executes comparison operations in parallel to perform more iterations and increase accuracy. When the number of iterations in the numerical comparison operation is (5, 3), the proposed three-level thresholding method shows an average peak signal-to-noise ratio of 28 dB compared to a conventional non-HE-based method and takes 3 minutes on a PC.