Homomorphic encryption is one of the main solutions for building secure and privacy-preserving solutions for Machine Learning as a Service, a major challenge in a society where AI becomes more and more pervasive. This motivates the development of homomorphic algorithms for the main building blocks of AI, typically for the components of the various types of neural networks architectures.

In a cyber-physical system (CPS), the interconnection of cyber and physical components occurs through a network. This structure, particularly cyber components and networks, makes it susceptible to malicious attacks. One of the solutions to this CPS security issue is to employ end-to-end homomorphic encryption (HE) that allows direct computations on encrypted data. Despite its promise, HE only supports basic operations, such as addition and multiplication, which limits its application areas. Numerical methods have been presented to perform a comparison operation in the HE domain. However, they suffer from a slow processing speed due to an inherently high number of iterations. To accelerate a homomorphic comparison operation, this paper introduces a novel approach that scales inputs using an asymmetric input range in thresholding. Additionally, parallelism in HE-based multilevel thresholding is explored and exploited through the use of a parallel processing application programming interface for further acceleration. Compared to a previous comparison operation method, the proposed method achieves comparable accuracy with fewer iterations, resulting in a 48% reduction in execution time on an edge computing device. Furthermore, employing an additional thread using parallelism increases this reduction to 63%.

Homomorphic Encryption (HE) schemes such as BGV, BFV, and CKKS consume some ciphertext modulus for each multiplication. Bootstrapping (BTS) restores the modulus and allows homomorphic computation to continue, but it is time-consuming and requires a significant amount of modulus. For these reasons, decreasing modulus consumption is crucial topic for BGV, BFV and CKKS, on which numerous studies have been conducted.

Bootstrapping, which enables the full homomorphic encryption scheme that can perform an infinite number of operations by restoring the modulus of the ciphertext with a small modulus, is an essential step in homomorphic encryption. However, bootstrapping is the most time and memory consuming of all homomorphic operations. As we increase the precision of bootstrapping, a large amount of computational resources is required. Specifically, for any of the previous bootstrap designs, the precision of bootstrapping is limited by rescaling precision.

Abstract A simultaneous Diophantine approximation (SDA) algorithm takes instances of the partial approximate common divisor (PACD) problem as input and outputs a solution. While several encryption schemes have been published and their securities depend on the presumed hardness of variant of the PACD problem, fewer studies have attempted to extend the SDA algorithm to be applicable to these variants. In this study, the SDA algorithm is extended to solve the general PACD problem. In order to proceed, first the variants of the PACD problem are classified and how to extend the SDA algorithm for each is suggested. Technically, the authors show that a short vector of some lattice used in the SDA algorithm gives an algebraic relation between secret parameters. Then, all the secret parameters can be recovered by finding this short vector. It is also confirmed experimentally that this algorithm works well.

Homomorphic encryption (HE) is one of the most promising solutions to secure cloud computing. The number theoretic transform (NTT) that is widely used for convolution operations in HE requires a large amount of computation and has high parallelism, and therefore it has been a good candidate for hardware acceleration. Nevertheless, prior NTT hardware solutions for HE-based applications are impractical in most applications because they do not seriously consider the critical bootstrapping procedure that allows unlimited homomorphic operations on encrypted data. In this paper, we suggest practical bootstrappable parameters, specifically for an established residue number system (RNS)based HE scheme, and apply them to our NTT hardware design. In addition, to limit the size of internal memory for roots of unity increased by the bootstrappable parameters, only a few roots of unity are stored and others are generated on the fly. In our NTT hardware architecture, multiple NTT butterfly units (BUs) are efficiently deployed for high throughput and high resource utilization. In particular, several groups of BUs for respective moduli work in a parallel and pipelined manner, which is effective in an RNS-based HE scheme with a number of moduli. Our implementation on a Xilinx UltraScale FPGA with the bootstrappable parameters achieves a $118 \times$ faster processing speed than a software implementation, and it further provides various trade-off choices such as the number of DSP slices against BRAMs based on available FPGA resources.