Polynomial multiplication across variable degrees has been a dominant bottleneck in cryptographic applications, including post-quantum cryptography (PQC) to fully homomorphic encryption (FHE). Yet, none of the existing approaches can achieve both high performance and scalability across variable degrees. Number theoretic transform (NTT) excels at large degrees but suffers from performance degradation when the polynomial is decomposed into small degrees, while the iterative convolution algorithm (ICA) scales poorly to large degrees. Reconciling two disparate approaches has been hindered by a mathematical barrier: NTT decomposition produces heterogeneous polynomial rings that defy uniform ICA processing. This work overcomes this barrier by establishing a new algorithm, based on isomorphic mapping, to transform heterogeneous rings into a fixed-size cyclic ring. Leveraging this algorithm, we propose a scalable architecture called MG-PM, a mixed-granularity polynomial multiplier that, for the first time, enables efficient acceleration across variable degrees. Implemented on a Xilinx VCU118 FPGA, MG-PM supports degrees from 256 to 65 536, achieving up to$\mathbf {9.1\times }$latency reduction and$\mathbf {2.2\times }$area reduction under different configurations, as well as 92.3% improvement in area-time product (ATP) compared with other state-of-the-art designs, establishing MG-PM as a highly efficient and scalable solution for accelerating polynomial multiplication across variable degrees.

The Virtual Private Cloud (VPC) service enables users to configure shared resources within public clouds on demand, providing isolation between users. However, configuring the VPC network is a complex and error-prone task, and misconfiguration has been the leading cause of cloud network security issues. The large number of complex network components and configurations makes it difficult to perform scalable, efficient, and accurate fault verification of the network behavior. To address this issue, we design a comprehensive and automated fault diagnosis and localization tool, calledLFVeri, which is built upon an innovative modular network model that accurately captures the logic functions of real components within VPC networks, and propose eleven functions to verify network reachability and security requirements. We conduct performance testing ofLFVerion various datasets and compared it with other verification tools. The experiments show thatLFVerioutperforms in modeling and analyzing real VPC scenarios while also possessing the fastest verification speed. It can model and analyze large VPC networks with tens of thousands of components and millions of configuration rules in less than half an hour.