Fully Homomorphic Encryption (FHE) enables privacy-preserving cloud services but incurs substantial computation overhead, making hardware acceleration essential. Among FHE operations, key-switching is a major performance bottleneck. Recent cryptographic advances introduce a novel key-switching method (i.e., KLSS) that reduces certain operational complexity but demands higher computational precision than the traditional Hybrid Key Switching (HKS) method. This trade-off leads to distinct computation and memory requirements, making the relative latency of KLSS and HKS highly dependent on hardware parallelism, FHE security parameters, and available on-chip memory capacity, particularly on FPGA platforms, where memory resources and parallelism must be carefully balanced. In this work, we first propose a memory-efficient KLSS datapath that eliminates off-chip ciphertext transfers. We then develop a performance model to analyze and compare the overheads of both KLSS and HKS. Our analysis reveals that an adaptive solution supporting both methods can achieve lower overall latency than a static method during FHE computation. Guided by the performance model, we design an adaptive FPGA-based FHE accelerator that dynamically selects between HKS and KLSS during computation. We implement the accelerator on an Alveo U280 and evaluate it across multiple FHE benchmarks. Experimental results demonstrate that our adaptive solution achieves a 1.84-3.31$\times$ speedup in bootstrapping latency and a 1.66-2.52$\times$ speedup in secure image classification compared to state-of-the-art FPGA accelerators.

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