Homomorphic encryption (HE) is a promising technique for privacy-preserving computations, especially the word-wise HE schemes that allow batching. However, the high computational overhead hinders the deployment of HE in real-word applications. GPUs are often used to accelerate execution, but a comprehensive performance comparison of different schemes on the same platform is still missing. In this work, we fill this gap by implementing three word-wise HE schemes BGV, BFV, and CKKS on GPU, with both theoretical and engineering optimizations. We enhance the hybrid key-switching technique, significantly reducing the computational and memory overhead. We explore several kernel fusing strategies to reuse data, resulting in reduced memory access and IO latency, and enhancing the overall performance. By comparing with the state-of-the-art works, we demonstrate the effectiveness of our implementation. Meanwhile, we introduce a unified framework that finely integrates our implementation of the three schemes, covering almost all scheme functions and homomorphic operations. We optimize the management of pre-computation, RNS bases, and memory in the framework, to provide efficient and low-latency data access and transfer. Based on this framework, we provide a thorough benchmark of the three schemes, which can serve as a reference for scheme selection and implementation in constructing privacy-preserving applications.

Digital signatures are fundamental building blocks in various protocols to provide integrity and authenticity. The development of the quantum computing has raised concerns about the security guarantees afforded by classical signature schemes. CRYSTALS-Dilithium is an efficient post-quantum digital signature scheme based on lattice cryptography and has been selected as the primary algorithm for standardization by the National Institute of Standards and Technology. In this work, we present a high-throughput GPU implementation of Dilithium. For individual operations, we employ a range of computational and memory optimizations to overcome sequential constraints, reduce memory usage and IO latency, address bank conflicts, and mitigate pipeline stalls. This results in high and balanced compute throughput and memory throughput for each operation. In terms of concurrent task processing, we leverage task-level batching to fully utilize parallelism and implement a memory pool mechanism for rapid memory access. We propose a dynamic task scheduling mechanism to improve multiprocessor occupancy and significantly reduce execution time. Furthermore, we apply asynchronous computing and launch multiple streams to hide data transfer latencies and maximize the computing capabilities of both CPU and GPU. Across all three security levels, our GPU implementation achieves over 160× speedups for signing and over 80× speedups for verification on both commercial and server-grade GPUs. This achieves microsecond-level amortized execution times for each task, offering a high-throughput and quantum-resistant solution suitable for a wide array of applications in real systems.

Homomorphic encryption (HE), which allows computation over encrypted data, has often been used to preserve privacy. However, the computationally heavy nature and complexity of network topologies make the deployment of HE schemes in the Internet of Things (IoT) scenario difficult. In this work, we propose CARM, the first optimized GPU implementation that covers BGV, BFV and CKKS, targeting for accelerating homomorphic multiplication using GPU in heterogeneous IoT systems. Our solution is suitable for accelerating RNS homomorphic multiplication on both high-performance and embedded GPUs, as it is a parametric and generic design and offers various trade-offs between resource and efficiency. We offer constant-time low-level arithmetic with minimum instructions and memory usage, as well as performance- and memory-prior configurations. Through this, we can provide more real-time evaluation results and relieve the computational pressure on cloud devices. We deploy our implementations on two GPUs. Compared to the CPU implementation, we achieve up to $378.4\times$ , $234.5\times$ , and $287.2\times$ speedup for homomorphic multiplication of BGV, BFV, and CKKS on Tesla V100S, and $8.8\times$ , $9.2\times$ , and $10.3\times$ on Jetson AGX Xavier, respectively.

The lattice-based signature scheme CRYSTALS-Dilithium is one of the two signature finalists in the third round NIST post-quantum cryptography (PQC) standardization project. For applications of low-power Internet-of-Things (IoT) devices, recent research efforts have been focusing on the performance optimization of PQC algorithms on embedded systems. In particular, performance optimization is more demanding for PQC signature algorithms that are usually significantly more time-consuming than PQC public-key encryption counterparts. For most cryptographic algorithms based on algebraic lattices including Dilithium, the fundamental and most time-consuming operation is polynomial multiplication over rings. For this computational task, number theoretic transform (NTT) is the most efficient multiplication method for NTT-friendly rings, and is now the typical technique for performing fast polynomial multiplications when implementing lattice-based PQC algorithms. The key observation of this work is that, besides multiplications of polynomials of standard forms, Dilithium involves a list of multiplications for polynomials of very small coefficients. Can we have more efficient methods for multiplying such polynomials of small coefficients? Under this motivation, we present in this work a parallel small polynomial multiplication algorithm to speed up the implementations of Dilithium. We complete both C reference implementation and ARM Neon implementation. Moreover, we conducted some speed tests in combination with Becker’s Neon NTT [4]. The results show that, in comparison with the C reference implementation of Dilithium submitted to the third round of the NIST PQC competition, our reference implementation with the proposed parallel small polynomial multiplication is faster: specifically, our Sign and Verify speed up 18% and 19% respectively for Dilithium-2 (30% and 7% for Dilithium-3, 27% and 3% for Dilithium-5, respectively). As for the Arm Neon implementation, we achieved a performance improvement of about 64% in Sign and 50% in Verify for Dilithium-2 (60% and 32% for Dilithium-3) compared with the C reference implementation of Dilithium submitted to the third round of the NIST PQC competition. We aslo compared our work with the state-of-the-art Arm Neon implementation of Dilithium [4], the results show our speed of Sign is 13.4% faster for Dilithium-2 and 8.0% faster for Dilithium-3, achieving a new record of fast Dilithium implementation.

The threat posed by quantum computing has precipitated an urgent need for post-quantum cryptography. Recently, the post-quantum digital signature draft FIPS 204 has been published, delineating the details of the ML-DSA, which is derived from the CRYSTALS-Dilithium. Despite these advancements, server environments, especially those equipped with GPU devices necessitating high- throughput signing, remain entrenched in classical schemes. A conspicuous void exists in the realm of GPU implementation or server-specific designs for ML-DSA. In this paper, we propose the first server-oriented GPU design tailored for the ML-DSA signing procedure in high- throughput servers. We introduce several innovative theoretical optimizations to bolster performance, including depth-prior sparse ternary polynomial multiplication, the branch elimination method, and the rejection-prioritized checking order. Furthermore, exploiting server-oriented features, we propose a comprehensive GPU hardware design, augmented by a suite of GPU implementation optimizations to further amplify performance. Additionally, we present variants for sampling sparse polynomials, thereby streamlining our design. The deployment of our implementation on both server-grade and commercial GPUs demonstrates significant speedups, ranging from 284.9× to 485.3× against the CPU baseline, and an improvement of up to 60.9% compared to related work, affirming the effectiveness and efficiency of the proposed GPU architecture for ML-DSA signing procedure

Homomorphic Encryption (HE) enhances data security by enabling computations on encrypted data, advancing privacy-focused computations. The BFV scheme, a promising HE scheme, raises considerable performance challenges. Graphics Processing Units (GPUs), with considerable parallel processing abilities, offer an effective solution. In this work, we present an in-depth study on accelerating and comparing BFV variants on GPUs, including Bajard-Eynard-Hasan-Zucca (BEHZ), Halevi-Polyakov-Shoup (HPS), and recent variants. We introduce a universal framework for all variants, propose optimized BEHZ implementation, and first support HPS variants with large parameter sets on GPUs. We also optimize low-level arithmetic and high-level operations, minimizing instructions for modular operations, enhancing hardware utilization for base conversion, and implementing efficient reuse strategies and fusion methods to reduce computational and memory consumption. Leveraging our framework, we offer comprehensive comparative analyses. Performance evaluation shows a 31.9X speedup over OpenFHE running on a multi-threaded CPU and 39.7% and 29.9% improvement for tensoring and relinearization over the state-of-the-art GPU BEHZ implementation. The leveled HPS variant records up to 4X speedup over other variants, positioning it as a highly promising alternative for specific applications.