It is widely acknowledged that efficient modular multiplication is a key to high-performance implementation of public-key cryptography, be it classical RSA, Diffie-Hellman, or (hyper-) elliptic curve algorithms. In the recent decade, practitioners have relied mainly on two popular methods: Montgomery Multiplication and regular long-integer multiplication in combination with Barrett's modular reduction technique. In this paper, we propose a modification to Barrett's algorithm that leads to a significant reduction (25% to 75%) in multiplications and additions.
As microprocessor designs integrate more cores, scalability of cache coherence protocols becomes a challenging problem. Most directory-based protocols avoid races by using blocking tag directories that can impact the performance of parallel applications. In this article, we first quantitatively demonstrate that state-of-the-art blocking protocols significantly constrain throughput at large core counts for several parallel applications. Nonblocking protocols address this throughput concern at the expense of scalability in the interconnection network or in the required resource overheads. To address this concern, we enhance nonblocking directory protocols by migrating the point of service of responses. Our approach uses in-flight chains of cores making parallel memory requests to incorporate scalability while maintaining high-throughput. The proposed cache coherence protocol called chained cache coherence, can outperform blocking protocols by up to 20% on scientific and 12% on commercial applications. It also has low resource overheads and simple address ordering requirements making it both a high-performance and scalable protocol. Furthermore, in-flight chains provide a scalable solution to building hierarchical and nonblocking tag directories as well as optimize communication latencies.
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