Optimizing scalar multiplication for Koblitz curves using hybrid FPGAs.

摘要

Elliptic curve cryptography (ECC) is a type of public-key cryptosystem which uses the additive group of points on a nonsingular elliptic curve as a cryptographic medium. Koblitz curves are special elliptic curves that have unique properties which allow scalar multiplication, the bottleneck operation in most ECC cryptosystems, to be performed very efficiently.Optimizing the scalar multiplication operation on Koblitz curves is an active area of research with many proposed algorithms for FPGA and software implementations. As of yet little to no research has been reported on using the capabilities of hybrid FPGAs, such as the Xilinx Virtex-4 FX series, which would allow for the design of a more flexible single-chip system that performs scalar multiplication and is not constrained by high communication costs between hardware and software.While the results obtained in this thesis were competitive with many other FPGA implementations, the most recent research efforts have produced significantly faster FPGA based systems. These systems were created by utilizing new and interesting approaches to improve the runtime of performing scalar multiplication on Koblitz curves and thus significantly outperformed the results obtained in this thesis. However, this thesis also functioned as a comparative study of the usage of different basis representations and proved that strict polynomial basis approaches can compete with strict normal basis implementations when performing scalar multiplication on Koblitz curves.