Abstract: Finite field inversion, an important module of elliptic curve cryptosystems and hyper-elliptic curve cryptosystems, is introduced. Through analyzing and comparing three inversion algorithms that have been invented, an optimized fast algorithm based on FPGA is put forward. According to the optimized algorithm, an inversion module is designed and realized with VHDL. This design is compiled, synthesized, and fitted into Altera’s EP1S10F780C6 FPGA, using its QuartusII software. The result indicates that this optimized algorithm has advanced performance in both speed and area than other past algorithms.

Key words: FPGA, ECC, HECC, Finite field, Inversion

摘要： 介绍了椭圆曲线密码和超椭圆曲线密码算法中一个重要的模块——求逆模块。分析并比较了现有的3种求逆算法，提出了针对FPGA快速实现的改进算法。根据改进的算法设计了求逆的硬件框图，并用VHDL实现了该设计。该设计使用Altera公司的Quartus II软件在EP1S10F780C6上进行编译、综合、布局布线。实验结果证明，该改进的算法无论在速度上还是在芯片面积上都具有比以往的算法更优秀的性能。

关键词: FPGA, 椭圆曲线密码, 超椭圆曲线密码, 有限域, 逆