High-performance elliptic curve cryptoprocessors over $$\varvec{GF(2^{m})}$$ G F ( 2 m ) on Koblitz curves

Abstract

Efficient computation of the finite field arithmetic is required for elliptic curve cryptography over $$G\!F\!(2^{m})$$GF(2m). In order to achieve the above, this paper presents the design of high-performance cryptoprocessors over $$G\!F\!(2^{163})$$GF(2163) and $$G\!F\!(2^{233})$$GF(2233) using finite field multipliers with digit-level processing. The arithmetic operations were implemented in hardware using Gaussian normal bases representation and the scalar point multiplication kP was performed on Koblitz curves using window-$$\tau $$?NAF algorithm with w = 2, 4 and 8. The cryptoprocessors were designed using VHDL description, synthesized on the FPGA Stratix-V using Quartus II 13.0, and verified using ModelSIM and Matlab. The simulation results show that the cryptoprocessors present a very good performance using low area. In this case, the processing times for calculating the scalar point multiplication over $$G\!F\!(2^{163})$$GF(2163) and $$G\!F\!(2^{233})$$GF(2233) for w = 2, 4 and 8 were 9.1, 6.7 and 5.1, and 12.9, 9.5 and 6.8 $$\upmu $$μs, respectively.