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This paper presents an efficient hardware implementation of cryptoprocessors that perform the scalar multiplication kP over a finite field GF(2163) using two digit-level multipliers. The finite field arithmetic operations were implemented using the Gaussian normal basis (GNB) representation, and the scalar multiplication kP was implemented using the Lopez-Dahab algorithm, the 2-non-adjacent form (2-NAF) halve-and-add algorithm and the wNAF method for Koblitz curves. The processors were designed using a VHDL description, synthesized on the Stratix-IV FPGA using Quartus II 12.0 and verified using SignalTAP II and Matlab. The simulation results show that the cryptoprocessors provide a very good performance when performing the scalar multiplication kP. In this case, the computation times of the multiplication kP using the Lopez-Dahab algorithm, 2-NAF halve-and-add algorithm and 16NAF method for Koblitz curves were 13.37 µs, 16.90 µs and 5.05 µs, respectively.

Design of elliptic curve cryptoprocessors over GF(2^163) using the Gaussian normal basis