Abstract
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.
Highlights
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
We present in this work efficient hardware implementations of cryptoprocessors over GF(2163) using a Gaussian normal basis (GNB) representation and the Lopez-Dahab algorithm, 2-non-adjacent form (2-NAF) halve-and-add algorithm and w- NAF method for Koblitz curves (Anomalous Binary Curves or ABC) with window sizes of 2, 4, 8 and 16 to perform the scalar multiplication kP
Our processor is based on the Koblitz curves and has a higher performance than the processor presented in Azarderakhsh (2013) because our design has a latency of 5M to compute the point addition, and it uses two digit-level multipliers and a window method that allows us to reduce the amount of point addition operations
Summary
It uses two register files, two parallel digit-level multipliers, one inversion block, several squaring and adder blocks, a main control and an FSM to perform the point addition, point double and conversion from the projective to affine coordinates. The architecture of the cryptoprocessor over GF(2163) using the w- NAF algorithm for Koblitz curves is shown, and it uses two register files, two digit-level finite multipliers, one Frobenius map block, one RAM that stores the expansion coefficients w- NAF of the scalar k, two ROMs that store the pre-computed points Pu in the affine coordinates, which were obtained from Matlab for w = 2, 4, 8 and 16, several squaring and adder blocks, a main control and an FSM to perform the point addition, point doubling and Q. One important remark is that the Koblitz curves are resistant to simple power analysis and to all the known special attacks (T. Juhas, 2007)
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