A Parallel Architecture for Koblitz Curve Scalar Multiplications on FPGA Platforms

Elliptic curve scalar multiplication is the central operation in elliptic curve cryptography. The paper presents a parallel architecture to accelerate scalar multiplications on Koblitz curves. The scalar multiplier architecture converts the scalar into τ-NAF representation and processes the zero digits of the scalar in parallel to point additions. Since the conversion from integer to τ-NAF is a time consuming operation, the proposed architecture uses recently developed double lazy reduction algorithm for conversion of scalar. The scalar multiplier processes two consecutive τ-NAF digits in every iteration. This facilitates parallel processing of large number of consecutive zero digits during a single point addition and practically no time is spent for processing the zero digits of the scalar. The proposed techniques are incorporated in a scalar multiplier and validated on Xilinx Virtex IV FPGA. Experimental results show that our architecture in F2163 has the best performance and has the computation time comparable with the fastest known implementation, which uses window based scalar multiplication algorithm.