Parallel Formulations of Scalar Multiplication on Koblitz Curves

Abstract: We present an algorithm that by using the ô and ô.1 Frobenius operators concur- rently allows us to obtain a parallelized version of the classical ô -and-add scalar multiplication algorithm for Koblitz elliptic curves. Furthermore, we report suitable irreducible polynomials that lead to efficient implementations of both ô and ô.1, thus showing that our algorithm can be effectively applied on all the NIST-recommended curves. We also present design details of software and hardware implementations of our procedure. In a two-processor workstation soft- ware implementation, we report experimental data showing that our parallel algorithm is able to achieve a speedup factor of almost 2 when compared with the standard sequential point multipli- cation. In our hardware implementation, the parallel version yields a more modest acceleration of 17% when compared with the traditional point multiplication algorithm. Although the focus is on Koblitz curves, analogous strategies are discussed for other curves, in particular for random curves over binary fields.