We present an algorithm that by using the τ and τ-1 Frobenius operators concurrently allows us to obtain a parallelized version of the classical τ-and-add scalar multiplicationalgorithm for Koblitz elliptic curves. Furthermore, we report suitable irreducible polynomials that lead to efficient implementations of both τ and τ-1, thus showing that our algorithm canbe 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 ison Koblitz curves, analogous strategies are discussed for other curves, in particular for random curves over binary fields.

elliptic curve cryptography, Koblitz curves, finite field arithmetic, fast cryptographic algorithms