Software Implementation of Koblitz Curves over Quadratic Fields

Abstract

In this work, we retake an old idea presented by Koblitz in his landmark paper [21], where he suggested the possibility of defining anomalous elliptic curves over the base field $$\mathbb {F}_4$$ . We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves. In order to achieve a fast reduction procedure, we adopted a redundant trinomial strategy that embeds elements of the field $$\mathbb {F}_{4^{m}},$$ with m a prime number, into a ring of higher order defined by an almost irreducible trinomial. We also report a number of techniques that allow us to take full advantage of the native vector instructions of high-end microprocessors. Our software library achieves the fastest timings reported for the computation of the timing-protected scalar multiplication on Koblitz curves, and competitive timings with respect to the speed records established recently in the computation of the scalar multiplication over prime fields.