In 2007 Shokrollahi proposed a new method of multiplying in a type-II normal basis. Shokrollahi's method efficiently transforms the normal-basis multiplication into a single multiplication of two size-$(n+1)$ polynomials.
This paper speeds up Shokrollahi's method in several ways. It first presents a simpler algorithm that uses only size-$n$ polynomials. It then explains how to reduce the transformation cost by dynamically switching to a `type-II optimal polynomial basis' and by using a new reduction strategy for multiplications that produce output in type-II polynomial basis.
As an illustration of its improvements, this paper explains in detail how the multiplication overhead in Shokrollahi's original method has been reduced by a factor of $1.4$ in a major cryptanalytic computation, the ongoing attack on the ECC2K-130 Certicom challenge. The resulting overhead is also considerably smaller than the overhead in a traditional low-weight-polynomial-basis approach. This is the first state-of-the-art binary-elliptic-curve computation in which type-II bases have been shown to outperform traditional low-weight polynomial bases.Category / Keywords: implementation / optimal normal basis, ONB, polynomial basis, transformation, elliptic-curve cryptography Date: received 9 Feb 2010, last revised 13 Apr 2010 Contact author: tanja at hyperelliptic org Available format(s): PDF | BibTeX Citation Version: 20100414:055551 (All versions of this report) Short URL: ia.cr/2010/069 Discussion forum: Show discussion | Start new discussion