Type-II Optimal Polynomial Bases

Abstract

In the 1990s and early 2000s several papers investigated the relative merits of polynomial-basis and normal-basis computations for F$_{2^n}$. Even for particularly squaring-friendly applications, such as implementations of Koblitz curves, normal bases fell behind in performance unless a type-I normal basis existed for F$_{2^n}$.

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.

Permanent ID of this document: 90995f3542ee40458366015df5f2b9de. Date of this document: 2010.04.12. This work has been supported in part by the European Commission through the ICT Programme under Contract ICT–2007–216676 ECRYPT-II and in part by the National Science Foundation under grant ITR–0716498.