Cryptology ePrint Archive: Report 2007/414

Optimizing double-base elliptic-curve single-scalar multiplication

Daniel J. Bernstein and Peter Birkner and Tanja Lange and Christiane Peters

Abstract: This paper analyzes the best speeds that can be obtained for single-scalar multiplication with variable base point by combining a huge range of options:

many choices of coordinate systems and formulas for individual group operations, including new formulas for tripling on Edwards curves;

double-base chains with many different doubling/tripling ratios, including standard base-2 chains as an extreme case;

many precomputation strategies, going beyond Dimitrov, Imbert, Mishra (Asiacrypt 2005) and Doche and Imbert (Indocrypt 2006).

The analysis takes account of speedups such as S-M tradeoffs and includes recent advances such as inverted Edwards coordinates.

The main conclusions are as follows. Optimized precomputations and triplings save time for single-scalar multiplication in Jacobian coordinates, Hessian curves, and tripling-oriented Doche/Icart/Kohel curves. However, even faster single-scalar multiplication is possible in Jacobi intersections, Edwards curves, extended Jacobi-quartic coordinates, and inverted Edwards coordinates, thanks to extremely fast doublings and additions; there is no evidence that double-base chains are worthwhile for the fastest curves. Inverted Edwards coordinates are the speed leader.

Category / Keywords: public-key cryptography / Edwards curves, double-base number systems, double-base chains, addition chains, scalar multiplication, tripling, quintupling

Date: received 28 Oct 2007, last revised 28 Oct 2007

Contact author: tanja at hyperelliptic org

Available format(s): PDF | BibTeX Citation

Version: 20071106:084026 (All versions of this report)

Discussion forum: Show discussion | Start new discussion


[ Cryptology ePrint archive ]