Optimizing double-base elliptic-curve single-scalar multiplication

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

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review


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.
Keyword: edwards curves,scalar multiplication,Algorithms,double-base chains,addition chains,Computation theory,Double-base number systems,quintupling,T,tripling,COMPUTER,double-base number systems,Problem solving,Optimization
Original languageEnglish
Title of host publicationPROGRESS IN CRYPTOLOGY - INDOCRYPT 2007
VolumeVolume 4859
PublisherSpringer Verlag, Berlin
Publication date2007
ISBN (Print)978-35-40-77025-1
Publication statusPublished - 2007
Externally publishedYes
Event8th International Conference on Cryptology in India: Progress in Cryptology - Chennai, India
Duration: 9 Dec 200713 Dec 2007
Conference number: 8


Conference8th International Conference on Cryptology in India
Internet address
SeriesLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberVolume 4859


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