Optimizing Double-Base Elliptic-Curve Single-Scalar Multiplication

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.