Fast scalar multiplication for ECC over GF(p) using division chains

Abstract

There have been many recent developments in formulae for efficient composite elliptic curve operations of the form dP+Q for a small integer d and points P and Q where the underlying field is a prime field. To make best use of these in a scalar multiplication kP, it is necessary to generate an efficient "division chain" for the scalar where divisions of k are by the values of d available through composite operations. An algorithm-generating algorithm for this is presented that takes into account the different costs of using various representations for curve points. This extends the applicability of methods presented by Longa & Gebotys at PKC 2009 to using specific characteristics of the target device. It also enables the transfer of some scalar recoding computation details to design time. An improved cost function also provides better evaluation of alternatives in the relevant addition chain. One result of these more general and improved methods includes a slight increase over the scalar multiplication speeds reported at PKC. Furthermore, by the straightforward removal of rules for unusual cases, some particularly concise yet efficient presentations can be given for algorithms in the target device. © 2011 Springer-Verlag.