Efficient Double Bases for Scalar Multiplication

In this paper we present efficient algorithms to take advantage of the double-base number system in the context of elliptic curve scalar multiplication. We propose a generalized version of Yao´s exponentiation algorithm allowing the use of general double-base expansions instead of the popular double base chains. We introduce a class of constrained double base expansions and prove that the average density of non-zero terms in such expansions is O( log k/ log log k) for any large integer k. We also propose an efficient algorithm for computing constrained expansions and finally provide a comprehensive comparison to double-base chain expansions, including a large variety of curve shapes and various key sizes.