On the expansion length of triple-base number systems

Abstract

Triple-base number systems are mainly used in elliptic curve cryptography to speed up scalar multiplication. We give an upper bound on the length of the canonical triple-base representation with base {2, 3, 5} of an integer x, which is by the greedy algorithm, and show that there are infinitely many integers x whose shortest triple-base representations with base {2, 3, 5} have length greater than where c is a positive constant, using the universal exponent method. This analysis gives a limit how much scalar multiplication on elliptic curves may be made faster. © 2013 Springer-Verlag Berlin Heidelberg.