On the Expansion Length Of Triple-Base Number Systems

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 O( log x log log x ) 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 c log x log log x log log log x , 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.