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 \(\mathcal{O}(\frac{\log x}{\log\log x})\) 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 \(\frac{c\log x}{\log\log x\log\log\log x},\) 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.

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