Two Lower Bounds for Shortest Double-Base Number System

Parinya CHALERMSOOK; Hiroshi IMAI; Vorapong SUPPAKITPAISARN

doi:10.1587/transfun.E98.A.1310

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Two Lower Bounds for Shortest Double-Base Number System

Parinya CHALERMSOOK, Hiroshi IMAI, Vorapong SUPPAKITPAISARN

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Keywords: analysis of algorithms, number representation, elliptic curve cryptography, double-base number system, double-base chain

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2015 Volume E98.A Issue 6 Pages 1310-1312

DOI https://doi.org/10.1587/transfun.E98.A.1310

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Abstract

In this letter, we derive two lower bounds for the number of terms in a double-base number system (DBNS), when the digit set is {1}. For a positive integer n, we show that the number of terms obtained from the greedy algorithm proposed by Dimitrov, Imbert, and Mishra [1] is $\Theta\left(\frac{\log n}{\log \log n}\right)$. Also, we show that the number of terms in the shortest double-base chain is Θ(log n).

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