**Scalar Multiplication on Koblitz Curves using Double Bases**

*Roberto Avanzi and Francesco Sica*

**Abstract: **The paper is an examination of double-base decompositions of
integers $n$, namely expansions loosely of the form
$$
n = \sum_{i,j} A^iB^j
$$
for some base $\{A,B\}$. This was examined in previous
works in the case when $A,B$ lie in
$\mathbb{N}$.

On the positive side, we show how to extend previous results of to Koblitz curves over binary fields. Namely, we obtain a sublinear scalar algorithm to compute, given a generic positive integer $n$ and an elliptic curve point $P$, the point $nP$ in time $O\left(\frac{\log n}{\log\log n}\right)$ elliptic curve operations with essentially no storage, thus making the method asymptotically faster than any know scalar multiplication algorithm on Koblitz curves.

On the negative side, we analyze scalar multiplication using double base numbers and show that on a generic elliptic curve over a finite field, we cannot expect a sublinear algorithm with double bases. Finally, we show that all algorithms used hitherto need at least $\frac{\log n}{\log\log n}$ curve operations.

**Category / Keywords: **implementation / elliptic curve cryptosystem, fast endomorphisms, number theory

**Date: **received 21 Feb 2006

**Contact author: **fsica at mta ca

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20060223:223826 (All versions of this report)

**Short URL: **ia.cr/2006/067

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