Paper 2006/067

Scalar Multiplication on Koblitz Curves using Double Bases

Roberto Avanzi and Francesco Sica


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.

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
elliptic curve cryptosystemfast endomorphismsnumber theory
Contact author(s)
fsica @ mta ca
2006-02-23: received
Short URL
Creative Commons Attribution


      author = {Roberto Avanzi and Francesco Sica},
      title = {Scalar Multiplication on Koblitz Curves using Double Bases},
      howpublished = {Cryptology ePrint Archive, Paper 2006/067},
      year = {2006},
      note = {\url{}},
      url = {}
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