**Efficient Scalar Multiplication by Isogeny Decompositions**

*Christophe Doche and Thomas Icart and David R. Kohel*

**Abstract: **On an elliptic curve, the degree of an isogeny corresponds essentially to
the degrees of the polynomial expressions involved in its application.
The multiplication--by--$\ell$ map $[\ell]$ has degree~$\ell^2$, therefore
the complexity to directly evaluate $[\ell](P)$ is $O(\ell^2)$.
For a small prime $\ell\, (= 2, 3)$ such that the additive binary
representation provides no better performance, this represents the true
cost of application of scalar multiplication.
If an elliptic curves admits an isogeny $\varphi$ of degree $\ell$ then
the costs of computing $\varphi(P)$ should in contrast be $O(\ell)$ field
operations. Since we then have a product expression $[\ell] =
\hat{\varphi}\varphi$, the existence of an $\ell$-isogeny $\varphi$ on
an elliptic curve yields a theoretical improvement from $O(\ell^2)$ to
$O(\ell)$ field operations for the evaluation of $[\ell](P)$ by na\"ive
application of the defining polynomials. In this work we investigate
actual improvements for small $\ell$ of this asymptotic complexity.
For this purpose, we describe the general construction of families of
curves with a suitable decomposition $[\ell] = \hat{\varphi}\varphi$,
and provide explicit examples of such a family of curves with simple
decomposition for~$[3]$. Finally we derive a new tripling algorithm
to find complexity improvements to triplication on a curve in certain
projective coordinate systems, then combine this new operation to non-adjacent forms
for $\ell$-adic expansions in order to obtain an improved
strategy for scalar multiplication on elliptic curves.

**Category / Keywords: **public-key cryptography / elliptic curve cryptosystem

**Date: **received 21 Nov 2005, last revised 22 Nov 2005

**Contact author: **doche at ics mq edu au

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

**Version: **20051123:064010 (All versions of this report)

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