Extending Scalar Multiplication Using Double Bases

It has been recently acknowledged [4, 6, 9] that the use of double bases representations of scalars n, that is an expression of the form n = ∑ e,s,t(−1) AB can speed up significantly scalar multiplication on those elliptic curves where multiplication by one base (say B) is fast. This is the case in particular of Koblitz curves and supersingular curves, where scalar multiplication can now be achieved in o(logn) curve additions. Previous literature dealt basically with supersingular curves (in characteristic 3, although the methods can be easily extended to arbitrary characteristic), where A,B ∈ N. Only [4] attempted to provide a similar method for Koblitz curves, where at least one basemust be non-real, although their method does not seem practical for cryptographic sizes (it is only asymptotic), since the constants involved are too large. We provide here a unifying theory by proposing an alternate recoding algorithm which works in all cases with optimal constants. Furthermore, it can also solve the until now untreatable case where bothA andB are nonreal. The resulting scalar multiplication method is then compared to standardmethods for Koblitz curves. It runs in less than logn/ log logn elliptic curve additions, and is faster than any given method with similar storage requirements already on the curve K-163, with larger improvements as the size of the curve increases, surpassing 50% with respect to the τ -NAF for the curves K-409 and K-571. With respect of windowed methods, that can approach our speed but requireO(log(n)/ log log(n)) precomputations for optimal parameters, we offer the advantage of a fixed, small memory footprint, as we need storage for at most two additional points. ? Partially supported by the European Commission through the IST Programme under Contract IST-2002-507932 ECRYPT. ?? This work was partially supported by a NSERC Discovery Grant