Extending scalar multiplication using double bases.

*(English)*Zbl 1172.94558
Lai, Xuejia (ed.) et al., Advances in cryptology – ASIACRYPT 2006. 12th international conference on the theory and application of cryptology and information security, Shanghai, China, December 3–7, 2006. Proceedings. Berlin: Springer (ISBN 978-3-540-49475-1/pbk). Lecture Notes in Computer Science 4284, 130-144 (2006).

Summary: It has been recently acknowledged that the use of double bases representations of scalars \(n\), that is an expression of the form \(n = \sum _{e,s,t} (-1)^{e} A^{s} B^{t}\) 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 \(o(\log n)\) curve additions.

Previous literature dealt basically with supersingular curves (in characteristic 3, although the methods can be easily extended to arbitrary characteristic), where \(A,B \in \mathbb N\). Only R. Avanzi and F. Sica [“Scalar multiplication on Koblitz curves using double bases”, Lect. Notes Comput. Sci. 4341, 131–146 (2006)] attempted to provide a similar method for Koblitz curves, where at least one base must 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 both \(A\) and \(B\) are non-real. The resulting scalar multiplication method is then compared to standard methods for Koblitz curves. It runs in less than \(\log n/\log \log n\) 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 \(\tau \)-NAF for the curves K-409 and K-571. With respect of windowed methods, that can approach our speed but require \(O(\log (n)/\log \log (n))\) precomputations for optimal parameters, we offer the advantage of a fixed, small memory footprint, as we need storage for at most two additional points.

For the entire collection see [Zbl 1133.94007].

Previous literature dealt basically with supersingular curves (in characteristic 3, although the methods can be easily extended to arbitrary characteristic), where \(A,B \in \mathbb N\). Only R. Avanzi and F. Sica [“Scalar multiplication on Koblitz curves using double bases”, Lect. Notes Comput. Sci. 4341, 131–146 (2006)] attempted to provide a similar method for Koblitz curves, where at least one base must 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 both \(A\) and \(B\) are non-real. The resulting scalar multiplication method is then compared to standard methods for Koblitz curves. It runs in less than \(\log n/\log \log n\) 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 \(\tau \)-NAF for the curves K-409 and K-571. With respect of windowed methods, that can approach our speed but require \(O(\log (n)/\log \log (n))\) precomputations for optimal parameters, we offer the advantage of a fixed, small memory footprint, as we need storage for at most two additional points.

For the entire collection see [Zbl 1133.94007].