Section: New Results

Point counting on curves with real multiplication

Pierrick Gaudry, David Kohel and Benjamin Smith have designed a new variant of the Schoof algorithm for point counting on hyperelliptic curves, that can take advantage of the presence of the knowledge of an explicit and efficient endomorphism coming from real multiplication. In that case, the overall complexity drops from $\tilde{O}\left({log}^{8}q\right)$ to $\tilde{O}\left({log}^{5}q\right)$. Using our algorithm we have computed a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian. The corresponding paper [19] obtained the Best Paper Award at the Asiacrypt 2011 conference.