Paper 2022/1704

Some applications of higher dimensional isogenies to elliptic curves (overview of results)

Abstract

We give some applications of the "embedding Lemma". The first one is a polynomial time (in $\log q$) algorithm to compute the endomorphism ring $\mathrm{End}(E)$ of an ordinary elliptic curve $E/{\mathbb{F}}_q$, provided we are given the factorisation of ${\Delta }_{\pi }$. In particular, this computation can be done in quantum polynomial time. The second application is an algorithm to compute the canonical lift of $E/{\mathbb{F}}_q$, $q=p^n$, (still assuming that $E$ is ordinary) to precision $m$ in time $\tilde{O}(nm{\log }^{O(1)}p)$. We deduce a point counting algorithm of complexity $\tilde{O}(n^2{\log }^{O(1)}p)$. In particular the complexity is polynomial in $\log p$, by contrast of what is usually expected of a $p$-adic cohomology computation. The third application is a quasi-linear CRT algorithm to compute Siegel modular polynomials of elliptic curves, which does not rely on any heuristic or conditional result (like GRH). We also outline how to generalize these algorithms to (ordinary) abelian varieties.

Note: Added a new section on computing modular polynomial. Also added a subsection explaining the link between canonical lifts and crystalline cohomology, and why lifts do not seem to help attack (commutative) isogeny based cryptography.