Welcome to the resource topic for
**2022/1704**

**Title:**

Some applications of higher dimensional isogenies to elliptic curves (preliminary version)

**Authors:**
Damien Robert

**Abstract:**

We give two 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 Δ_π.

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}(n m \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. This algorithm generalizes

to ordinary abelian varieties.

**ePrint:**
https://eprint.iacr.org/2022/1704

See all topics related to this paper.

Feel free to post resources that are related to this paper below.

**Example resources include:**
implementations, explanation materials, talks, slides, links to previous discussions on other websites.

For more information, see the rules for Resource Topics .