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
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