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Preprints, Working Papers, ... Year : 2022

Towards computing canonical lifts of ordinary elliptic curves in medium characteristic

Abstract

Let $p$ be a prime; using modular polynomial $\Phi_p$, T.~Satoh and al \cite{satoh2000canonical,harley2002,vercau} developed several algorithms to compute the canonical lift of an ordinary elliptic curve $E$ over $\F_{p^n}$ with $j$-invariant not in $\F_{p^2}$. When $p$ is constant, the best variant has a complexity $\Otilde(n m)$ to lift $E$ to $p$-adic precision~$m$. As an application, lifting $E$ to precision $m=O(n)$ allows to recover its cardinality in time $\Otilde(n^2)$. However, taking $p$ into account the complexity is $\Otilde(p^2 n m)$, so Satoh's algorithm can only be applied to small~$p$. We propose in this paper two variants of these algorithms, which do not rely on the modular polynomial, for computing the canonical lift of an ordinary curve. Our new method yield a complexity of $\Otilde(p n m)$ to lift at precision~$m$, and even $\Otilde(\sqrt{p} nm)$ when we are provided a rational point of $p$-torsion on the curve. This allows to extend Saoth's point counting algorithm to larger~$p$.
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Dates and versions

hal-03702658 , version 1 (26-07-2022)

Identifiers

  • HAL Id : hal-03702658 , version 1

Cite

Abdoulaye Maiga, Damien Robert. Towards computing canonical lifts of ordinary elliptic curves in medium characteristic. 2022. ⟨hal-03702658⟩
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