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Conference Papers Year : 2013

## Improved CRT Algorithm for Class Polynomials in Genus $2$

Kristin Lauter
• Function : Author
Damien Robert

#### Abstract

We present a generalization to genus~2 of the probabilistic algorithm of Sutherland for computing Hilbert class polynomials. The improvement over the Br{ö}ker-Gruenewald-Lauter algorithm for the genus~2 case is that we do not need to find a curve in the isogeny class whose endomorphism ring is the maximal order; rather, we present a probabilistic algorithm for ''going up'' to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of $(\ell,\ell)$-isogenies to compute all isogenous maximal curves from an initial one. This is an extended version of the article published at ANTS~X.

#### Domains

Computer Science [cs] Symbolic Computation [cs.SC]

### Dates and versions

hal-00734450 , version 1 (21-09-2012)
hal-00734450 , version 2 (17-04-2013)

### Identifiers

• HAL Id : hal-00734450 , version 2
• DOI :

### Cite

Kristin Lauter, Damien Robert. Improved CRT Algorithm for Class Polynomials in Genus $2$. ANTS X - Algorithmic Number Theory 2012, Jul 2012, San Diego, United States. pp.437-461, ⟨10.2140/obs.2013.1.437⟩. ⟨hal-00734450v2⟩

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