1. A CM construction of curves of genus 2 with p-rank 1
- Author
-
Laura Hitt O'Connor, Marco Streng, Michael Naehrig, Gary McGuire, Discrete Mathematics, and Coding Theory and Cryptology
- Subjects
Rank (linear algebra) ,Genus-2 curves ,Mathematics::Number Theory ,Complex multiplication ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Genus (mathematics) ,Weil numbers ,p-rank ,FOS: Mathematics ,Order (group theory) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Discrete mathematics ,14H40 ,Algebra and Number Theory ,Group (mathematics) ,Explicit CM constructions ,010102 general mathematics ,Embedding degree ,Finite field ,Family of curves ,Embedding - Abstract
We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field $\F_{p^2}$ of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of $\F_{p^2}$-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over $\F_{p^2}$ out of necessity: we show that curves of $p$-rank 1 over $\F_p$ for large $p$ cannot be efficiently constructed using explicit CM constructions., Comment: 19 pages
- Published
- 2011