On Curves over Finite Fields with Jacobians of Small Exponent
نویسندگان
چکیده
We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g = 1. We also show when g = 1 or g = 2 that our bounds are best possible.
منابع مشابه
Constructing Pairing-Friendly Genus 2 Curves with Ordinary Jacobians
We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large prime-order subgroups, and have small embedding degree. Our algorithm is modeled on the Cocks-Pinch method for constructing pairing-friendly elliptic curves [5], and works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are define...
متن کاملConstructing pairing-friendly genus 2 curves over prime fields with ordinary Jacobians
We provide the first explicit construction of genus 2 curves over finite fields whose Jacobians are ordinary, have large prime-order subgroups, and have small embedding degree. Our algorithm works for arbitrary embedding degrees k and prime subgroup orders r. The resulting abelian surfaces are defined over prime fields Fq with q ≈ r. We also provide an algorithm for constructing genus 2 curves ...
متن کاملConstructing pairing-friendly hyperelliptic curves using Weil restriction
A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields Fq whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over Fq that become ...
متن کاملComputing in Picard groups of projective curves over finite fields
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that various desirable operations can be performed efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite mor...
متن کاملFields of definition of torsion points on the Jacobians of genus 2 hyperelliptic curves over finite fields
This paper deals with fields of definition of the l-torsion points on the Jacobians of genus 2 hyperelliptic curves over finite fields in order to speed Gaudry and Schost’s point counting algorithm for genus 2 hyperelliptic curves up. A result in this paper shows that the extension degrees of the fields of difinition of the l-torsion points can be in O(l) instead of O(l). The effects of the res...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007