# A note on elliptic curves over finite fields

@article{Rck1987ANO, title={A note on elliptic curves over finite fields}, author={Hans-Georg R{\"u}ck}, journal={Mathematics of Computation}, year={1987}, volume={49}, pages={301-304} }

Let E be an elliptic curve over a finite field k and let E(k) be the group of k-rational points on E. We evaluate all the possible groups E(k) where E runs through all the elliptic curves over a given fixed finite field k.

## 65 Citations

Group Structure of Elliptic Curves over Finite Fields

- Mathematics
- 2001

Abstract Let A be a finite abelian group such that there is an elliptic curve defined over a finite field F q with E( F q)≅A. We will determine the possible group structures E( F qk) as E varies over…

Topic In Elliptic Curves Over Finite Fields: The Groups of Points

- Mathematics, Computer Science
- 2011

This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves…

Counting Prime Divisors on Elliptic Curves and Multiplication in Finite Fields

- Mathematics
- 2000

Let K/F q be an elliptic function field. For every natural number n we determine the number of prime divisors of degree n of K/F q which lie in a given divisor class of K.

A note on divisor class groups of degree zero of algebraic function fields over finite fields

- Mathematics
- 2003

On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields

- MathematicsExp. Math.
- 2012

Some of the results are rigorous and are based on recent advances in analytic number theory; some are conditional under certain widely believed conjectures; and others are purely heuristic in nature.

Elliptic curves over finite fields : number theoretic and cryptographic aspects

- Mathematics, Computer Science
- 2013

Several natural questions about elliptic curves are presented, mostly over finite fields, that have led to some interesting number theoretic questions and whose solutions require rather involved techniques from various area of number theory.

On the lattices from elliptic curves over finite fields

- Mathematics, Computer ScienceFinite Fields Their Appl.
- 2015

On Group Structures Realized by Elliptic Curves over a Fixed Finite Field

- MathematicsExp. Math.
- 2012

These formulas are used to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves according to their group structure.

ON CURVES OVER FINITE FIELDS WITH JACOBIANS OF SMALL EXPONENT

- Mathematics, Computer Science
- 2006

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 Duke in the case of g = 1. We…

Constructions of Sequences from Algebraic Curves over Finite Fields

- Mathematics, Computer ScienceSETA
- 2001

This paper surveys some recent constructions of various sequences based on algebraic curves over finite fields and includes sequences with good linear complexity profiles and sequence families with both large linear complexities and low correlation.

## References

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We shall generalize to Abelian varieties the well-known fact that an elliptic curve over a finite field always has a rational point (see Theorem 3). Our first theorem is purely algebraic and…

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Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its…

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- Variétés Abèliennes et Courbes Algébriques
- 1948

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

- 1948