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

@article{Farashahi2012OnGS,
  title={On Group Structures Realized by Elliptic Curves over a Fixed Finite Field},
  author={Reza Rezaeian Farashahi and Igor E. Shparlinski},
  journal={Experimental Mathematics},
  year={2012},
  volume={21},
  pages={1 - 10}
}
We obtain explicit formulas for the number of nonisomorphic elliptic curves with a given group structure (considered as an abstract abelian group) and the number of distinct group structures of all elliptic curves over a finite field. We use these formulas 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. Finally, we present results of some numerical tests that… 
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References

SHOWING 1-10 OF 16 REFERENCES
On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields
TLDR
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.
Nonsingular plane cubic curves over finite fields
  • R. Schoof
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1987
A note on elliptic curves over finite fields
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
Factoring integers with elliptic curves
TLDR
This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2.
Abelian Varieties over ℂ
These lecture notes present, in outline, the theory of abelian varieties over the complex numbers. They focus mainly on the analytic side of the subject. In the first section we prove some basic
ABELIAN VARIETIES OVER FINITE FIELDS.
  • S. Lang
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1955
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
Elliptic Curves: Number Theory and Cryptography
TLDR
This book discusses Elliptic Curve Cryptography, a Cryptosystem Based on Factoring and its Applications, and some of the techniques used to develop such systems.
Handbook of Elliptic and Hyperelliptic Curve Cryptography
TLDR
The introduction to Public-Key Cryptography explains the development of algorithms for computing Discrete Logarithms and their applications in Pairing-Based Cryptography and its applications in Fast Arithmetic Hardware Smart Cards.
On Artin's Conjecture for Primitive Roots
Various generalizations of the Artin’s Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform
A note on elliptic curves over finite fields
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