论文信息 - On Group Structures Realized by Elliptic Curves over a Fixed Finite Field

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

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 exhibit several interesting phenomena in the distribution of group structures.

Igor E. Shparlinski | Reza Rezaeian Farashahi | I. Shparlinski | R. R. Farashahi

[1] L. Washington. Elliptic Curves: Number Theory and Cryptography , 2003 .

[2] Hans-Georg Rück. A note on elliptic curves over finite fields , 1987 .

[3] S. Lang,et al. Abelian varieties over finite fields , 2005 .

[4] H. W. Lenstra,et al. Factoring integers with elliptic curves , 1987 .

[5] E. T.. An Introduction to the Theory of Numbers , 1946, Nature.

[6] C. Pomerance,et al. Prime Numbers: A Computational Perspective , 2002 .

[7] S. G. Vladut,et al. Cyclicity Statistics for Elliptic Curves over Finite Fields , 1999 .

[8] J. Voloch. A note on elliptic curves over finite fields , 1988 .

[9] D. R. Heath-Brown. ARTIN'S CONJECTURE FOR PRIMITIVE ROOTS , 1986 .

[10] René Schoof,et al. Nonsingular plane cubic curves over finite fields , 1987, J. Comb. Theory A.

[11] Tanja Lange,et al. Handbook of Elliptic and Hyperelliptic Curve Cryptography , 2005 .

[12] Francesco Pappalardi,et al. On Artin's Conjecture for Primitive Roots , 1993 .

[13] Igor E. Shparlinski,et al. On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields , 2010, Exp. Math..

[14] Michael Rosen. Abelian Varieties over ℂ , 1986 .