## Elliptic curves over finite fields with fixed

Citations: | 1 - 1 self |

### BibTeX

@MISC{Najman_ellipticcurves,

author = {Filip Najman},

title = {Elliptic curves over finite fields with fixed},

year = {}

}

### OpenURL

### Abstract

subgroups

### Citations

822 | The Arithmetic of Elliptic Curves - Silverman - 1986 |

186 |
Speeding the Pollard and elliptic curve methods of factorization
- Montgomery
- 1987
(Show Context)
Citation Context ...tic curve, and see that not all elliptic curves have the same properties. In the elliptic curve factoring method [19], one hopes that the order of E(Fp) is smooth. Atkin and Morain [1] and Montgomery =-=[23]-=- suggested using elliptic curves with large rational torsion, because the torsion subgroup injects into E(Fp) for all except a few p. This makes the order of the elliptic curve divisible by the order ... |

30 | Finding suitable curves for the elliptic curve method of factorization
- Atkin, Morain
- 1993
(Show Context)
Citation Context ...at we fix the elliptic curve, and see that not all elliptic curves have the same properties. In the elliptic curve factoring method [19], one hopes that the order of E(Fp) is smooth. Atkin and Morain =-=[1]-=- and Montgomery [23] suggested using elliptic curves with large rational torsion, because the torsion subgroup injects into E(Fp) for all except a few p. This makes the order of the elliptic curve div... |

25 |
Torsion points on elliptic curves and q-coefficients of modular forms
- Kamienny
- 1992
(Show Context)
Citation Context ...s way get an elliptic curve E such that for a large density of the primes p ∈ P, G is isomorphic to a subgroup of E(Fp). Currently all the possible torsion groups over quadratic fields are known (see =-=[16]-=- and [17]) and all the torsion groups over cubic (see [12]) and quartic (see [13]) fields that appear infinitely often. One can find how to construct elliptic curves with given torsion over cubic and ... |

23 |
Primality of the number of points on an elliptic curve over a finite field, Pacific J.Math.131
- Koblitz
- 1988
(Show Context)
Citation Context ...ic curves for integer factorization with elliptic curves. 1 Introduction The order and group structure of an elliptic curve over a finite field is of great theoretical and practical interest. Koblitz =-=[18]-=- considered, for a fixed rational elliptic curve E, the probability of |E(Fp)| to be prime as p varies through the primes. Galbraith and McKee [9] examined the probability for |E(Fp)| to be prime or a... |

21 |
Lenstra Jr. Factoring integers with elliptic curves
- W
- 1987
(Show Context)
Citation Context ...ical properties are computed. Our paper is different in the sense that we fix the elliptic curve, and see that not all elliptic curves have the same properties. In the elliptic curve factoring method =-=[19]-=-, one hopes that the order of E(Fp) is smooth. Atkin and Morain [1] and Montgomery [23] suggested using elliptic curves with large rational torsion, because the torsion subgroup injects into E(Fp) for... |

19 | Torsion points on elliptic curves defined over quadratic fields
- Kenku, Momose
- 1988
(Show Context)
Citation Context ... an elliptic curve E such that for a large density of the primes p ∈ P, G is isomorphic to a subgroup of E(Fp). Currently all the possible torsion groups over quadratic fields are known (see [16] and =-=[17]-=-) and all the torsion groups over cubic (see [12]) and quartic (see [13]) fields that appear infinitely often. One can find how to construct elliptic curves with given torsion over cubic and quartic n... |

15 |
On the group orders of elliptic curves over finite fields
- Howe
- 1993
(Show Context)
Citation Context ... elliptic curve E, the probability of |E(Fp)| to be prime as p varies through the primes. Galbraith and McKee [9] examined the probability for |E(Fp)| to be prime or a small multiple of a prime. Howe =-=[11]-=- fixed the finite field Fp and studied the probability of E(Fp) to be a given group. In [7], [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E... |

15 |
On the torsion of elliptic curves over quartic number fields
- Jeon, Kim, et al.
(Show Context)
Citation Context ...G is isomorphic to a subgroup of E(Fp). Currently all the possible torsion groups over quadratic fields are known (see [16] and [17]) and all the torsion groups over cubic (see [12]) and quartic (see =-=[13]-=-) fields that appear infinitely often. One can find how to construct elliptic curves with given torsion over cubic and quartic number fields in [14] and [15]. For larger groups, that not appear over f... |

14 | Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
- Cojocaru, Murty
(Show Context)
Citation Context ...ed the probability of E(Fp) to be a given group. In [7], [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], =-=[6]-=-, [25] and [26]. A method for constructing curves with a given number of points are given in [3]. This paper is probably most similar to [10], where probabilities for a random curve over a random fini... |

13 |
On the torsion of elliptic curves over cubic number fields, Acta Arithmetica 113 (2004), 291–301
- Jeon, Kim, et al.
(Show Context)
Citation Context ...y of the primes p ∈ P, G is isomorphic to a subgroup of E(Fp). Currently all the possible torsion groups over quadratic fields are known (see [16] and [17]) and all the torsion groups over cubic (see =-=[12]-=-) and quartic (see [13]) fields that appear infinitely often. One can find how to construct elliptic curves with given torsion over cubic and quartic number fields in [14] and [15]. For larger groups,... |

10 | The probability that the number of points on an elliptic curve over a finite field is prime
- McKee
(Show Context)
Citation Context ...of great theoretical and practical interest. Koblitz [18] considered, for a fixed rational elliptic curve E, the probability of |E(Fp)| to be prime as p varies through the primes. Galbraith and McKee =-=[9]-=- examined the probability for |E(Fp)| to be prime or a small multiple of a prime. Howe [11] fixed the finite field Fp and studied the probability of E(Fp) to be a given group. In [7], [20] and [21] bo... |

8 |
Cyclicity statistics for elliptic curves over finite fields
- Vlădut¸
- 1999
(Show Context)
Citation Context ...e probability of E(Fp) to be a given group. In [7], [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], [6], =-=[25]-=- and [26]. A method for constructing curves with a given number of points are given in [3]. This paper is probably most similar to [10], where probabilities for a random curve over a random finite fie... |

7 |
On the cyclicity of the group of Fp-rational points of non-CM elliptic curves
- Cojocaru
- 2002
(Show Context)
Citation Context ... and studied the probability of E(Fp) to be a given group. In [7], [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied =-=[4]-=-, [5], [6], [25] and [26]. A method for constructing curves with a given number of points are given in [3]. This paper is probably most similar to [10], where probabilities for a random curve over a r... |

6 | The distribution of group structures on elliptic curves over finite prime fields, Documenta Mathematica 11
- Gekeler
- 2006
(Show Context)
Citation Context .... The probability that E(Fp) is cyclic is studied [4], [5], [6], [25] and [26]. A method for constructing curves with a given number of points are given in [3]. This paper is probably most similar to =-=[10]-=-, where probabilities for a random curve over a random finite field to have certain group-theoretical properties are computed. Our paper is different in the sense that we fix the elliptic curve, and s... |

6 | On the exponent of the group of points on elliptic curves in extension fields
- Luca, Shparlinski
(Show Context)
Citation Context ...th and McKee [9] examined the probability for |E(Fp)| to be prime or a small multiple of a prime. Howe [11] fixed the finite field Fp and studied the probability of E(Fp) to be a given group. In [7], =-=[20]-=- and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], [6], [25] and [26]. A method for constructing curves with a g... |

5 |
Cojocaru, Cyclicity of CM elliptic curves modulo p
- C
(Show Context)
Citation Context ...studied the probability of E(Fp) to be a given group. In [7], [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], =-=[5]-=-, [6], [25] and [26]. A method for constructing curves with a given number of points are given in [3]. This paper is probably most similar to [10], where probabilities for a random curve over a random... |

5 |
Almost all reductions modulo p of an elliptic curve have a large exponent
- Duke
(Show Context)
Citation Context ...lbraith and McKee [9] examined the probability for |E(Fp)| to be prime or a small multiple of a prime. Howe [11] fixed the finite field Fp and studied the probability of E(Fp) to be a given group. In =-=[7]-=-, [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], [6], [25] and [26]. A method for constructing curves wi... |

5 | Torsion subgroups of elliptic curves in elementary abelian 2-extensions of Q - Fujita |

4 | Equations for the modular curve X1(N) and models of elliptic curves with torsion points
- Baaziz
(Show Context)
Citation Context ...moment is to try to find points of relatively low degree on X1(m, n), the modular curve characterizing elliptic curves with torsion subgroup Zm ⊕ Zn. Note that one can find nice models of X1(1, N) in =-=[2]-=-. Note that the standard heuristics is that larger torsion of E(Q) implies a greater probability that |E(Fp)| is smooth. From the proof of Theorem 2, one can see that this is not necessary so, as a cu... |

4 | Efficient CM-constructions of elliptic curves over finite fields
- Bröker, Stevenhagen
(Show Context)
Citation Context ...largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], [6], [25] and [26]. A method for constructing curves with a given number of points are given in =-=[3]-=-. This paper is probably most similar to [10], where probabilities for a random curve over a random finite field to have certain group-theoretical properties are computed. Our paper is different in th... |

4 |
Small exponent point groups on elliptic curves
- Luca, McKee, et al.
(Show Context)
Citation Context ...Kee [9] examined the probability for |E(Fp)| to be prime or a small multiple of a prime. Howe [11] fixed the finite field Fp and studied the probability of E(Fp) to be a given group. In [7], [20] and =-=[21]-=- bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], [6], [25] and [26]. A method for constructing curves with a given numb... |

4 | Subtleties in the distribution of the numbers of points on elliptic curves over a finite prime field - McKee - 1999 |

2 |
On the cyclicity of elliptic curves over finite field extensions
- Vlădut¸
- 1999
(Show Context)
Citation Context ...lity of E(Fp) to be a given group. In [7], [20] and [21] bounds for the exponent (largest order of a point) of E(Fp) are given. The probability that E(Fp) is cyclic is studied [4], [5], [6], [25] and =-=[26]-=-. A method for constructing curves with a given number of points are given in [3]. This paper is probably most similar to [10], where probabilities for a random curve over a random finite field to hav... |