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Some remarks on the construction of class polynomials
1. | Department of Information and Communication Systems Engineering, University of the Aegean, 83200, Samos, Greece |
2. | Department of Mathematics, University of Athens, Panepistimioupolis 15784, Athens, Greece |
References:
[1] |
L. V. Ahlfors, "Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable,'' $3^{rd}$ edition, McGraw-Hill Book Co., New York, 1978. |
[2] |
J. Belding, R. Bröker, A. Enge and K. Lauter, Computing Hilbert class polynomials, in "Algorithmic Number Theory Symposium - ANTS 2008,'' Springer-Verlag, (2008), 282-295. |
[3] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: the user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[4] |
R. Bröker, A $p$-adic algorithm to compute the Hilbert class polynomial, Math. Compu., 77 (2008), 2417-2435.
doi: 10.1090/S0025-5718-08-02091-7. |
[5] |
A. Enge, The complexity of class polynomial computation via floating point approximations, Math. Compu., 78 (2009), 1089-1107.
doi: 10.1090/S0025-5718-08-02200-X. |
[6] |
A. Enge and F. Morain, Comparing invariants for class fields of imaginary quadratic fields, in "Algorithmic Number Theory Symposium - ANTS 2002,'' Springer-Verlag, (2002), 252-266. |
[7] |
A. Enge and R. Schertz, Constructing elliptic curves over finite fields using double eta-quotients, J. Théor. Nombres Bordeaux, 16 (2004), 555-568. |
[8] |
A. Enge and A. V. Sutherland, Class invariants for the CRT method, in "Algorithmic Number Theory Symposium - ANTS 2010,'' Springer-Verlag, (2010), 142-156. |
[9] |
E. Konstantinou and A. Kontogeorgis, Computing polynomials of the Ramanujan $t_n$ class invariants, Canadian Math. Bull., 52 (2009), 583-597.
doi: 10.4153/CMB-2009-058-6. |
[10] |
E. Konstantinou and A. Kontogeorgis, Ramanujan's class invariants and their use in elliptic curve cryptography, Comput. Math. Appl., 59 (2010), 2901-2917.
doi: 10.1016/j.camwa.2010.02.008. |
[11] |
G. J. Lay and H. Zimmer, Constructing elliptic curves with given group order over large finite fields, in "Algorithmic Number Theory - ANTS-I,'' Springer-Verlag, (1994), 250-263. |
[12] |
F. Morain, Modular curves and class invariants,, preprint., ().
|
[13] |
F. Morain, Advances in the CM method for elliptic curves, in "Slides of Fields Cryptography Retrospective Meeting,'' (2009); available online at http://www.lix.polytechnique.fr/~morain/Exposes/fields09.pdf |
[14] |
W. Narkiewicz, "Elementary and Analytic Theory of Algebraic Numbers,'' $2^{nd}$ edition, Springer-Verlag, 1990. |
[15] |
R. Schertz, Weber's class invariants revisited, J. Théor. Nombres Bordeaux, 4 (2002), 325-343. |
[16] |
A. V. Sutherland, Computing Hilbert class polynomials with the Chinese Remainder Theorem,, Math. Compu., ().
|
show all references
References:
[1] |
L. V. Ahlfors, "Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable,'' $3^{rd}$ edition, McGraw-Hill Book Co., New York, 1978. |
[2] |
J. Belding, R. Bröker, A. Enge and K. Lauter, Computing Hilbert class polynomials, in "Algorithmic Number Theory Symposium - ANTS 2008,'' Springer-Verlag, (2008), 282-295. |
[3] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: the user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[4] |
R. Bröker, A $p$-adic algorithm to compute the Hilbert class polynomial, Math. Compu., 77 (2008), 2417-2435.
doi: 10.1090/S0025-5718-08-02091-7. |
[5] |
A. Enge, The complexity of class polynomial computation via floating point approximations, Math. Compu., 78 (2009), 1089-1107.
doi: 10.1090/S0025-5718-08-02200-X. |
[6] |
A. Enge and F. Morain, Comparing invariants for class fields of imaginary quadratic fields, in "Algorithmic Number Theory Symposium - ANTS 2002,'' Springer-Verlag, (2002), 252-266. |
[7] |
A. Enge and R. Schertz, Constructing elliptic curves over finite fields using double eta-quotients, J. Théor. Nombres Bordeaux, 16 (2004), 555-568. |
[8] |
A. Enge and A. V. Sutherland, Class invariants for the CRT method, in "Algorithmic Number Theory Symposium - ANTS 2010,'' Springer-Verlag, (2010), 142-156. |
[9] |
E. Konstantinou and A. Kontogeorgis, Computing polynomials of the Ramanujan $t_n$ class invariants, Canadian Math. Bull., 52 (2009), 583-597.
doi: 10.4153/CMB-2009-058-6. |
[10] |
E. Konstantinou and A. Kontogeorgis, Ramanujan's class invariants and their use in elliptic curve cryptography, Comput. Math. Appl., 59 (2010), 2901-2917.
doi: 10.1016/j.camwa.2010.02.008. |
[11] |
G. J. Lay and H. Zimmer, Constructing elliptic curves with given group order over large finite fields, in "Algorithmic Number Theory - ANTS-I,'' Springer-Verlag, (1994), 250-263. |
[12] |
F. Morain, Modular curves and class invariants,, preprint., ().
|
[13] |
F. Morain, Advances in the CM method for elliptic curves, in "Slides of Fields Cryptography Retrospective Meeting,'' (2009); available online at http://www.lix.polytechnique.fr/~morain/Exposes/fields09.pdf |
[14] |
W. Narkiewicz, "Elementary and Analytic Theory of Algebraic Numbers,'' $2^{nd}$ edition, Springer-Verlag, 1990. |
[15] |
R. Schertz, Weber's class invariants revisited, J. Théor. Nombres Bordeaux, 4 (2002), 325-343. |
[16] |
A. V. Sutherland, Computing Hilbert class polynomials with the Chinese Remainder Theorem,, Math. Compu., ().
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