| [1] |
Fadwa S. Abu Muriefah, Florian Luca, and Alain Togbé.
On the Diophantine equation x² + 5a13b = yn.
Glasg. Math. J., 50(1):175–181, 2008. |
| [2] |
Fatima K. Abu Salem and Kamal Khuri-Makdisi.
Fast Jacobian group operations for C3, 4 curves over a
large finite field.
LMS J. Comput. Math., 10:307–328 (electronic), 2007. |
| [3] |
Ali Akhavi and Damien Stehlé.
Speeding-up lattice reduction with random projections (extended
abstract).
In LATIN 2008: Theoretical informatics, volume 4957 of Lecture Notes in Comput. Sci., pages 293–305. Springer, Berlin, 2008. |
| [4] |
Bill Allombert.
An efficient algorithm for the computation of Galois automorphisms.
Math. Comp., 73(245):359–375 (electronic), 2004. |
| [5] |
Roberto Maria Avanzi.
Another look at square roots (and other less common operations) in
fields of even characteristic.
In Selected Areas in Cryptography, volume 4876/2007 of Lecture Notes in Computer Science, pages 138–154. Springer Berlin /
Heidelberg, 2007. |
| [6] |
Eric Bach and Denis Charles.
The hardness of computing an eigenform.
In Computational arithmetic geometry, volume 463 of Contemp. Math., pages 9–15. Amer. Math. Soc., Providence, RI, 2008. |
| [7] |
Werner Backes and Susanne Wetzel.
An efficient LLL gram using buffered transformations.
In Computer Algebra in Scientific Computing, volume 4770/2007
of Lecture Notes in Computer Science, pages 31–44. Springer Berlin /
Heidelberg, 2007. |
| [8] |
David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, and Eric W. Weisstein.
Ten problems in experimental mathematics.
Amer. Math. Monthly, 113(6):481–509, 2006. |
| [9] |
Stéphane Ballet.
Quasi-optimal algorithms for multiplication in the extensions of
F16 of degree 13, 14 and 15.
J. Pure Appl. Algebra, 171(2-3):149–164, 2002. |
| [10] |
M. Bauer, M. J. Jacobson, Jr., Y. Lee, and R. Scheidler.
Construction of hyperelliptic function fields of high three-rank.
Math. Comp., 77(261):503–530 (electronic), 2008. |
| [11] |
Michael Beck, Eric Pine, Wayne Tarrant, and Kim Yarbrough Jensen.
New integer representations as the sum of three cubes.
Math. Comp., 76(259):1683–1690 (electronic), 2007. |
| [12] |
Daniel J. Bernstein, Peter Birkner, Tanja Lange, and Christiane Peters.
Optimizing double-base elliptic-curve single-scalar multiplication.
In Progress in Cryptology - INDOCRYPT 2007, volume 4859/2007 of
Lecture Notes in Computer Science, pages 167–182. Springer Berlin /
Heidelberg, 2007. |
| [13] |
Daniel J. Bernstein, Peter Birkner, Tanja Lange, and Christiane Peters.
ECM using Edwards curves.
IACR eprint:2008:016, 18 pages, 2008. |
| [14] |
Daniel J. Bernstein and Tanja Lange.
Faster addition and doubling on elliptic curves.
In Advances in Cryptology - ASIACRYPT 2007, volume 4833/2007 of
Lecture Notes in Computer Science, pages 29–50. Springer Berlin /
Heidelberg, 2007. |
| [15] |
Amnon Besser and Rob De Jeu.
li(p)-service? an algorithm for computing p-adic
polyalgorithms.
Math. Comp., 77(262):1105–1134, 2008. |
| [16] |
Peter Birkner.
Efficient divisor class halving on genus two curves.
In Selected Areas in Cryptography, volume 4356 of Lecture
Notes in Computer Science, pages 317–326. Springer, Berlin/Heidelberg. |
| [17] |
Werner Bley and Robert Boltje.
Computation of locally free class groups.
In Algorithmic Number Theory, volume 4076 of Lecture
Notes in Comput. Sci., pages 72–86. Springer, Berlin, 2006. |
| [18] |
Jonathan Borwein and David Bailey.
Mathematics by Experiment.
A K Peters Ltd., Natick, MA, 2004. |
| [19] |
Wieb Bosma.
Some computational experiments in number theory.
In Discovering Mathematics with Magma, volume 19 of Algorithms Comput. Math., pages 1–30. Springer, Berlin, 2006. |
| [20] |
Wieb Bosma, John Cannon, and Allan Steel.
Lattices of compatibly embedded finite fields.
J. Symbolic Comput., 24(3-4):351–369, 1997. |
| [21] |
Wieb Bosma and Bart de Smit.
Class number relations from a computational point of view.
J. Symbolic Comput., 31(1-2):97–112, 2001. |
| [22] |
Wieb Bosma and Bart de Smit.
On arithmetically equivalent number fields of small degree.
In Algorithmic Number Theory (Sydney, 2002), volume 2369 of
Lecture Notes in Comput. Sci., pages 67–79. Springer, Berlin, 2002. |
| [23] |
Wieb Bosma and Arjen K. Lenstra.
An implementation of the elliptic curve integer factorization method.
In Computational Algebra and Number Theory (Sydney,
1992), volume 325 of Math. Appl., pages 119–136. Kluwer Acad. Publ.,
Dordrecht, 1995. |
| [24] |
Wieb Bosma and Peter Stevenhagen.
Density computations for real quadratic units.
Math. Comp., 65(215):1327–1337, 1996. |
| [25] |
Johan Bosman.
On the computation of Galois representations associated to level
one modular forms.
arXiv:0710.1237v1, 15 pages, 2007. |
| [26] |
Alin Bostan, Pierrick Gaudry, and Éric Schost.
Linear recurrences with polynomial coefficients and computation of
the Cartier-Manin operator on hyperelliptic curves.
In Finite Fields and Applications, volume 2948 of Lecture Notes in Comput. Sci., pages 40–58. Springer, Berlin, 2004. |
| [27] |
Aaron Bradord, Michael Monagan, and Colin Percival.
Integer factorization and computing discrete logarithms in Maple.
In Proceedings of the 2006 Maple Conference, pages 2–13, 2006. |
| [28] |
Richard P. Brent.
Factorization of the tenth Fermat number.
Math. Comp., 68(225):429–451, 1999. |
| [29] |
Richard P. Brent.
Recent progress and prospects for integer factorisation algorithms.
In Computing and Combinatorics (Sydney, 2000), volume 1858 of
Lecture Notes in Comput. Sci., pages 3–22. Springer, Berlin, 2000. |
| [30] |
Richard P. Brent.
Note on Marsaglia's xorshift random number generators.
J. Stat. Soft, 11(5):1–5, 2004. |
| [31] |
Nils Bruin and Michael Stoll.
Deciding existence of rational points on curves: an experiment.
Experiment. Math., 17(2):181–189, 2008. |
| [32] |
Nils Bruin and Michael Stoll.
Two-cover descent on hyperelliptic curves.
arXiv:0803.2052v1 [math.NT], 19 pages, 2008. |
| [33] |
David G. Cantor and Daniel M. Gordon.
Factoring polynomials over p-adic fields.
In Algorithmic Number Theory (Leiden, 2000), volume 1838 of
Lecture Notes in Comput. Sci., pages 185–208. Springer, Berlin, 2000. |
| [34] |
Wouter Castryck, Hendrik Hubrechts, and Frederik Vercauteren.
Computing zeta functions in families of Ca, b curves using
deformation.
In Algorithmic Number Theory, volume 5011 of Lecture Notes
in Computer Science, pages 296–311. Springer, 2008. |
| [35] |
Antoine Chambert-Loir.
Compter (rapidement) le nombre de solutions d'equations dans les
corps finis.
arXiv:math.NT/0611584, 46 pages, 2006. |
| [36] |
Hugo Chapdelaine.
Computation of p-units in ray class fields of real quadratic number
fields.
Math. Comp., 78:2307–2345, 2009. |
| [37] |
J. E. Cremona, T. A. Fisher, C. O'Neil, D. Simon, and M. Stoll.
Explicit n-descent on elliptic curves. I. Algebra.
J. Reine Angew. Math., 615:121–155, 2008. |
| [38] |
J. E. Cremona and D. Rusin.
Efficient solution of rational conics.
Math. Comp., 72(243):1417–1441 (electronic), 2003. |
| [39] |
M. Daberkow.
Computing with subfields.
J. Symbolic Comput., 24(3-4):371–384, 1997. |
| [40] |
M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig,
and K. Wildanger.
KANT V4.
J. Symbolic Comput., 24(3-4):267–283, 1997. |
| [41] |
Lassina Dembélé.
Quaternionic Manin symbols, Brandt matrices, and Hilbert
modular forms.
Math. Comp., 76(258):1039–1057 (electronic), 2007. |
| [42] |
Lassina Dembélé and Steve Donnelly.
Computing Hilbert modular forms over fields with nontrivial class
group.
In Algorithmic Number Theory, volume 5011 of Lecture Notes
in Computer Science, pages 371–386. Springer Berlin / Heidelberg, 2008. |
| [43] |
Francisco Diaz y Diaz, Jean-François Jaulent, Sebastian Pauli, Michael
Pohst, and Florence Soriano-Gafiuk.
A new algorithm for the computation of logarithmic l-class groups
of number fields.
Experiment. Math., 14(1):65–74, 2005. |
| [44] |
Claus Diem.
Index calculus in class groups of plane curves of small degree.
Preprint, 43 pages, 2005. |
| [45] |
Claus Diem.
An index calculus algorithm for plane curves of small degree.
In Algorithmic Number Theory, volume 4076 of Lecture
Notes in Comput. Sci., pages 543–557. Springer, Berlin, 2006. |
| [46] |
Jintai Ding, Jason E. Gower, and Dieter S. Schmidt.
Zhuang-Zi: A new algorithm for solving multivariate polynomial
equations over a finite field.
Preprint, 14 pages, 2006. |
| [47] |
Jacques Dubrois and Jean-Guillaume Dumas.
Efficient polynomial time algorithms computing industrial-strength
primitive roots.
Inform. Process. Lett., 97(2):41–45, 2006. |
| [48] |
Sylvain Duquesne.
Montgomery ladder for all genus 2 curves in characteristic 2.
In Arithmetic of Finite Fields, volume 5130 of Lecture
Notes in Computer Science, pages 174–188. Springer, 2008. |
| [49] |
I. Duursma, P. Gaudry, and F. Morain.
Speeding up the discrete log computation on curves with
automorphisms.
In Advances in Cryptology—Asiacrypt'99 (Singapore), volume
1716 of Lecture Notes in Comput. Sci., pages 103–121. Springer,
Berlin, 1999. |
| [50] |
Claus Fieker.
Applications of the class field theory of global fields.
In Discovering Mathematics with Magma, volume 19 of Algorithms Comput. Math., pages 31–62. Springer, Berlin, 2006. |
| [51] |
Claus Fieker.
Sparse representation for cyclotomic fields.
Experiment. Math., 16(4):493–500, 2007. |
| [52] |
Claus Fieker and Willem A. de Graaf.
Finding integral linear dependencies of algebraic numbers and
algebraic Lie algebras.
LMS J. Comput. Math., 10:271–287 (electronic), 2007. |
| [53] |
Claus Fieker and Michael E. Pohst.
Dependency of units in number fields.
Math. Comp., 75(255):1507–1518 (electronic), 2006. |
| [54] |
Tom Fisher.
The Hessian of a genus one curve.
arXiv:math.NT/0610403, 28 pages, 2006. |
| [55] |
Tom Fisher.
The invariants of a genus one curve.
Proc. Lond. Math. Soc. (3), 97(3):753–782, 2008. |
| [56] |
E. V. Flynn and C. Grattoni.
Descent via isogeny on elliptic curves with large rational torsion
subgroups.
J. Symbolic Comput., 43(4):293–303, 2008. |
| [57] |
Felix Fontein.
The infrastructure of a global field of arbitrary unit rank.
arXiv:0809.1685, 36 pages, 2008. |
| [58] |
Robert Fraatz.
Computation of Maximal Orders of Cyclic Extensions of Function
Fields.
PhD Thesis, Technischen Universtät Berlin, 2005. |
| [59] |
David Freeman.
Constructing pairing-friendly genus 2 curves with ordinary
Jacobians.
In Pairing-based cryptography—Pairing 2007, volume 4575 of
Lecture Notes in Comput. Sci., pages 152–176. Springer, Berlin, 2007. |
| [60] |
Pierrick Gaudry.
An algorithm for solving the discrete log problem on hyperelliptic
curves.
In Advances in Cryptology—Eurocrypt 2000 (Bruges), volume
1807 of Lecture Notes in Comput. Sci., pages 19–34. Springer, Berlin,
2000. |
| [61] |
Pierrick Gaudry, Alexander Kruppa, and Paul Zimmermann.
A GMP-based implementation of Schönhage-Strassen's large
integer multiplication algorithm.
In ISSAC 2007, pages 167–174. ACM, New York, 2007. |
| [62] |
Willi Geiselmann, Jörn Müller-Quade, and Rainer Steinwandt.
Comment on: ``A new representation of elements of finite fields
GF(2m) yielding small complexity arithmetic circuits'' by G.
Drolet.
IEEE Trans. Comput., 51(12):1460–1461, 2002. |
| [63] |
Willi Geiselmann and Rainer Steinwandt.
A redundant representation of GF(qn) for designing
arithmetic circuits.
IEEE Trans Comp, 52(7):848–853, 2003. |
| [64] |
Willi Geiselmann and Rainer Steinwandt.
Non-wafer-scale sieving hardware for the NFS: another attempt to
cope with 1024-bit.
In Advances in cryptology—EUROCRYPT 2007, volume 4515 of
Lecture Notes in Comput. Sci., pages 466–481. Springer, Berlin, 2007. |
| [65] |
Martine Girard and Leopoldo Kulesz.
Computation of sets of rational points of genus-3 curves via the
Demjanenko-Manin method.
LMS J. Comput. Math., 8:267–300 (electronic), 2005. |
| [66] |
Norbert Goeb.
Computing the automorphism groups of hyperelliptic function fields.
arXiv:math.NT/0305284, 16 pages, 2003. |
| [67] |
Edray Goins, Florian Luca, and Alain Togbé.
On the diophantine equation x ² + 2α5β13γ = y n .
In Algorithmic Number Theory, volume 5011 of Lecture Notes
in Computer Science, pages 430–442. Springer Berlin / Heidelberg, 2008. |
| [68] |
Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein, and Corina
Tarnita.
Computational verification of the birch and swinnerton-dyer
conjecture for individual elliptic curves.
Math. Comp, 78:2397–2425, 2009. |
| [69] |
Jordi Guardia, Jesus Montes, and Enric Nart.
Higher Newton polygons in the computation of discriminants and
prime ideal decomposition in number fields.
arXiv:0807.4065v3 [math.NT], 24 pages, 2008. |
| [70] |
G. Hanrot and F. Morain.
Solvability by radicals from an algorithmic point of view.
In Proceedings of the 2001 International Symposium on Symbolic
and Algebraic Computation, pages 175–182 (electronic), New York, 2001. ACM. |
| [71] |
Guillaume Hanrot and Damien Stehlé.
Improved analysis of Kannan's shortest lattice vector algorithm
(extended abstract).
In Advances in cryptology—CRYPTO 2007, volume 4622 of Lecture Notes in Comput. Sci., pages 170–186. Springer, Berlin, 2007. |
| [72] |
David Harvey.
A cache-friendly truncated FFT.
Theor. Comput. Sci., 410(27-29):2649–2658, 2009. |
| [73] |
Lenwood S. Heath and Nicholas A. Loehr.
New algorithms for generating Conway polynomials over finite
fields.
J. Symbolic Comput., 38(2):1003–1024, 2004. |
| [74] |
Florian Hess, Sebastian Pauli, and Michael E. Pohst.
Computing the multiplicative group of residue class rings.
Math. Comp., 72(243):1531–1548 (electronic), 2003. |
| [75] |
Hendrik Hubrechts.
Point counting in families of hyperelliptic curves.
Found. Comput. Math., 8(1):137–169, 2008. |
| [76] |
Hendrik Hubrechts.
Quasi-quadratic elliptic curve point counting using rigid cohomology.
J. Symb. Comput., 44(9):1255–1267, 2009. |
| [77] |
Jean-François Jaulent, Sebastian Pauli, Michael E. Pohst, and Florence
Soriano-Gafiuk.
Computation of 2-groups of positive classes of exceptional number
fields.
J. Théor. Nombres Bordeaux, 20(3):715–732, 2008. |
| [78] |
Antoine Joux and Reynald Lercier.
Counting points on elliptic curves in medium characteristic.
Preprint, page 15, 2006. |
| [79] |
Markus Kirschmer and John Voight.
Algorithmic enumeration of ideal classes for quaternion orders.
arXiv:0808.3833v1 [math.NT], 29 pages, 2008. |
| [80] |
Jürgen Klüners.
Algorithms for function fields.
Experiment. Math., 11(2):171–181, 2002. |
| [81] |
Grégoire Lecerf.
Fast separable factorization and applications.
Appl. Algebra Engrg. Comm. Comput., 19(2):135–160, 2008. |
| [82] |
Reynald Lercier and Thomas Sirvent.
On Elkies subgroups of l-torsion points in elliptic curves
defined over a finite field.
J. Théor. Nombres Bordeaux, 20(3):783–797, 2008. |
| [83] |
J.M. Miret, R. Moreno, J. Pujolas, and A. Rio.
Halving for the 2-Sylow subgroup of genus 2 curves over binary
fields.
Finite Fields Appl., 15(5):569–579, 2009. |
| [84] |
Michael Monagan and Mark van Hoeij.
A modular algorithm for computing polynomial GCDs over number
fields presented with multiple extensions.
http://www.cecm.sfu.ca/CAG/papers/HoeijMonGCD.pdf, 36 pages. |
| [85] |
I. Morel, D. Stehlé, and G. Villard.
Analyse numerique et reduction de reseaux.
Technique et Science Informatiques, To appear, 29 pages, 2009. |
| [86] |
J.-M. Muller, N. Brisebarre, F. de Dinechin, C.-P. Jeannerod, L. Vincent,
G. Melquiond, N. Revol, D. Stehlé, and S. Torres.
Handbook of Floating-point Arithmetic.
Birkhäuser, Boston, MA, 2009. |
| [87] |
Siguna Müller.
On the computation of square roots in finite fields.
Des. Codes Cryptogr., 31(3):301–312, 2004. |
| [88] |
Phong Q. Nguyen and Damien Stehlé.
Floating-point LLL revisited.
In Advances in cryptology—EUROCRYPT 2005, volume 3494 of
Lecture Notes in Comput. Sci., pages 215–233. Springer, Berlin, 2005. |
| [89] |
Harris Nover.
Computation of Galois groups associated to the 2-class towers of
some imaginary quadratic fields with 2-class group c2 ×c2 ×c2.
Journal of Number Theory, 129(1):231 – 245, 2009. |
| [90] |
Titus Piezas.
Solving solvable sextics using polynomial decomposition.
Preprint, 22 pages, 2004. |
| [91] |
M. E. Pohst.
Computational aspects of Kummer theory.
In Algorithmic number theory (Talence, 1996), volume 1122 of
Lecture Notes in Comput. Sci., pages 259–272. Springer, Berlin, 1996. |
| [92] |
Xavier-François Roblot.
Polynomial factorization algorithms over number fields.
J. Symbolic Comput., 38(5):1429–1443, 2004. |
| [93] |
Tanaka Satoru and Nakamula Ken.
More constructing pairing-friendly elliptic curves for cryptography.
arXiv:0711.1942, 11 pages, 2007. |
| [94] |
René Schoof.
Computing Arakelov class groups.
In Algorithmic number theory: lattices, number fields, curves
and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages
447–495. Cambridge Univ. Press, Cambridge, 2008. |
| [95] |
Nigel P. Smart.
The Algorithmic Resolution of Diophantine Equations,
volume 41 of London Mathematical Society Student Texts.
Cambridge University Press, Cambridge, 1998. |
| [96] |
B. Smith.
Isogenies and the discrete logarithm problem in Jacobians of genus
3 hyperelliptic curves.
J. Cryptology, 22(4):505–529, 2009. |
| [97] |
Benjamin Smith.
Isogenies and the discrete logarithm problem in Jacobians of genus
3 hyperelliptic curves.
In Advances in Cryptology, Eurocrypt 2008, volume 4965 of Lecture Notes in Computer Science, pages 163–180. Springer
Berlin/Heidelberg, 2008. |
| [98] |
Damien Stehlé.
Floating-point LLL: Theoretical and practical aspects.
Proceedings of LLL+25 Conference, 2007, 36 pages, 2009. |
| [99] |
Damien Stehlé and Paul Zimmermann.
A binary recursive GCD algorithm.
In Algorithmic Number Theory, volume 3076 of Lecture
Notes in Comput. Sci., pages 411–425. Springer, Berlin, 2004. |
| [100] |
Katsuyuki Takashima.
A new type of fast endomorphisms on Jacobians of hyperelliptic
curves and their cryptographic application.
IEICE Trans. Fundamentals, E89-A(1):124–133, 2006. |
| [101] |
Hans-Christian Graf v. Bothmer.
Finite field experiments (with an appendix by Stefan Wiedmann).
In Higher-Dimensional Geometry over Finite Fields, volume 16 of
NATO Science for Peace and Security Series, D: Information and
Communication Security, pages 1–62. 2008. |
| [102] |
Mark van Hoeij.
Factoring polynomials and the knapsack problem.
J. Number Theory, 95(2):167–189, 2002. |
| [103] |
Gilles Villard.
Certification of the QR factor R and of lattice basis
reducedness.
In ISSAC 2007, pages 361–368. ACM, New York, 2007. |
| [104] |
P. G. Walsh.
On a very particular class of Ramanujan-Nagell type equations.
Far East J. Math. Sci. (FJMS), 24(1):55–58, 2007. |
| [105] |
Paul Zimmermann and Bruce Dodson.
20 years of ECM.
In Algorithmic Number Theory, volume 4076 of Lecture
Notes in Comput. Sci., pages 525–542. Springer, Berlin, 2006. |
