Number Theory

Computational Methods

11-04 and 11Yxx

  1. Fadwa S. Abu Muriefah, Florian Luca, and Alain Togbé, On the Diophantine equation x2+5a13b=yn, Glasg. Math. J. 50 (2008), no. 1, 175–181.[MR]
  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 (2007), 307–328 (electronic).[MR]
  3. Ali Akhavi and Damien Stehlé, Speeding-up lattice reduction with random projections (extended abstract), LATIN 2008: Theoretical informatics, Lecture Notes in Comput. Sci., vol. 4957, Springer, Berlin, 2008, pp. 293–305.[MR]
  4. Bill Allombert, An efficient algorithm for the computation of Galois automorphisms, Math. Comp. 73 (2004), no. 245, 359–375 (electronic).[MR]
  5. Roberto Maria Avanzi, Another look at square roots (and other less common operations) in fields of even characteristic, Selected Areas in Cryptography, Lecture Notes in Computer Science, vol. 4876/2007, Springer Berlin / Heidelberg, 2007, pp. 138–154.[eprint]
  6. Eric Bach and Denis Charles, The hardness of computing an eigenform, Computational arithmetic geometry, Contemp. Math., vol. 463, Amer. Math. Soc., Providence, RI, 2008, pp. 9–15.[MR/arXiv]
  7. Werner Backes and Susanne Wetzel, An efficient LLL gram using buffered transformations, Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, vol. 4770/2007, Springer Berlin / Heidelberg, 2007, pp. 31–44.
  8. David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, and Eric W. Weisstein, Ten problems in experimental mathematics, Amer. Math. Monthly 113 (2006), no. 6, 481–509.[MR]
  9. Stéphane Ballet, Quasi-optimal algorithms for multiplication in the extensions of \bf F16 of degree 13, 14 and 15, J. Pure Appl. Algebra 171 (2002), no. 2-3, 149–164.[MR]
  10. M. Bauer, M. J. Jacobson, Jr., Y. Lee, and R. Scheidler, Construction of hyperelliptic function fields of high three-rank, Math. Comp. 77 (2008), no. 261, 503–530 (electronic).[MR]
  11. Michael Beck, Eric Pine, Wayne Tarrant, and Kim Yarbrough Jensen, New integer representations as the sum of three cubes, Math. Comp. 76 (2007), no. 259, 1683–1690 (electronic).[MR]
  12. Daniel J. Bernstein, Batch binary edwards, Advances in Cryptology - CRYPTO 2009, Lecture Notes in Comput. Sci., vol. 5677, Springer, Berlin, 2009, pp. 317–336.[doi]
  13. Daniel J. Bernstein, Peter Birkner, Tanja Lange, and Christiane Peters, Optimizing double-base elliptic-curve single-scalar multiplication, Progress in cryptology—INDOCRYPT 2007, Lecture Notes in Comput. Sci., vol. 4859, Springer, Berlin, 2007, pp. 167–182.[MR/doi]
  14. Daniel J. Bernstein, Peter Birkner, Tanja Lange, and Christiane Peters, ECM using Edwards curves, IACR (2008), 18 pages.[eprint]
  15. Daniel J. Bernstein and Tanja Lange, Faster addition and doubling on elliptic curves, Advances in Cryptology - ASIACRYPT 2007, Lecture Notes in Computer Science, vol. 4833/2007, Springer Berlin / Heidelberg, 2007, pp. 29–50.
  16. Amnon Besser and Rob De Jeu, Li(p)-service? an algorithm for computing p-adic polyalgorithms, Math. Comp. 77 (2008), no. 262, 1105–1134.[MR]
  17. Peter Birkner, Efficient divisor class halving on genus two curves, Selected Areas in Cryptography, Lecture Notes in Computer Science, vol. 4356, Springer, Berlin/Heidelberg, pp. 317–326.[link]
  18. Werner Bley and Robert Boltje, Computation of locally free class groups, Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 72–86.[MR/link]
  19. Jonathan Borwein and David Bailey, Mathematics by Experiment, A K Peters Ltd., Natick, MA, 2004, pp. x+288.[MR]
  20. Wieb Bosma, Some computational experiments in number theory, Discovering Mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 1–30.[MR]
  21. Wieb Bosma, John Cannon, and Allan Steel, Lattices of compatibly embedded finite fields, J. Symbolic Comput. 24 (1997), no. 3-4, 351–369.[MR]
  22. Wieb Bosma and Ben Kane, The Aliquot constant, preprint (2009), 16 pages.[arXiv]
  23. Wieb Bosma and Arjen K. Lenstra, An implementation of the elliptic curve integer factorization method, Computational Algebra and Number Theory (Sydney, 1992), Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 119–136.[MR]
  24. Wieb Bosma and Bart de Smit, Class number relations from a computational point of view, J. Symbolic Comput. 31 (2001), no. 1-2, 97–112.[MR]
  25. Wieb Bosma and Bart de Smit, On arithmetically equivalent number fields of small degree, Algorithmic Number Theory (Sydney, 2002), Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 67–79.[MR]
  26. Wieb Bosma and Peter Stevenhagen, Density computations for real quadratic units, Math. Comp. 65 (1996), no. 215, 1327–1337.[MR]
  27. Johan Bosman, On the computation of Galois representations associated to level one modular forms, preprint (2007), 15 pages.[arXiv]
  28. Alin Bostan, Pierrick Gaudry, and Éric Schost, Linear recurrences with polynomial coefficients and computation of the Cartier-Manin operator on hyperelliptic curves, Finite Fields and Applications, Lecture Notes in Comput. Sci., vol. 2948, Springer, Berlin, 2004, pp. 40–58.[MR]
  29. Hans-Christian Graf v. Bothmer, Finite field experiments (with an appendix by Stefan Wiedmann), Higher-Dimensional Geometry over Finite Fields, NATO Science for Peace and Security Series, D: Information and Communication Security, vol. 16, IOS Press, 2008, pp. 1–62.
  30. Aaron Bradord, Michael Monagan, and Colin Percival, Integer factorization and computing discrete logarithms in Maple, in Proceedings of the 2006 Maple Conference, 2006, pp. 2–13.
  31. Richard P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), no. 225, 429–451.[MR]
  32. Richard P. Brent, Recent progress and prospects for integer factorisation algorithms, Computing and Combinatorics (Sydney, 2000), Lecture Notes in Comput. Sci., vol. 1858, Springer, Berlin, 2000, pp. 3–22.[MR]
  33. Richard P. Brent, Note on Marsaglia's xorshift random number generators, J. Stat. Soft 11 (2004), no. 5, 1-5.
  34. Nils Bruin and Michael Stoll, Deciding existence of rational points on curves: an experiment, Experiment. Math. 17 (2008), no. 2, 181–189.[MR/arXiv]
  35. Nils Bruin and Michael Stoll, Two-cover descent on hyperelliptic curves, preprint (2008), 19 pages.[arXiv]
  36. Nils Bruin and Michael Stoll, The Mordell-Weil sieve: Proving non-existence of rational points on curves, LMS J. Comput. Math 13 (2010), 272–306.[arXiv]
  37. David G. Cantor and Daniel M. Gordon, Factoring polynomials over p-adic fields, Algorithmic Number Theory (Leiden, 2000), Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 185–208.[MR]
  38. Robert Carls, Explicit Frobenius lifts on elliptic curves, preprint (2009), 26 pages.[arXiv]
  39. Robert Carls, Fast point counting on genus two curves in characteristic three, preprint (2010).[arXiv]
  40. Wouter Castryck, Hendrik Hubrechts, and Frederik Vercauteren, Computing zeta functions in families of Ca, b curves using deformation, Algorithmic Number Theory, Lecture Notes in Computer Science, vol. 5011, Springer, 2008, pp. 296-311.
  41. Antoine Chambert-Loir, Compter (rapidement) le nombre de solutions d'equations dans les corps finis, preprint (2006), 46 pages.[arXiv]
  42. Hugo Chapdelaine, Computation of p-units in ray class fields of real quadratic number fields, Math. Comp. 78 (2009), 2307–2345.
  43. 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 (2008), 121–155.[MR]
  44. J. E. Cremona and D. Rusin, Efficient solution of rational conics, Math. Comp. 72 (2003), no. 243, 1417–1441 (electronic).[MR]
  45. M. Daberkow, Computing with subfields, J. Symbolic Comput. 24 (1997), no. 3-4, 371–384.[MR]
  46. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283.[MR]
  47. Lassina Dembélé, Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms, Math. Comp. 76 (2007), no. 258, 1039–1057 (electronic).[MR]
  48. Lassina Dembélé, On the computation of algebraic modular forms on compact inner forms of GSp4, preprint (2009), 21 pages.[arXiv]
  49. Lassina Dembélé and Steve Donnelly, Computing Hilbert modular forms over fields with nontrivial class group, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 371–386.[MR/doi]
  50. 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 (2005), no. 1, 65–74.[MR]
  51. Claus Diem, The GHS attack in odd characteristic, J. Ramanujan Math. Soc. 18 (2003), no. 1, 1–32.[MR]
  52. Claus Diem, Index calculus in class groups of plane curves of small degree, Preprint (2005), 43 pages.
  53. Claus Diem, An index calculus algorithm for plane curves of small degree, Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 543–557.[MR]
  54. Jintai Ding, Jason E. Gower, and Dieter S. Schmidt, Zhuang-Zi: a new algorithm for solving multivariate polynomial equations over a finite field, Preprint (2006), 14 pages.[link]
  55. Jacques Dubrois and Jean-Guillaume Dumas, Efficient polynomial time algorithms computing industrial-strength primitive roots, Inform. Process. Lett. 97 (2006), no. 2, 41–45.[MR]
  56. Sylvain Duquesne, Montgomery ladder for all genus 2 curves in characteristic 2, Arithmetic of Finite Fields, Lecture Notes in Computer Science, vol. 5130, Springer, 2008, pp. 174–188.
  57. I. Duursma, P. Gaudry, and F. Morain, Speeding up the discrete log computation on curves with automorphisms, Advances in Cryptology—Asiacrypt'99 (Singapore), Lecture Notes in Comput. Sci., vol. 1716, Springer, Berlin, 1999, pp. 103–121.[MR]
  58. Luca De Feo, Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic, J. Number Theory, to appear (2010), 21 pages.[arXiv]
  59. Claus Fieker, Applications of the class field theory of global fields, Discovering Mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 31–62.[MR]
  60. Claus Fieker, Sparse representation for cyclotomic fields, Experiment. Math. 16 (2007), no. 4, 493–500.[MR]
  61. Claus Fieker and Willem A. de Graaf, Finding integral linear dependencies of algebraic numbers and algebraic Lie algebras, LMS J. Comput. Math. 10 (2007), 271–287 (electronic).[MR]
  62. Claus Fieker and Michael E. Pohst, Dependency of units in number fields, Math. Comp. 75 (2006), no. 255, 1507–1518 (electronic).[MR]
  63. Tom Fisher, The Hessian of a genus one curve, preprint (2006), 28 pages.[arXiv]
  64. Tom Fisher, Some improvements to 4-descent on an elliptic curve, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 125–138.[MR/doi]
  65. Tom Fisher, The invariants of a genus one curve, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 753–782.[MR/arXiv]
  66. E. V. Flynn and C. Grattoni, Descent via isogeny on elliptic curves with large rational torsion subgroups, J. Symbolic Comput. 43 (2008), no. 4, 293–303.[MR]
  67. Felix Fontein, The infrastructure of a global field of arbitrary unit rank, preprint (2008), 36 pages.[arXiv]
  68. Robert Fraatz, Computation of maximal orders of cyclic extensions of function fields, PhD Thesis, Technischen Universtät Berlin, 2005.
  69. David Freeman, Constructing pairing-friendly genus 2 curves with ordinary Jacobians, Pairing-based cryptography—Pairing 2007, Lecture Notes in Comput. Sci., vol. 4575, Springer, Berlin, 2007, pp. 152-176.[MR]
  70. David Mandell Freeman and Takakazu Satoh, Constructing pairing-friendly hyperelliptic curves using Weil restriction, Preprint (2010), 1–31.[link]
  71. P. Gaudry, F. Hess, and N. P. Smart, Constructive and destructive facets of Weil descent on elliptic curves, J. Cryptology 15 (2002), no. 1, 19–46.[MR]
  72. Pierrick Gaudry, An algorithm for solving the discrete log problem on hyperelliptic curves, Advances in Cryptology—Eurocrypt 2000 (Bruges), Lecture Notes in Comput. Sci., vol. 1807, Springer, Berlin, 2000, pp. 19–34.[MR]
  73. Pierrick Gaudry and Nicolas Gürel, An extension of Kedlaya's point-counting algorithm to superelliptic curves, Advances in Cryptology—Asiacrypt 2001 (Gold Coast), Lecture Notes in Comput. Sci., vol. 2248, Springer, Berlin, 2001, pp. 480–494.[MR]
  74. Pierrick Gaudry, Alexander Kruppa, and Paul Zimmermann, A GMP-based implementation of Schönhage-Strassen's large integer multiplication algorithm, ISSAC 2007, ACM, New York, 2007, pp. 167–174.[MR]
  75. 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 (2002), no. 12, 1460–1461.[MR]
  76. Willi Geiselmann and Rainer Steinwandt, A redundant representation of GF(qn) for designing arithmetic circuits, IEEE Trans. Comp 52 (2003), no. 7, 848–853.
  77. Willi Geiselmann and Rainer Steinwandt, Non-wafer-scale sieving hardware for the NFS: another attempt to cope with 1024-bit, Advances in cryptology—EUROCRYPT 2007, Lecture Notes in Comput. Sci., vol. 4515, Springer, Berlin, 2007, pp. 466–481.[MR/link]
  78. 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 (2005), 267–300 (electronic).[MR]
  79. Norbert Goeb, Computing the automorphism groups of hyperelliptic function fields, preprint (2003), 16 pages.[arXiv]
  80. 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 (2009), 2397–2425.
  81. J. Guardia, J. Montes, and E. Nart, Higher Newton polygons and integral bases, preprint (2009).[arXiv]
  82. Jordi Guardia, Jesus Montes, and Enric Nart, Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields, preprint (2008), 24 pages.[arXiv]
  83. Lajos Hajdu, Optimal systems of fundamental S-units for LLL-reduction, Period. Math. Hungar. 59 (2009), no. 1, 53–79.[MR/doi]
  84. 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, ACM, New York, 2001, pp. 175–182 (electronic).[MR]
  85. Guillaume Hanrot and Damien Stehlé, Improved analysis of Kannan's shortest lattice vector algorithm (extended abstract), Advances in cryptology—CRYPTO 2007, Lecture Notes in Comput. Sci., vol. 4622, Springer, Berlin, 2007, pp. 170–186.[MR]
  86. David Harvey, Kedlaya's algorithm in larger characteristic, Int. Math. Res. Not. IMRN (2007), no. 22, Art. ID rnm095, 29.[MR]
  87. David Harvey, A cache-friendly truncated FFT, Theor. Comput. Sci. 410 (2009), no. 27-29, 2649–2658.[arXiv]
  88. Lenwood S. Heath and Nicholas A. Loehr, New algorithms for generating Conway polynomials over finite fields, J. Symbolic Comput. 38 (2004), no. 2, 1003–1024.[MR]
  89. F. Hess, Weil descent attacks, Advances in Elliptic Curve Cryptography, London Math. Soc. Lecture Note Ser., vol. 317, Cambridge Univ. Press, Cambridge, 2005, pp. 151–180.[MR]
  90. Florian Hess, Sebastian Pauli, and Michael E. Pohst, Computing the multiplicative group of residue class rings, Math. Comp. 72 (2003), no. 243, 1531–1548 (electronic).[MR]
  91. Mark van Hoeij, Factoring polynomials and the knapsack problem, J. Number Theory 95 (2002), no. 2, 167–189.[MR]
  92. Hendrik Hubrechts, Point counting in families of hyperelliptic curves, Found. Comput. Math. 8 (2008), no. 1, 137–169.[MR/arXiv]
  93. Hendrik Hubrechts, Quasi-quadratic elliptic curve point counting using rigid cohomology, J. Symb. Comput. 44 (2009), no. 9, 1255–1267.[arXiv]
  94. David Jao and Vladimir Soukharev, A subexponential algorithm for evaluating large degree isogenies, Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 219–233.[doi/arXiv]
  95. 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 (2008), no. 3, 715–732.[MR]
  96. Antoine Joux and Reynald Lercier, Counting points on elliptic curves in medium characteristic, Preprint (2006), 15.[link]
  97. Ben Kane, CM liftings of supersingular elliptic curves, preprint (2009), 26 pages.[arXiv]
  98. Markus Kirschmer and John Voight, Algorithmic enumeration of ideal classes for quaternion orders, SIAM J. Comput 39 (2010), no. 5, 1714–1747.
  99. Jürgen Klüners, Algorithms for function fields, Experiment. Math. 11 (2002), no. 2, 171–181.[MR]
  100. Alan G. B. Lauder, Degenerations and limit Frobenius structures in rigid cohomology, preprint (2009), 41 pages.[arXiv]
  101. Grégoire Lecerf, Fast separable factorization and applications, Appl. Algebra Engrg. Comm. Comput. 19 (2008), no. 2, 135–160.[MR]
  102. D. Lehavi and C. Ritzenthaler, An explicit formula for the arithmetic-geometric mean in genus 3, Experiment. Math. 16 (2007), no. 4, 421–440.[MR/arXiv]
  103. 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 (2008), no. 3, 783–797.[MR/arXiv]
  104. Rudolf Lidl, Computational problems in the theory of finite fields, Appl. Algebra Engrg. Comm. Comput. 2 (1991), no. 2, 81–89.[MR]
  105. J. M. Miret, R. Moreno, J. Pujolàs, and A. Rio, Halving for the 2-Sylow subgroup of genus 2 curves over binary fields, Finite Fields Appl. 15 (2009), no. 5, 569–579.[MR/doi]
  106. Marcel Mohyla and Gabor Wiese, A computational study of the asymptotic behaviour of coefficient fields of modular forms, preprint (2009), 19 pages.[arXiv]
  107. Michael Monagan and Mark van Hoeij, A modular algorithm for computing polynomial GCDs over number fields presented with multiple extensions, preprint http://www.cecm.sfu.ca/CAG/papers/HoeijMonGCD.pdf, 36 pages.
  108. I. Morel, D. Stehlé, and G. Villard, Analyse numerique et reduction de reseaux, Technique et Science Informatiques, to appear (2009), 29 pages.
  109. 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, pp. 660.
  110. Siguna Müller, On the computation of square roots in finite fields, Des. Codes Cryptogr. 31 (2004), no. 3, 301–312.[MR]
  111. Phong Q. Nguên and Damien Stehlé, Floating-point LLL revisited, Advances in cryptology—EUROCRYPT 2005, Lecture Notes in Comput. Sci., vol. 3494, Springer, Berlin, 2005, pp. 215–233.[MR]
  112. 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 (2009), no. 1, 231–245.[doi]
  113. Titus Piezas, Solving solvable sextics using polynomial decomposition, Preprint (2004), 22 pages.
  114. M. E. Pohst, Computational aspects of Kummer theory, Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 259–272.[MR]
  115. Xavier-François Roblot, Polynomial factorization algorithms over number fields, J. Symbolic Comput. 38 (2004), no. 5, 1429–1443.[MR]
  116. Tanaka Satoru and Nakamula Ken, More constructing pairing-friendly elliptic curves for cryptography, preprint (2007), 11 pages.[arXiv]
  117. René Schoof, Computing Arakelov class groups, Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, pp. 447–495.[MR/arXiv]
  118. Nigel P. Smart, The Algorithmic Resolution of Diophantine Equations, London Mathematical Society Student Texts, vol. 41, Cambridge University Press, Cambridge, 1998, pp. xvi+243.[MR]
  119. B. Smith, Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves, J. Cryptology 22 (2009), no. 4, 505–529.
  120. Benjamin Smith, Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves, Advances in Cryptology, Eurocrypt 2008, Lecture Notes in Computer Science, vol. 4965, Springer Berlin/Heidelberg, 2008, pp. 163–180.[eprint]
  121. Damien Stehlé, Floating-point LLL: Theoretical and practical aspects, Proceedings of LLL+25 Conference, 2007 (2009), 36 pages.
  122. Damien Stehlé, Floating-point LLL: Theoretical and practical aspects, in Information Security and Cryptography: The LLL Algorithm, Information Security and Cryptography, Springer, Berlin Heidelberg, 2010, pp. 179–213.[doi]
  123. Damien Stehlé and Paul Zimmermann, A binary recursive GCD algorithm, Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 411–425.[MR]
  124. William A. Stein, An introduction to computing modular forms using modular symbols, Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, pp. 641–652.[MR]
  125. Katsuyuki Takashima, A new type of fast endomorphisms on Jacobians of hyperelliptic curves and their cryptographic application, IEICE Trans. Fundamentals E89-A (2006), no. 1, 124–133.
  126. Nicolas M. Thiéry, Computing minimal generating sets of invariant rings of permutation groups with SAGBI-Gröbner basis, in Discrete Mathematics and Theoretical Computer Science: 4th International Conference, DMTCS 2003, Dijon, France, July 7-12, 2003: Proceedings, Lecture Notes in Computer Science, vol. 2731, Springer, Berlin, 2003, pp. 315–328.
  127. Xavier Taixes i Ventosa and Gabor Wiese, Computing congruences of modular forms and Galois representations modulo prime powers, preprint (2009), 26 pages.[arXiv]
  128. Gilles Villard, Certification of the QR factor R and of lattice basis reducedness, ISSAC 2007, ACM, New York, 2007, pp. 361–368.[MR/arXiv]
  129. P. G. Walsh, On a very particular class of Ramanujan-Nagell type equations, Far East J. Math. Sci. (FJMS) 24 (2007), no. 1, 55–58.[MR]
  130. Kenneth Koon-Ho Wong, Applications of finite field computation to cryptology: Extension field arithmetic in public key systems and algebraic attacks on stream ciphers, PhD Thesis, Queensland University of Technology, 2008.[link]
  131. Paul Zimmermann and Bruce Dodson, 20 years of ECM, Algorithmic Number Theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 525–542.[MR]