Compter (rapidement) le nombre de solutions d'équations dans les corps finis
Séminaire Bourbaki - Volume 2006/2007 - Exposés 967-981, Astérisque, no. 317 (2008), Talk no. 968, p. 39-90
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     author = {Chambert-Loir, Antoine},
     title = {Compter (rapidement) le nombre de solutions d'\'equations dans les corps finis},
     booktitle = {S\'eminaire Bourbaki - Volume 2006/2007  - Expos\'es 967-981},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {317},
     year = {2008},
     note = {talk:968},
     pages = {39-90},
     zbl = {1189.11059},
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Chambert-Loir, Antoine. Compter (rapidement) le nombre de solutions d'équations dans les corps finis, in Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981, Astérisque, no. 317 (2008), Talk no. 968, pp. 39-90. http://www.numdam.org/item/AST_2008__317__39_0/

[1] T. G. Abott, K. S. Kedlaya & D. Roe - Bounding Picard numbers of surfaces using p-adic cohomology, in Arithmetic, Geometry, and Coding Theory (AGCT-10), 2006, arXiv:math.NT/0601508. | MR 2856564 | Zbl 1214.14007

[2] L. M. Adleman & M.-D. Huang - Primality testing and abelian varieties over finite fields, Lecture Notes in Mathematics, vol. 1512, Springer, 1992. | MR 1176511 | Zbl 0744.11065

[3] L. M. Adleman & M.-D. Huang, Counting points on curves and abelian varieties over finite fields, J. Symbolic Comput. 32 (2001), p. 171-189. | Article | MR 1851164 | Zbl 0986.11039

[4] A. O. L. Atkin - The number of points of an elliptic curve modulo a prime, 1988.

[5] A. O. L. Atkin, The number of points of an elliptic curve modulo a prime (II), 1992.

[6] E. Bach - Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), p. 355-380. | Article | MR 1023756 | Zbl 0701.11075

[7] F. Baldassarri & B. Chiarellotto - Algebraic versus rigid cohomology with logarithmic coefficients, in Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., vol. 15, Académie Press, 1994, p. 11-50. | Article | MR 1307391 | Zbl 0833.14010

[8] P. Berthelot - Géométrie rigide et cohomologie des variétés algébriques de caractéristique p, Mém. Soc. Math. France (1986), p. 7-32, Introductions aux cohomologies p-adiques (Luminy, 1984). | Article | Numdam | MR 865810 | Zbl 0606.14017

[9] P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres. Première partie, Prépublication, IRMAR, Université Rennes 1, 1996.

[10] P. Berthelot, Dualité de Poincaré et formule de Kùnneth en cohomologie rigide, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), p. 493-498. | Article | MR 1692313 | Zbl 0908.14006

[11] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), p. 329-377, avec un appendice en anglais par A. J. de Jong. | Article | MR 1440308 | Zbl 0908.14005

[12] I. F. Blake, G. Seroussi & N. Smart - Elliptic curves in cryptography, London Mathematical Society Lecture Note Series, vol. 265, Cambridge University Press, 2000, Reprint of the 1999 original. | MR 1771549 | Zbl 0937.94008

[13] I. F. Blake, G. Seroussi & N. Smart (éds.) - Advances in elliptic curve cryptography, London Mathematical Society Lecture Note Series, vol. 317, Cambridge University Press, 2005. | Zbl 1089.94018

[14] E. Bombieri - On exponential sums in finite fields. II, Invent. Math. 47 (1978), p. 29-39. | Article | MR 506272 | Zbl 0396.14001

[15] S. Bosch - A rigid analytic version of M. Artin's theorem on analytic équations, Math. Ann. 255 (1981), p. 395-404. | Article | MR 615859 | Zbl 0462.14002

[16] W. Bosma, J. Cannon & C. Playoust - The Magma algebra System. I. The user language, J. Symbolic Comput. 24 (1997), p. 235-265, Computational algebra and number theory (London, 1993). | Article | MR 1484478 | Zbl 0898.68039

[17] J.-B. Bost & J.-F. Mestre - Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2, Gaz. Math. (1988), p. 36-64. | MR 970659 | Zbl 0682.14031

[18] A. Bostan, P. Gaudry & É. Schost - Linear récurrences with polynomial coefficients and application to integer factorization and Cartier-Manin operator, SIAM Journal on Computing 36 (2007), p. 1777-1806. | Article | MR 2299425 | Zbl 1210.11126

[19] C. Breuil, B. Conrad, F. Diamond & R. Taylor - On the modularity of elliptic curves over 𝐐 : wild 3-adic exercises, Amer. J. Math. 14 (2001), p. 843-939. | Article | MR 1839918 | Zbl 0982.11033

[20] D. G. Cantor - On the analogue of the division polynomials for hyperelliptic curves, J. reine angew. Math. 447 (1994), p. 91-145. | MR 1263171 | Zbl 0788.14026

[21] R. Carls - A generalized arithmetico-geometric mean, Thèse, Rijksuniversiteit Groningen, 2004.

[22] R. Carls, Canonical coordinates on the canonical lift, The Ramanujan Journal 22 (2007), p. 1-14, arXiv:math/0508007. | MR 2312545 | Zbl 1138.14031

[23] R. Carls & D. Lubicz - A p-adic quasi-quadratic point counting algorithm, 2007, arXiv:math/0706.0234. | MR 2480098 | Zbl 1160.11034

[24] W. Castryck. J. Denef & F. Vercauteren - Computing Zeta Functions of Nondegenerate Curves, 2006, arXiv:math.NT/0607308. | MR 2268492 | Zbl 1161.14302

[25] B. Chiarellotto - Weights in rigid cohomology applications to unipotent F-isocrystals, Ann. Sci. École Norm. Sup. 31 (1998), p. 683-715. | Article | Numdam | MR 1643966 | Zbl 0933.14008

[26] B. Chiarellotto & B. Le Stum - A comparison theorem for weights, J. reine angew. Math. 546 (2002), p. 159-176. | MR 1900997 | Zbl 0993.14006

[27] H. Cohen - A course in computational algebraic number theory, Graduate Texts in Mathématics, vol. 138, Springer, 1993. | MR 1228206 | Zbl 0786.11071

[28] H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen & F. Vercauteren (éds.) - Handbook of elliptic and hyperelliptic curve cryptography, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2006. | MR 2162716 | Zbl 1082.94001

[29] P. Cohen - On the coefficients of the transformation polynomials for the elliptic modular function, Math. Proc. Cambridge Philos. Soc. 95 (1984), p. 389-402. | Article | MR 755826 | Zbl 0541.10026

[30] J.-M. Couveignes - Computing -isogenies using the p-torsion, in Algorithme number theory (Talence, 1996), Lecture Notes in Comput. Sci., vol. 1122, Springer, 1996, p. 59-65. | Article | MR 1446498 | Zbl 0903.11030

[31] P. Deligne - Équations différentielles à points singuliers réguliers, Springer, 1970, Lecture Notes in Mathematics, Vol. 163. | MR 417174 | Zbl 0244.14004

[32] P. Deligne, Formes modulaires et représentations -adiques, in Séminaire Bourbaki 1968/69, Lecture Notes in Math., 1971, Exp. 355, p. 139-172. | Article | Numdam | MR 3077124 | Zbl 0206.49901

[33] P. Deligne, Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), p. 5-57. | Article | Numdam | MR 498551 | Zbl 0219.14007

[34] P. Deligne, La conjecture de Weil, I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), p. 273-307. | Article | Numdam | MR 340258 | Zbl 0287.14001

[35] P. Deligne, Cohomologie étale (Séminaire de géométrie algébrique du Bois-Marie - SGA 4 1 2 ), Lecture Notes in Mathematics, vol. 569, Springer, 1977, Avec la collaboration de J.-F. Boutot, A. Grothendieck, L. Illusie et J.-L. Verdier. | MR 463174 | Zbl 0345.00010

[36] P. Deligne, La conjecture de Weil, II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), p. 137-252. | Article | Numdam | MR 601520 | Zbl 0456.14014

[37] J. Denef & F. Vercauteren - Counting points on C ab curves using Monsky-Washnitzer cohomology, Finite Fields Appl. 12 (2006), p. 78-102. | Article | MR 2190188 | Zbl 1104.11032

[38] J. Denef & F. Vercauteren, An extension of Kedlaya's algorithm to hyperelliptic curves in characteristic 2, J. Cryptology 19 (2006), p. 1-25. | Article | MR 2210897 | Zbl 1100.14017

[39] M. Deuring - Die Typen der Multiplikatorringe elliptischer Funktionenkörper, Abh. Math. Sem. Hannsischen Univ. 14 (1941), p. 197-272. | Article | JFM 67.0107.01 | MR 5125

[40] W. Diffie & M. Hellman - New directions in cryptography, IEEE Trans. Inform. Theory 22 (1976), p. 644-654. | Article | MR 437208 | Zbl 0435.94018

[41] B. Dwork - On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), p. 631-648. | Article | MR 140494 | Zbl 0173.48501

[42] B. Dwork & A. Ogus - Canonical liftings of Jacobians, Compositio Math. 58 (1986), p. 111-131. | Numdam | MR 834049 | Zbl 0603.14029

[43] B. Edixhoven - Point counting after Kedlaya, EIDMA-Stieltjes Graduate course, Leiden, September 22-26, 2003, http://www.math.leidenuniv.nl/edix/oww/mathofcrypt/carls_edixhoven/kedlaya.pdf, 2003.

[44] B. Edixhoven, J.-M. Couveignes, R. De Jong, F. Merkl & J. Bosman - On the computation of coefficients of a modular form, Tech. report, 2006, arXiv:math.NT/0605244.

[45] N. Elkies - Explicit isogenies, 1991.

[46] R. Elkik - Solutions d'équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. 6 (1973), p. 553-603. | Article | Numdam | MR 345966 | Zbl 0327.14001

[47] A. Enge - Computing modular polynomials in quasi-linear time, Prépublication, http://www.lix.polytechnique.fr/Labo/Andreas.Enge/vorabdrucke/modcomp.pdf, 2006. | Zbl 1215.11121

[48] A. Enge, P. Gaudry & F. Morain - A new record for SEA, http://www.lix.polytechnique.fr/~morain/SEA/d2100x.annonce, 2006.

[49] M. Fouquet, P. Gaudry & R. Harley - An extension of Satoh's algorithm and its implementation, J. Ramanujan Math. Soc. 15 (2000), p. 281-318. | MR 1801223 | Zbl 1009.11048

[50] J. Fresnel & M. Van Der Put - Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser, 2004. | MR 2014891 | Zbl 1096.14014

[51] G. Frey & H.-G. Rüjck - A remark concerning m-divisibility and the discrète logarithm problem in the divisor class group of curves, Math. Comp. 62 (1994), p. 865-874. | MR 1218343 | Zbl 0813.14045

[52] S. Galbraith & A. Menezes - Algebraic curves and cryptography, Finite Fields Appl. 11 (2005), p. 544-577. | Article | MR 2158776 | Zbl 1079.94014

[53] P. Gaudry & N. Gürel - An extension of Kedlaya's point-counting algorithm to superelliptic curves, in Advances in cryptology-ASIACRYPT 2001 (Gold Coast), Lecture Notes in Comput. Sci., vol. 2248, Springer, 2001, p. 480-494. | Article | MR 1934859 | Zbl 1064.11080

[54] P. Gaudry & N. Gürel, Counting points in médium characteristic using Kedlaya's algorithm, Experiment. Math. 12 (2003), p. 395-402. | Article | MR 2043990 | Zbl 1076.11038

[55] P. Gaudry & R. Harley - Counting points on hyperelliptic curves over finite fields, in Algorithmic number theory (Leiden, 2000), Lecture Notes in Comput. Sci., vol. 1838, Springer, 2000, p. 313-332. | Article | MR 1850614 | Zbl 1011.11041

[56] R. Gerkmann - Relative rigid cohomology and point counting in families of elliptic curves, 2005, http://joguinf.informatik.uni-mainz.de/~gerkmann/legendre.pdf. | Zbl 1247.14023

[57] S. Goldwasser & J. Kilian - Primality testing using elliptic curves, J. ACM 46 (1999), p. 450-472. | Article | MR 1812127 | Zbl 1064.11503

[58] A. Grothendieck - On the de Rham cohomology of algebraic varieties, Publ. Math. I.H.É.S. (1966), p. 95-103. | Article | Numdam | MR 199194 | Zbl 0145.17602

[59] A. Grothendieck, Cohomologie -adique et fonctions L, Lecture Notes in Math., Springer, 1972-73, SGA 5.

[60] A. Grothendieck, Formule de Lefschetz et rationalité des fonctions L, in Séminaire Bourbaki, Vol. 9, Soc. Math. France, 1995, Exp. No. 279, Année 1964-1965, p. 41-55. | Numdam | MR 1608788 | Zbl 0199.24802

[61] R. Harley - Asymptotically optimal p-adic point-counting, e-mail à la liste NMBRTHRY, décembre 2002.

[62] D. Harvey - Kedlaya's Algorithm in Larger Characteristic, Internat. Math. Res. Notices (2006), à paraître, arXiv:math.NT/0610973. | MR 2376210 | Zbl 1206.11080

[63] H. Hasse - Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F. K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen. Vorläufige Mitteilung, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 42 (1933), p. 253-262, http://gdzdoc.sub.uni-goettingen.de/sub/digbib/loader?did=D64289. | JFM 59.0212.01 | Zbl 0007.39702

[64] H. Hasse, Zur Théorie der abstrakten elliptischen Funktionenkörper III. Die Struktur des Meromorphismenrings. Die Riemannsche Vermutung., J. reine angew. Math. 175 (1936), p. 193-208, http://gdzdoc.sub.uni-goettingen.de/sub/digbib/loader?did=D255900. | JFM 62.0117.02 | MR 1581508 | Zbl 0014.24902

[65] W. V. D. Hodge & M. F. Atiyah - Integrais of the second kind on an algebraic variety, Ann. of Math. 62 (1955), p. 56-91. | Article | MR 74082 | Zbl 0068.34401

[66] H. Hubrechts - Memory efficient hyperelliptic curve point counting, 2006, arXiv:math. NT/0609032. | MR 2776016 | Zbl 1222.11079

[67] H. Hubrechts, Point counting in families of hyperelliptic curves, 2006, arXiv:math.NT/0601438. | MR 2403533 | Zbl 1141.11310

[68] H. Hubrechts, Point counting in families of hyperelliptic curves in characteristic 2, 2006, arXiv:math.NT/0607346. | MR 2320829 | Zbl 1222.11080

[69] G. C. Kato & S. Lubkin - Zeta matrices of elliptic curves, J. Number Theory 15 (1982), p. 318-330. | Article | MR 680536 | Zbl 0494.14009

[70] N. M. Katz & W. Messing - Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), p. 73-77. | Article | MR 332791 | Zbl 0275.14011

[71] K. S. Kedlaya - Counting points on hyperelliptic curves using Monsky- Washnitzer cohomology, J. Ramanujan Math. Soc. 16 (2001), p. 323-338, | MR 1877805 | Zbl 1066.14024

K. S. Kedlaya - Counting points on hyperelliptic curves using Monsky- Washnitzer cohomology, Erratum : J. Ramanujan Math. Soc. 18 (2003), no. 4, 417-418. | MR 2043934 | Zbl 1053.14507

[72] K. S. Kedlaya, Finiteness of rigid cohomology with coefficients, Duke Math. J. 134 (2006), p. 15-97. | Article | MR 2239343 | Zbl 1133.14019

[73] N. Koblitz & A. Menezes - A survey of public-key cryptosystems, SIAM Rev. 46 (2004), p. 599-634. | Article | MR 2124678 | Zbl 1071.94013

[74] D. R. Kohel - The AGM-X 0 (N) Heegner point lifting algorithm and elliptic curve point counting, in Advances in cryptology-ASIACRYPT 2003, Lecture Notes in Comput. Sci., vol. 2894, Springer, 2003, p. 124-136. | Article | MR 2093256 | Zbl 1205.11071

[75] A. G. B. Lauder - Deformation theory and the computation of zeta functions, Proc. London Math. Soc. 88 (2004), p. 565-602. | Article | MR 2044050 | Zbl 1119.11053

[76] A. G. B. Lauder, Rigid cohomology and p-adic point counting, J. Théor. Nombres Bordeaux 17 (2005), p. 169-180. | Article | Numdam | MR 2152218 | Zbl 1087.14020

[77] A. G. B. Lauder, A recursive method for Computing zeta functions of varieties, LMS J. Comp. Math. 9 (2006), p. 222-267, http://www.lms.ac.uk/jcm/9/lms2006-005. | Article | MR 2261044 | Zbl 1108.14018

[78] A. G. B. Lauder & D. Wan - Counting points on varieties over finite fields of small characteristic, in Algorithmic Number Theory : Lattices, Number Fields, Curves and Cryptography (J. Buhler & P. Stevenhagen, éds.), Mathematical Sciences Research Institute Publications, Cambridge University Press, 2007, à paraître, http://www.maths.ox.ac.uk/. | Zbl 1188.11069

[79] B. Le Stum - A course on rigid cohomology, Cambridge Tracts in Mathematics, Cambridge Univ. Press, 2007, à paraître. | MR 2358812 | Zbl 1131.14001

[80] R. Lercier - Computing isogenies in 𝐅 2 n , in Algorithmic number theory (Talence, 1996), Lecture Notes in Comput. Sci., vol. 1122, Springer, 1996, p. 197-212. | Article | MR 1446512 | Zbl 0911.11029

[81] R. Lercier & D. Lubicz - Counting points in elliptic curves over finite fields of small characteristic in quasi quadratic time, in Advances in cryptology- EUROCRYPT 2003, Lecture Notes in Comput. Sci., vol. 2656, Springer, 2003, p. 360-373. | Article | MR 2090429 | Zbl 1035.11067

[82] R. Lercier & D. Lubicz, A Quasi Quadratic Time Algorithm for Hyperelliptic Curve Point Counting, The Ramanujan Journal 12 (2006), p. 399-423, http://medicis.polytechnique.fr/~lercier/file/LL05.pdf. | Article | MR 2293798 | Zbl 1166.11021

[83] R. Lercier & F. Morain - Computing isogenies between elliptic curves over 𝐅 p n using Couveignes's algorithm, Math. Comp. 69 (2000), p. 351-370. | Article | MR 1642770 | Zbl 0963.11032

[84] J. Lubin, J-P. Serre & J. Tate - Elliptic curves and formai groups, 1964, Lecture notes prepared in connection with the seminars held at the Summer institute on Algebraic Geometry, Whitney State, Woods Hole, Massachusetts, http://ma.utexas.edu/users/voloch/lst.html.

[85] U. M. Maurer & S. Wolf - The relationship between breaking the Diffie- Hellman protocol and Computing discrète logarithms, SIAM J. Comput. 28 (1999), p. 1689-1721. | Article | MR 1694168 | Zbl 1053.94014

[86] Z. Mebkhout - Sur le théorème de finitude de la cohomologie p-adique d'une variété affine non singulière, Amer. J. Math. 119 (1997), p. 1027-1081. | Article | MR 1473068 | Zbl 0926.14007

[87] W. Messing - The crystals associated to Barsotti-Tate groups : with applications to abelian schemes, Lecture Notes in Math., Springer, 1976. | MR 347836 | Zbl 0243.14013

[88] J.-F. Mestre - Utilisation de l'AGM pour le calcul de E ( 𝐅 2 n ) , Lettre à Harley et Gaudry, décembre 2000, http://www.math.jussieu.fr/~mestre.

[89] J.-F. Mestre, Algorithmes pour compter des points de courbes en petite caractéristique et en petit genre, rédigé par David Lubicz, http://www.math.jussieu.fr/~mestre/rennescrypto.ps, 2002.

[90] P. Monsky & G. Washnitzer - Formal cohomology I, Ann. of Math. 88 (1968), p. 181-217. | Article | MR 248141 | Zbl 0162.52504

[91] F. Morain - Calcul du nombre de points sur une courbe elliptique dans un corps fini : aspects algorithmiques, J. Théor. Nombres Bordeaux 7 (1995), p. 255-282, Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993). | Article | Numdam | MR 1413579 | Zbl 0843.11030

[92] F. Morain, La primalité en temps polynomial (d'après Adleman, Huang ; Agrawal, Kayal, Saxena), Astérisque (2004), p. 205-230. | Numdam | MR 2111645 | Zbl 1097.11059

[93] V. Müller - Ein Algorithmus zur Bestimmung der Punktanzahl elliptischer Kurven über endlichen Körpern der Charakteristik gröfier drei, Thèse, University of Saarland, Saarbrücken, 1995.

[94] D. Mumford - Tata lectures on Thêta II, Birkhäuser, 1984. | MR 742776 | Zbl 0549.14014

[95] A. Muzereau, N. Smart & F. Vercauteren - The equivalence between the DHP and DLP for elliptic curves used in practical applications, LMS J. Comput. Math. 7 (2004), p. 50-72. | Article | MR 2047214 | Zbl 1093.14504

[96] A. Nitaj - L'algorithme de Cornacchia, Exposition. Math. 13 (1995), p. 358-365. | MR 1358213 | Zbl 0852.11074

[97] The Pari Group - PARI/GP, version 2.3.1, 2006, http://pari.math.u-bordeaux.fr/.

[98] J. Pila - Frobenius maps of abelian varieties and finding roots of unity in finite fields, Math. Comp. 55 (1990), p. 745-763. | Article | MR 1035941 | Zbl 0724.11070

[99] J. Pila, Counting points on curves over families in polynomial time, 1991, arXiv:math.NT/0504570.

[100] S. C. Pohlig & M. Hellman - An improved algorithm for Computing logarithms over GF(p) and its cryptographic significance, IEEE Trans. Information Theory IT-24 (1978), p. 106-110. | Article | MR 484737 | Zbl 0375.68023

[101] J. M. Pollard - Monte Carlo methods for index computation (mod p), Math. Comp. 32 (1978), p. 918-924. | MR 491431 | Zbl 0382.10001

[102] M. Van Der Put - The cohomology of Monsky and Washnitzer, Mém. Soc. Math. France N.S. (1986), p. 4, 33-59, Introductions aux cohomologies p -adiques (Luminy, 1984). | Numdam | MR 865811 | Zbl 0606.14018

[103] C. Ritzenthaler - Point counting on genus 3 non hyperelliptic curves, in Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, 2004, p. 379-394. | Article | MR 2138009 | Zbl 1125.11331

[104] T. Satoh - The canonical lift of an ordinary elliptic curve over a finite field and its point counting, J. Ramanujan Math. Soc. 15 (2000), p. 247-270. | MR 1801221 | Zbl 1009.11051

[105] T. Satoh, B. Skjernaa & Y. Taguchi - Fast computation of canonical lifts of elliptic curves and its application to point counting, Finite Fields Appl. 9 (2003), p. 89-101. | Article | MR 1954785 | Zbl 1106.14302

[106] R. Schoof - Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp. 44 (1985), p. 483-494. | MR 777280 | Zbl 0579.14025

[107] R. Schoof, Counting points on elliptic curves over finite fields, J. Théor. Nombres Bordeaux 7 (1995), p. 219-254, Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993). | Article | Numdam | MR 1413578 | Zbl 0852.11073

[108] D. Shanks - Class number, a theory of factorization, and genera, in 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., 1971, p. 415-440. | MR 316385 | Zbl 0223.12006

[109] A. Shiho - Crystalline fundamental groups. II Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), p. 1-163. | MR 1889223 | Zbl 1057.14025

[110] V. Shoup - Lower bounds for discrète logarithms and related problems, in Advances in cryptology-EUROCRYPT '97, Lecture Notes in Comput. Sci., vol. 1233, Springer, 1997, p. 256-266. | MR 1603068

[111] B. Skjernaa - Satoh's algorithm in characteristic 2, Math. Comp. 72 (2003), p. 477-487. | Article | MR 1933832 | Zbl 1027.11045

[112] N. Tsuzuki - Bessel F-isocrystals and an algorithm for Computing Kloosterman sums, 2003.

[113] J. Vélu - Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 273 (1971), p. 238-241. | MR 294345 | Zbl 0225.14014

[114] F. Vercauteren - The SEA algorithm in characteristic 2, http://homes.esat.kuleuven.be/~fvercaut/papers/SEA.pdf.gz, 2000.

[115] F. Vercauteren, Computing zeta functions of hyperelliptic curves over finite fields of characteristic 2, in Advances in cryptology-CRYPTO 2002, Lecture Notes in Comput. Sci., vol. 2442, Springer, 2002, p. 369-384. | Article | MR 2054832 | Zbl 1023.14007

[116] F. Vercauteren, B. Preneel & J. Vandewalle - A memory efficient version of Satoh's algorithm, in Advances in cryptology-EUROCRYPT 2001 (Innsbruck), Lecture Notes in Comput. Sci., vol. 2045, Springer, 2001, p. 1-13. | Article | MR 1895422 | Zbl 1009.11052

[117] D. Wan - Algorithmic theory of zeta functions over finite fields, in Algorithmic Number Theory : Lattices, Number Fields, Curves and Cryptography (J. Buhler & P. Stevenhagen, éds.), Mathematical Sciences Research Institute Publications, Cambridge University Press, 2007, à paraître, http://www.maths.ox.ac.uk/. | MR 2467557 | Zbl 1188.11075

[118] W. C. Waterhouse - Abelian varieties over finite fields, Ann. Sci. Ecole Norm. Sup. 2 (1969), p. 521-560, http://www.numdam.org/item?id=ASENS_1969_4_2_4_521_0. | Article | Numdam | MR 265369 | Zbl 0188.53001

[119] A. Weil - Variétés abéliennes et courbes algébriques, Actualités Sci. Ind., Hermann & Cie., Paris, 1948. | MR 29522 | Zbl 0037.16202