nous pouvonségalementpouvonségalement demander de calculer l'image d'un point par l'isogénie. Les variétés initiales et finales peuventêtrepeuventêtre données de différentes façons : 1, Thêta constantes de niveau n ,
nous supposons données les thêta constantes d'unepremì ere variété et une description d'un noyau d'une isogénie Nous allons calculer alors les thêta constantes de ladeuxì eme variété. Par ailleurs nous expliquons comment calculer l'image d'un point. Nous nous plaçons en caractéristique différente de 2. Par ailleurs, nous nous intéressons au calcul de d-isogénies séparablesséparablesétant donnés leurs noyaux ,
A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields editeurs : ANTS-I, Lecture Notes in Computer Science, vol.877, pp.28-40, 1994. ,
Complete addition laws on abelian varieties. eprint arXiv:1102.2349, Février 2011 ,
Finding suitable curves for the elliptic curve method of factorization, Mathematics of Computation, vol.60, issue.201, pp.399-405, 1993. ,
DOI : 10.1090/S0025-5718-1993-1140645-1
URL : https://hal.archives-ouvertes.fr/inria-00075015
Atkin : The Number of Points on an Elliptic Curve modulo a Prime. Ensemble d'e-mailsàmailsà la liste de diffusion NMBRTHRY, 1992. ,
On the Theory of ?-Functions, the Moduli of Abelian Varieties, and the Moduli of Curves, Annals of Mathematics. Second Series, vol.75, issue.2, pp.342-381, 1962. ,
Starfish on Strike, Progress in Cryptology ? LATINCRYPT 2010, pp.61-80, 2010. ,
DOI : 10.1007/978-3-642-14712-8_4
New Families of ECM Curves for Cunningham Numbers, Guillaume Hanrot 9th International Symposium, pp.96-109, 2010. ,
DOI : 10.1007/978-3-642-14518-6_11
URL : https://hal.archives-ouvertes.fr/hal-00947685
Identity-Based Encryption from the Weil Pairing, Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology, CRYPTO '01, pp.213-229, 2001. ,
Lauter : Explicit CM-theory in dimension 2. eprint arXiv:0910.1848, Octobre, 2009. ,
How the number of points of an elliptic curve over a fixed prime field varies ,
Endomorphism rings in cryptography, Thèse de doctorat, 2011. ,
The Jacobi Model of an Elliptic Curve and Side-Channel Analysis ,
DOI : 10.1007/3-540-44828-4_5
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, vol.2643, pp.34-42, 2003. ,
Complex abelian varieties ,
DOI : 10.1007/978-3-662-06307-1
Faster Addition and Doubling on Elliptic Curves, Advances in Cryptology ? ASIACRYPT 2007, pp.29-50, 2007. ,
Short Signatures from the Weil Pairing, Advances in Cryptology ? ASIACRYPT 2001, pp.514-532, 2001. ,
DOI : 10.1007/3-540-45682-1_30
Modular polynomials via isogeny volcanoes, Mathematics of Computation, vol.81, issue.278, 2010. ,
DOI : 10.1090/S0025-5718-2011-02508-1
Some integer factorization algorithms using elliptic curves Réécrit et appendice ajouté en, Australian Computer Science Communications, vol.8, pp.149-163, 1986. ,
Factorization of the tenth and eleventh Fermat numbers, Mathematics of Computation, vol.68, issue.225, pp.429-451, 1999. ,
Factoring with cyclotomic polynomials, Mathematics of Computation, vol.52, issue.185, pp.201-219, 1989. ,
DOI : 10.1090/S0025-5718-1989-0947467-1
Computing the endomorphism ring of an ordinary elliptic curve over a finite field, Journal of Number Theory, vol.131, issue.5, pp.815-831, 2011. ,
DOI : 10.1016/j.jnt.2009.11.003
URL : https://hal.archives-ouvertes.fr/inria-00383155
Factorizations of a n ± 1, 13 ? a < 100, 1992. ,
Modern Computer Arithmetic, volume 18 de Cambridge Monographs on Applied and Computational Mathematics, 2010. ,
Computing in the Jacobian of a hyperelliptic curve, Mathematics of Computation, vol.48, issue.177, pp.95-101, 1987. ,
Galois theory of the canonical theta structure. eprint arXiv:math/0509092, p.509092, 2009. ,
Sequences of numbers generated by addition in formal groups and new primality and factorization tests, Advances in Applied Mathematics, vol.7, issue.4, pp.385-434, 1986. ,
DOI : 10.1016/0196-8858(86)90023-0
Handbook of Elliptic and Hyperelliptic Curve Cryptography, 2005. ,
DOI : 10.1201/9781420034981
Hubrechts : The distribution of the number of points modulo an integer on elliptic curves over finite fields, 2011. ,
On compact complex analytic varieties, American Journal of Mathematics, vol.71, pp.893-914, 1949. ,
Higher-dimensional 3-adic CM construction, Journal of Algebra, vol.319, issue.3, pp.971-1006, 2008. ,
DOI : 10.1016/j.jalgebra.2007.11.016
URL : https://hal.archives-ouvertes.fr/hal-00383119
Computing $(\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $2$ curves, Mathematics of Computation, vol.84, issue.294, 2011. ,
DOI : 10.1090/S0025-5718-2014-02899-8
Arithmetic Geometry, 1986. ,
DOI : 10.1007/978-1-4613-8655-1
The Schottky Problem: An Update Current Topics in Algebraic Geometry, de Mathematical Sciences Research Institute Publications, pp.57-64, 1994. ,
New directions in cryptography, IEEE Transactions on Information Theory, vol.22, issue.6, pp.644-654, 1976. ,
DOI : 10.1109/TIT.1976.1055638
Quelques aspects de l'arithmétique des courbes hyperelliptiques de genre 2, Thèse de doctorat, 2010. ,
An index calculus algorithm for plane curves of small degree, Lecture Notes in Computer Science, vol.4076, pp.543-557, 2006. ,
On isogenous principally polarized abelian surfaces. eprint arXiv:0710.1298, Mars, 2008. ,
Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three, Journal of Cryptology, vol.38, issue.192, pp.593-611, 2008. ,
DOI : 10.1007/978-1-4757-1920-8
URL : https://hal.archives-ouvertes.fr/inria-00107290
Moyenne arithmético-géométrique, suites de Borchardt et applications, Thèse de doctorat, ´ Ecole polytechnique, 2006. ,
Montgomery scalar multiplication for genus 2 curves, Lecture Notes in Computer Science, vol.3076, pp.153-168, 2004. ,
Improving the arithmetic of elliptic curve in the Jacobi model, Information Processing Letters, vol.104, pp.101-105, 2007. ,
Farkas : An Elementary Proof of Thomae's Formulae, Online Journal of Analytic Combinatorics, vol.3, 2008. ,
An L(1/3) Discrete Logarithm Algorithm for Low Degree Curves, Journal of Cryptology, vol.68, issue.226, pp.24-41, 2011. ,
DOI : 10.1090/S0025-5718-01-01352-7
URL : https://hal.archives-ouvertes.fr/inria-00135324
Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation, Lecture Notes in Computer Science, vol.4547, pp.202-218, 2007. ,
DOI : 10.1007/978-3-540-73074-3_16
ElGamal : A public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE Transactions on Information Theory, IT?, vol.31, issue.4, pp.469-472, 1985. ,
Elkies : Explicit isogenies, 1991. ,
Elkies : Elliptic and modular curves over finite fields and related computational issues ,
Computational Perspectives on Number Theory, Proceedings of a Conference in Honor of A.O.L. Atkin, pp.21-76, 1998. ,
Discrete logarithms in curves over finite fields, Eighth International Conference on Finite Fields and Applications -Fq8 Finite Fields and Applications de Contemporary Mathematics, pp.119-139, 2008. ,
Efficient Arithmetic on Hessian Curves, Lecture Notes in Computer Science, vol.6056, pp.243-260, 2010. ,
DOI : 10.1007/978-3-642-13013-7_15
Riemann Surfaces, volume 71 de Graduate Texts in Mathematics, 1980. ,
Theta constants, Riemann surfaces and the modular group, de Graduate studies in Mathematics, p.179, 2001. ,
DOI : 10.1090/gsm/037
Computing modular correspondences for abelian varieties. eprint arXiv:0910.4668, Octobre, 2009. ,
Morain : Isogeny volcanoes and the SEA algorithm Algorihmic Number Theory, volume 2369 de Lecture Notes in Computer Science, 5th International Symposium, ANTS-V, pp.276-291, 2002. ,
Rück : A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves, Mathematics of Computation, vol.62, issue.206, pp.865-874, 1994. ,
Faster integer multiplication, Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, STOC '07, pp.57-66, 2007. ,
Mathematics of Public Key Cryptography. Livre en préparation, 2011. ,
Algorithmique des courbes hyperelliptiques et applicationsàapplicationsà la cryptologie, Thèse de doctorat, ´ Ecole Polytechnique, 2000. ,
Algorithmes de comptage de points d'une courbe définie sur un corps fini, Gau07] P. Gaudry : Fast genus 2 arithmetic based on theta functions, pp.243-265, 2004. ,
Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem, Journal of Symbolic Computation, vol.44, issue.12, pp.1690-1702, 2009. ,
Weng : The 2-adic CM method for genus 2 curves with application to cryptography, Advances in Cryptology ? ASIACRYPT 2006, pp.114-129, 2006. ,
Efficient Hyperelliptic Arithmetic Using Balanced Representation for Divisors, Alfred van der Poorten et Andreas Stein, pp.342-356, 2008. ,
DOI : 10.1007/978-3-540-79456-1_23
The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines. Finite Fields and Their Applications, pp.246-260, 2009. ,
URL : https://hal.archives-ouvertes.fr/inria-00266565
Explicit Algorithms for Humbert Surfaces, Thèse de doctorat, 2008. ,
The Schottky problem. eprint arXiv:1009.0369, Septembre http: //arxiv.org/abs/1009.0369. [GS01] P. Gaudry etÉet´etÉ. Schost : On the invariants of the quotients of the Jacobian of a curve of genus 2, S. Bozta¸sBozta¸s et I. Shparlinski, ´ editeurs : Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, volume 2227 de Lecture Notes in Computer Science, pp.373-386, 2001. ,
A double large prime variation for small genus hyperelliptic index calculus, Mathematics of Computation, vol.76, issue.257, pp.475-492, 2007. ,
DOI : 10.1090/S0025-5718-06-01900-4
URL : https://hal.archives-ouvertes.fr/inria-00077334
Algebraic Geometry, volume 52 de Graduate Texts in Mathematics, 1977. ,
Poonen : Large torsion subgroups of split Jacobians of curves of genus two or three, Forum Mathematicum, vol.12, pp.315-364, 2000. ,
Isogeny classes of abelian varieties over finite fields, Journal of the Mathematical Society of Japan, vol.20, issue.12, pp.83-95, 1968. ,
On the group orders of elliptic curves over finite fields, Compositio Mathematica, vol.85, pp.229-247, 1993. ,
Comparing the MOV and FR Reductions in Elliptic Curve Cryptography, Lecture Notes in Computer Science, vol.1592, pp.190-205, 1999. ,
DOI : 10.1007/3-540-48910-X_14
Elliptic curves, volume 111 de Graduate Texts in Mathematics, 1987. ,
Arithmetic variety of moduli for genus two, Annals of Mathematics. Second Series, vol.72, pp.612-649, 1960. ,
On Siegel modular forms of genus two, American Journal of Mathematics, vol.84, pp.175-200, 1962. ,
Igusa : Theta functions, volume 194 de Grundlehren der mathematischen Wissenschaften, 1972. ,
Algebraic geometry, volume 76 de Graduate Texts in Mathematics, 1981. ,
Factorization of a 768-bit rsa modulus, Advances in Cryptology ? Crypto'2010, pp.333-350, 2010. ,
Linear systems on abelian varieties, American Journal of Mathematics, vol.111, issue.1, pp.65-94, 1989. ,
Introductory lectures on Siegel modular forms, volume 20 de Cambridge studies in advanced mathematics, 1990. ,
The Art of Computer Programming: Seminumerical Algorithms, 1981. ,
Endomorphism rings of elliptic curves over finite fields, Thèse de doctorat, 1996. ,
Addition law structure of elliptic curves, Journal of Number Theory, vol.131, issue.5, pp.894-919, 2011. ,
Theta relations and projective normality of abelian varieties, American Journal of Mathematics, pp.865-889, 1976. ,
Remarks on Takase's paper " A generalization of Rosenhain's normal form with an application, Japan Academy. Proceedings. Series A. Mathematical Sciences, pp.12-13, 1997. ,
Factoring into large primes with P-1, P+1 and ECM Slides au workshop CADO, disponiblè a l'adresse http://cado.gforge.inria.fr/workshop/slides/kruppa. pdf. [Kru10] A. Kruppa : Speeding up Integer Multiplication and Factorization, Thèse de doctorat, 2008. ,
Hyperelliptic curves, l-polynomials, and random matrices. eprint arXiv:0803.4462, Mars, 2010. ,
Curves of genus 2 with split jacobian. Transactions of the, pp.41-49, 1988. ,
Reciprocity and Correspondences, American Journal of Mathematics, vol.80, issue.2, pp.431-440, 1958. ,
Abelian varieties. Numéro 7 de Interscience Tracts in Pure and Applied Mathematics, 1959. ,
Introduction to algebraic and abelian functions, 1972. ,
Formulae for arithmetic on genus 2 hyperelliptic curves Applicable Algebra in Engineering, Communication and Computing, vol.15, issue.5, pp.295-328, 2005. ,
Factoring integers with elliptic curves, Annals of Mathematics. Second Series, vol.126, pp.649-673, 1987. ,
Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 2002. ,
A rigorous time bound for factoring integers, Journal of the American Mathematical Society, vol.5, issue.3, pp.483-516, 1992. ,
DOI : 10.1090/S0894-0347-1992-1137100-0
A Hyperelliptic Smoothness Test. I, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.345, issue.1676, pp.397-408, 1993. ,
DOI : 10.1098/rsta.1993.0138
URL : https://openaccess.leidenuniv.nl/bitstream/handle/1887/3840/346_118.pdf%3Bjsessionid%3D7ECAEA2F9C83D4AEA42CDA85CB610518?sequence%3D1
Pomerance : A hyperelliptic smoothness test, II, Proc. London Math. Soc, pp.105-146, 2002. ,
DOI : 10.1112/plms/84.1.105
URL : http://cr.yp.to/bib/2002/lenstra-hecm.pdf
Complete systems of addition laws on abelian varieties, Inventiones Mathematicae, vol.29, issue.3, pp.603-610, 1985. ,
DOI : 10.1007/BF01388526
Computing isogenies between abelian varieties. eprint arXiv:1001.2016, Mars 2010 ,
Efficient pairing computation with theta functions Algorithmic Number Theory, Juillet 2010. 9th International Symposium, 2010. ,
Construction de courbes de genre 2 ` a partir de leurs modules Effective methods in algebraic geometry, Proc. Congress in, pp.313-334, 1990. ,
Complex Multiplication Abelian varieties http://www.jmilne.org/math/CourseNotes/av : Evaluating recurrences of form x m+n = f (x m , x n , x m?n ) via Lucas chains, 1983. ,
Speeding the Pollard and Elliptic Curve Methods of Factorization, Mathematics of Computation, vol.48, issue.177, pp.243-264, 1987. ,
An FFT extension of the Elliptic Curve Method of factorization, Thèse de doctorat, 1992. ,
A Survey of Modern Integer Factorization Algorithms, CWI Quarterly, vol.7, pp.337-366, 1994. ,
Vanstone : Reducing elliptic curves logarithms to logarithms in a finite field, Proc. 23rd Annual ACM Symposium on Theory of Computing (STOC), pp.80-89, 1991. ,
On the equations defining abelian varieties. I, Inventiones Mathematicae, vol.111, issue.4, pp.287-354, 1966. ,
DOI : 10.1007/978-3-662-00095-3
On the equations defining abelian varieties, III. Inventiones Mathematicae, vol.3, pp.215-244, 1967. ,
Varieties defined by quadratic equations. Questions on Algebraic Varieties, pp.29-100, 1969. ,
Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, issue.5, 1970. ,
Tata lectures on theta I, de Progress in Mathematics. Birkhäuser, 1983. ,
Tata lectures on theta II, de Progress in Mathematics. Birkhäuser, 1984. ,
Tata lectures on theta III, 1991. ,
Discrete logarithms: the past and the future, Designs, Codes, and Cryptography, vol.19, pp.129-145, 1999. ,
An improved algorithm for computing logarithms over<tex>GF(p)</tex>and its cryptographic significance (Corresp.), IEEE Transactions on Information Theory, vol.24, issue.1, pp.106-110, 1978. ,
DOI : 10.1109/TIT.1978.1055817
Pollard : Theorems on factorization and primality testing, pp.521-528, 1974. ,
Poor : The Hyperelliptic locus, Duke Mathematical Journal, vol.76, issue.3, pp.809-884, 1994. ,
Essai sur une méthode générale pour déterminer la valeur des intégrales ultraelliptiques , fondée sur des transformations remarquables de ces transcendantes. Comptes Rendus de l, Académie des Sciences. Série I, vol.2, pp.622-627, 1836. ,
De transformatione Integralium Abelianorum primiordinis commentation, Journal fur die Reine und Angewandte Mathematik, pp.221-341, 1837. ,
Theta functions and cryptographic applications, Thèse de doctorat, 2010. ,
A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM, vol.21, issue.2, pp.120-126, 1978. ,
DOI : 10.1145/359340.359342
The canonical lift of an ordinary elliptic curve over a finite field and its point counting, Journal of the Ramanujan Mathematical Society, vol.15, pp.247-270, 2000. ,
Morphismes d'une courbe de genre 2 vers une courbe de genre 1, Arithmétique des revêtements algébriques -Actes du colloque de Saint-´ Etienne de Séminaires et Congrès Zur Theorie der Abel schen Functionen vor vier Variablen. Journal fur die Reine und Angewandte Mathematik, pp.133-146304, 1888. ,
G??om??trie alg??brique et g??om??trie analytique, Annales de l???institut Fourier, vol.6, pp.1-42 ,
DOI : 10.5802/aif.59
Lectures on the Mordell-Weil theorem. Friedr, 1989. ,
Curves of genus 2 with (n, n)-split Jacobians, Journal of Symbolic Computation, vol.31, pp.603-617, 2001. ,
Genus 2 curves with (3, 3)-split Jacobian and large automorphism group, ANTS V, volume 2369 de Lecture Notes in Computer Science, pp.100-113, 2002. ,
Genus 2 fields with degree 3 elliptic subfields, Forum Mathematicum, vol.16, pp.263-280, 2004. ,
Genus 2 curves covering elliptic curves, a computational approach, Lecture Notes in Computer Science, vol.13, pp.151-195, 2005. ,
Abelian Varieties with Complex Multiplication and Modular Functions, 1998. ,
The arithmetic of elliptic curves, volume 106 de Graduate Texts in Mathematics, 1986. ,
Advanced Topics in the Arithmetic of Elliptic Curves, volume 151 de Graduate Texts in Mathematics, 1994. ,
Neue Sätzë uber Symmetralfunktionen und die Abel'schen Funktionen der Riemann'schen Theorie, Smi09] B. Smith : Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves, pp.282-297505, 1909. ,
Algebraic function fields and codes, 1993. ,
Complex multiplication of abelian surfaces, Thèse de doctorat, 2010. ,
Computing Hilbert class polynomials with the Chinese remainder theorem, Mathematics of Computation, vol.80, pp.501-538, 2011. ,
Informal preliminary report (8) Letter to Richard P, pp.25-1985 ,
A Generalization of Rosenhain's Normal Form for Hyperelliptic Curves with an Application, Japan Academy. Proceedings. Series A. Mathematical Sciences, pp.162-165, 1996. ,
Endomorphisms of Abelian varieties over finite fields, Inventiones Mathematicae, vol.2, pp.134-144, 1966. ,
Beitrag zur Bestimmung von ? (0, 0, . . . 0) durch die Klassbmoduln agebraischer Functionnen, Journal fur die Reine und Angewandte Mathematik, vol.70, pp.201-222, 1870. ,
Darstellung des Quotienten zweier Thetafunctionen, deren Argumente sich um Drittel ganzer Periodicitätsmoduln unterscheiden, durch algebraische Fiunctionen, Mathematische Annalen, vol.6, pp.603-612, 1873. ,
Abelian varieties. Livre en préparation, Vél71] J. Vélu : Isogénies entre courbes elliptiques. Comptes Rendus de l'Académie des Sciences. Série A, pp.238-241, 1971. ,
Equations for the Jacobian of a hyperelliptic curve. Transactions of the, pp.3083-3106, 1998. ,
Sur certains groupes d'opérateurs unitaires, Acta Mathematica, vol.111, issue.1, pp.143-211, 1964. ,
Constructing hyperelliptic curves of genus 2 suitable for cryptography, Mathematics of Computation, vol.72, issue.241, pp.435-458, 2003. ,
DOI : 10.1090/S0025-5718-02-01422-9
On Hyperelliptic ?-Functions with Rational Characteristics, American Journal of Mathematics, vol.50, issue.3, pp.315-344, 1928. ,
years of ECM, ANTS VII, pp.20-525, 2006. ,
URL : https://hal.archives-ouvertes.fr/inria-00070192