]. D. Ber06 and . Bernstein, Differential addition chains, 2006.

D. J. Bernstein, C. Chuengsatiansup, T. Lange, and P. Schwabe, Kummer Strikes Back: New DH Speed Records
DOI : 10.1007/978-3-662-45611-8_17

J. W. Bos, C. Costello, H. Hisil, and K. Lauter, Fast cryptography in genus 2, Advances in Cryptology?EUROCRYPT 2013, pp.194-210, 2013.

]. D. Bro06 and . Brown, Multi-dimensional Montgomery ladders for elliptic curves, 2006.

]. D. Can87 and . Cantor, Computing in the Jacobian of a hyperelliptic curve, In: Math. Comp, vol.48177, pp.95-101, 1987.

]. R. Cos11 and . Cosset, Application des fonctions thêta à la cryptographie sur courbes hyperelliptiques, 2011.

D. [. Cosset and . Robert, An algorithm for computing (, )-isogenies in polynomial time on Jacobians of hyperelliptic curves of genus 2 " . Accepted for publication in Mathematics of computation, pp.2011-143, 2013.

]. R. Dup06 and . Dupont, Moyenne arithmetico-geometrique, suites de Borchardt et applications, 2006.

R. [. Gallant, S. A. Lambert, and . Vanstone, Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms, In: CRYPTO. Ed. by J. Kilian. Lecture Notes in Computer Science, vol.2139, issue.13, pp.190-200, 2001.
DOI : 10.1007/3-540-44647-8_11

]. P. Gau07 and . Gaudry, Fast genus 2 arithmetic based on Theta functions, Journal of Mathematical Cryptology, vol.13, issue.2, pp.243-265, 2007.

D. [. Gaudry and . Lubicz, The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines, Finite Fields and Their Applications, vol.15, issue.2, pp.246-260, 2009.
DOI : 10.1016/j.ffa.2008.12.006
URL : https://hal.archives-ouvertes.fr/inria-00266565

K. [. Goren and . Lauter, Genus 2 Curves with Complex Multiplication, International Mathematics Research Notices, vol.2012, issue.5, pp.1068-1142, 2012.
DOI : 10.1093/imrn/rnr052

]. E. Got59 and . Gottschling, Explizite bestimmung der randflächen des fundamentalbereiches der modulgruppe zweiten grades, Mathematische Annalen 138, pp.103-124, 1959.

]. R. Har00 and . Hartshorne, Algebraic geometry, 2000.

H. Hisil and C. Costello, Jacobian Coordinates on Genus 2 Curves

]. Igu72 and . Igusa, Theta functions. Die Grundlehren der mathematischen Wissenschaften, Band 194, p.232, 1972.

]. G. Kem88 and . Kempf, Multiplication over abelian varieties, American Journal of Mathematics, vol.1104, pp.765-773, 1988.

]. G. Kem89 and . Kempf, Linear systems on abelian varieties, American Journal of Mathematics, vol.1111, issue.10, pp.65-94, 1989.

]. D. Koh11 and . Kohel, Arithmetic of split Kummer surfaces: Montgomery endomorphism of Edwards products, Coding and Cryptology, pp.238-245, 2011.

]. S. Koi76 and . Koizumi, Theta relations and projective normality of abelian varieties, American Journal of Mathematics, vol.14, pp.865-889, 1976.

]. T. Lan05 and . Lange, Formulae for arithmetic on genus 2 hyperelliptic curves, Applicable Algebra in Engineering, Communication and Computing, vol.155, pp.295-328, 2005.

D. [. Lubicz and . Robert, Efficient pairing computation with theta functions Algorithmic Number Theory, Thomé. 9th International Symposium Proceedings. doi: 10.1007, pp.978-981, 2010.

D. [. Lubicz and . Robert, Computing isogenies between abelian varieties, Compositio Mathematica, vol.2, issue.05, pp.1483-1515, 2012.
DOI : 10.1515/crll.1837.16.221
URL : https://hal.archives-ouvertes.fr/hal-00446062

D. [. Lubicz and . Robert, A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties, Journal of Symbolic Computation, vol.67, issue.2, pp.11-13, 2013.
DOI : 10.1016/j.jsc.2014.08.001

]. P. Mon87 and . Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Mathematics of computation 48, pp.243-264, 1987.

]. P. Mon92 and . Montgomery, Evaluating recurrences of form Xm+n= f (Xm, Xn, Xm-n) via Lucas chains, 1992.

D. Mumford, On the equations defining abelian varieties. I, Inventiones Mathematicae, vol.111, issue.4, pp.287-354, 1966.
DOI : 10.1007/BF01389737

D. Mumford, On the equations defining abelian varieties. I, Inventiones Mathematicae, vol.111, issue.4, pp.287-354, 1966.
DOI : 10.1007/BF01389737

]. D. Mum67a and . Mumford, On the equations defining abelian varieties. II, In: Invent. Math, vol.3, pp.75-135, 1967.

D. Mumford, On the equations defining abelian varieties. III, Inventiones Mathematicae, vol.80, issue.3, pp.215-244, 1967.
DOI : 10.1007/BF01425401

]. D. Mum69 and . Mumford, Varieties defined by quadratic equations, In: Questions on Algebraic Varieties, pp.29-100, 1969.

]. D. Mum70 and . Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, p.242, 1970.

]. D. Mum83, C. Mumford, M. Musili, E. Nori, M. Previato et al., Tata lectures on theta I, Progress in Mathematics. With the assistance of, pp.235-238, 1983.

]. D. Mum84, C. Mumford, M. Musili, E. Nori, M. Previato et al., Tata lectures on theta II, Progress in Mathematics. Jacobian theta functions and differential equations, pp.272-272, 1984.

R. Mumford, Tata lectures on theta III With the collaboration of Madhav Nori and Peter Norman, Progress in Mathematics, pp.202-202, 1991.

]. D. Rob10 and . Robert, Fonctions thêta et applications à la cryptographie, 2010.

D. Générale-de-l-'armement and . Celar-bp, 35174 Bruz Cedex INRIA Bordeaux?Sud-Ouest, 200 avenue de la Vieille Tour, 33405 Talence Cedex FRANCE E-mail address: damien