Isogenies and the discrete logarithm problem in jacobians of genus 3 hyperelliptic Curves

Journal of Cryptology

Volumn 22, Issue 4, 2009, Pages 505-529

  Smith, Benjamin ^a  

^a ÉCOLE POLYTECHNIQUE   (France)

Author keywords

Discrete logarithm problem; Genus 3; Hyperelliptic curve cryptography; Isogeny

Indexed keywords

DISCRETE LOGARITHM PROBLEM; DISCRETE LOGARITHM PROBLEMS; EXPLICIT FORMULA; FINITE FIELDS; GENUS 3; GENUS 3 HYPERELLIPTIC CURVES; HYPERELLIPTIC CURVE CRYPTOGRAPHY; INDEX CALCULUS; ISOGENY; JACOBIANS;

CRYPTOGRAPHY;

ALGEBRA;

EID: 68549121006     PISSN: 09332790     EISSN: 14321378     Source Type: Journal    
DOI: 10.1007/s00145-009-9038-1     Document Type: Article

References (25)

1. C. Birkenhake, H. Lange, Complex Abelian Varieties (2e), Grundlehren der mathematischen Wissenschaften, vol. 302 (Springer, Berlin, 2004)
2. W. Bosma J. Cannon C. Playoust 2006 The Magma computational algebra system. I. The user language J. Symb. Comput. 24 3-4 235 265
3. J.-B. Bost J.-F. Mestre 1988 Moyenne arithmético- géométrique et périodes des courbes de genre 1 et 2 Gaz. Math. Soc. Fr. 38 36 64
4. P. Cartier 1960 Isogenies and duality of abelian varieties Ann. Math. 71 2 315 351
5. C. Diem, An index calculus algorithm for plane curves of small degree, in ANTS-VII, ed. by F. Hess, S. Pauli, M. Pohst. LNCS, vol. 4076 (Springer, Berlin, 2006), pp. 543-557
6. R. Donagi, The fibres of the Prym map, in Curves, Jacobians, and Abelian Varieties, Amherst, MA, 1990. Contemp. Math. 136, 55-125 (1992)
7. R. Donagi R. Livné 1999 The arithmetic-geometric mean and isogenies for curves of higher genus Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 28 2 323 339
8. P. Gaudry E. Thomé N. Thériault C. Diem 2007 A double large prime variation for small genus hyperelliptic index calculus Math. Comput. 76 475 492
9. P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)
10. J. Harris, Algebraic Geometry: A First Course (Springer, Berlin, 1992)
11. F. Hess 2002 Computing Riemann-Roch spaces in algebraic function fields and related topics J. Symb. Comput. 33 4 425 445
12. M. Hindry, J. Silverman, Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, vol. 201 (Springer, Berlin, 2000)
13. S. Lang A. Weil 1954 Number of points of varieties in finite fields Am. J. Math. LXXVI 4 819 827
14. D. Lehavi C. Ritzenthaler 2007 An explicit formula for the arithmetic geometric mean in genus 3 Exp. Math. 16 421 440
15. The Magma computational algebra system, http://magma.maths.usyd.edu.au/
16. J.S. Milne, Abelian varieties, in Arithmetic Geometry, Storrs, Conn., 1984 (Springer, New York, 1986), pp. 103-150
17. D. Mumford, Tata Lectures on Theta II (Birkhäuser, Basel, 1984)
18. F. Oort K. Ueno 1973 Principally polarized abelian varieties of dimension two or three are Jacobian varieties J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 377 381
19. G. Pohlig M. Hellman 1978 An improved algorithm for computing logarithms over GF(p) and its cryptographic significance IEEE Trans. Inf. Theory 24 106 110
20. 1 's are the Prym's of trigonal curves Bol. Soc. Mat. Mex. (2) 19 1 9 13
21. J.-P. Serre, Algebraic Curves and Class Fields, Graduate Texts in Mathematics, vol. 117 (Springer, Berlin, 1988)
22. J.-P. Serre, Galois Cohomology (Springer, Berlin, 2002)
23. B. Smith, Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves, in EUROCRYPT 2008, ed. by N. Smart. LNCS, vol. 4965 (Springer, Berlin, 2008), pp. 163-180
24. R. Vakil 2001 Twelve points on the projective line, branched covers, and rational elliptic fibrations Math. Ann. 320 1 33 54
25. J. Vélu 1971 Isogénies entre courbes elliptiques C. R. Acad. Sci. Paris, Sér. A 273 305 347