A weil descent attack against elliptic curve cryptosystems over quartic extension fields

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

Volumn E89-A, Issue 5, 2006, Pages 1246-1254

  Arita, Seigo ^a,d   Matsuo, Kazuto ^a,b   Nagao, Koh Ichi ^c   Shimura, Mahoro ^b  

^a INSTITUTE OF INFORMATION SECURITY   (Japan)

^b CHUO UNIVERSITY   (Japan)

^c KANTO GAKUIN UNIVERSITY   (Japan)

^d JMS   (Japan)

Author keywords

Elliptic curve cryptosystems; Hyperelliptic curve cryptosystems; Scholten form; Weil descent attack

Indexed keywords

DISCRETE TIME CONTROL SYSTEMS; MATHEMATICAL MODELS; PROBLEM SOLVING; QUADRATIC PROGRAMMING;

ELLIPTIC CURVE CRYPTOSYSTEMS; HYPERELLIPTIC CURVE CRYPTOSYSTEMS; SCHOLTEN FORM; WEIL DESCENT ATTACK;

CRYPTOGRAPHY;

EID: 33646803467     PISSN: 09168508     EISSN: 17451337     Source Type: Journal    
DOI: 10.1093/ietfec/e89-a.5.1246     Document Type: Article

References (26)

1. ab curves," IEICE Trans. Fundamentals, vol.E83-A, no.9, pp.1809-1814, Sept. 2000.
2. S. Arita, "Weil descent of elliptic curves over finite fields of characteristic three," ASIACRYPT 2000, LNCS 1976, pp.248-258, Springer-Verlag, 2000.
3. S. Arita, "A Weil descent attack against genus two hyperelliptic curve cryptosystems over quadratic extention fields," IEICE Technical Report, ISEC2002-62, 2002.
4. ab curves," Discrete Appl. Math., vol.130, no.1, pp. 13-31, 2003.
5. S. Arita, "A Weil descent attack against elliptic curve cryptosystems over quartic fields II," Proc. SCIS2004, pp.903-908, 2004.
6. S. Arita, K. Matsuo, K. Nagao, and M. Shimura, "A weil descent attack against elliptic curve cryptosystems over quartic extension fields," Cryptology ePrint Archive, Report 2004/240, 2004.
7. C. Diem, "The GHS attack in odd characteristic," J. Ramanujan Math. Soc., vol.18, no.1, pp.1-32, 2003.
8. C. Diem, "Index calculus in class groups of plane curves of small degree," Cryptology ePrint Archive, Report 2005/119, 2005.
9. G. Frey and H. Gangl, "How to disguise an elliptic curve (Weil descent)," Talk at ECC'98, 1998.
10. G. Frey and E. Kani, "Curves of genus 2 covering elliptic curves and an arithmetical application," Arithmetic Algebraic Geometry, PM 89, pp. 153-176, Birkhäuser, 1991.
11. G. Frey and H. Rück, "A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves," Math. Comp., vol.62, pp.865-874, 1994.
12. S. Galbraith and N. Smart, "A cryptographic application of Weil descent," Cryptography and Coding, LNCS 1746, pp. 191-200, Springer-Verlag, 1999.
13. S. Galbraith, "Weil descent of Jacobians," Discrete Appl. Math., vol.128, no.1, pp.165-180, 2003.
14. P. Gaudry, "An algorithm for solving the discrete log problem on hyperelliptic curves," EUROCRYPT 2000, LNCS 1807, pp. 19-34, Springer-Verlag, 2000.
15. P. Gaudry, F. Hess, and N. Smart, "Constructive and destructive facets of Weil descent on elliptic curves," J. Cryptol., vol.15, no.1, pp. 19-46, 2002.
16. P. Gaudry, N. Thériault, and E. Thomé, "A double large prime variation for small genus hyperelliptic index calculus," Cryptology ePrint Archive, Report 2004/153, 2004.
17. A. Menezes, T. Okamoto, and S. Vanstone, "Reducing elliptic curve logarithms to logarithms in a finite fields," Proc. STOC, pp.80-89, 1991.
18. S. Miura, "Linear codes on affine algebraic curves," IEICE Trans. Fundamentals (Japanese Edition), vol.J81-A, no. 10, pp. 1398-1421, Oct. 1998.
19. K. Nagao, "Improvement of Thériault algorithm of index calculus for Jacobian of hyperelliptic curves of small genus," Cryptology ePrint Archive, Report 2004/161, 2004.
20. K. Nagao, S. Arita, K. Matsuo, and M. Shimura, "A Weil descent attack against elliptic curve cryptosystems over quartic fields I," Proc. SCIS2004, pp.897-902, 2004.
21. T. Satoh and K. Araki, "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves," Commentarii Mathematici Universitatis Sancti Pauli, vol.47, no.1, 1998.
22. J. Scholten, "Weil restriction of an elliptic curve over a quadratic extension," preprint, http://www.esat.kuleuven.ac.be/~jscholte/weilres.ps, 2003.
23. I. Semaev, "Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p," Math. Comp., vol.67, pp.353-356, 1998.
24. N. Smart, "The discrete logarithm problem on elliptic curves of trace one," J. Cryptology, vol.12, no.3, pp.193-196, 1999.
25. N. Thériault, "Index calculus attack for hyperelliptic curves of small genus," ASIACRYPT 2003, LNCS 2894, pp.75-92, Springer-Verlag, 2003.
26. N. Thériault, "Weil descent attack for Kummer extensions," J. Ramanujan Math. Soc., vol.18, no.3, pp.281-312, 2003.