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Factorization with genus 2 curves
Author(s):
Romain
Cosset.
Journal:
Math. Comp.
79
(2010),
1191-1208.
MSC (2000):
Primary 11Y05;
Secondary 11Y16, 11Y40
Posted:
August 20, 2009
MathSciNet review:
2600562
Retrieve article in:
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Abstract |
References |
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Additional information
Abstract:
The elliptic curve method (ECM) is one of the best factorization methods available. It is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower. We use special hyperelliptic curves and Kummer surfaces to reduce the complexity of the algorithm. Our implementation GMP-HECM is faster than GMP-ECM for factoring large numbers.
References:
-
- 1.
- D. J. Bernstein, P. Birkner, T. Lange, and C. Peters, ECM using Edwards curves, Cryptology ePrint Archive, 2008, http://eprint.iacr.org/2008/016.
- 2.
- S. Duquesne, Improving the arithmetic of elliptic curve in the Jacobi model, Inform. Process. Lett. 104 (2007), 101-105. MR 2348218 (2008h:94055)
- 3.
- P. Gaudry, Fast genus
 arithmetic based on theta functions, J. Math. Cryptol. 1 (2007), 243-265. MR 2372155 (2009f:11156) - 4.
- P. Gaudry and É. Schost, On the invariants of the quotients of the Jacobian of a curve of genus
 , Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (S. Boztaş and I. Shparlinski, eds.), Lecture Notes in Comput. Sci., vol. 2227, Springer-Verlag, 2001, pp. 373-386. MR 1913484 (2003e:14020) - 5.
- A. Kruppa, Factoring into large primes with
 ,  and ECM, 2008, Available at http://cado.gforge.inria.fr/workshop/slides/kruppa.pdf. - 6.
- H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), 649-673. MR 916721 (89g:11125)
- 7.
- P. L. Montgomery, Evaluating recurrences of form
 via Lucas chains, 1983, Available at ftp.cwi.nl/pub/pmontgom/Lucas.ps.gz. - 8.
- -, Speeding the Pollard and Elliptic Curve Methods of Factorization, Math. Comp. 48 (1987), no. 177, 243-264. MR 866113 (88e:11130)
- 9.
- D. Mumford, Tata lectures on theta. I, Progr. Math., vol. 28, Birkhäuser, 1983. MR 688651 (85h:14026)
- 10.
- -, Tata lectures on theta. II, Progr. Math., vol. 43, Birkhäuser, 1984. MR 742776 (86b:14017)
- 11.
- T. Shaska, Genus
 curves covering elliptic curves, a computational approach, Lecture Notes Ser. Comput. 13 (2005), 151-195. MR 2182041 (2006g:14051) - 12.
- P. van Wamelen, Equations for the Jacobian of a hyperelliptic curve, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3083-3106. MR 1432144 (98k:14038)
- 13.
- P. Zimmermann and B. Dodson, 20 years of ECM, Algorithmic Number Theory Symposium VII, Lecture Notes in Comput. Sci., vol. 4076, Springer-Verlag, 2006, pp. 525-542. MR 2282947 (2007j:11172)
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Additional Information:
Romain
Cosset
Affiliation:
LORIA, Campus Scientifique - BP 239, 54506 Vandoeuvre-lès-Nancy, France
Email:
romain.cosset@loria.fr
DOI:
10.1090/S0025-5718-09-02295-9
PII:
S 0025-5718(09)02295-9
Received by editor(s):
February 10, 2009
Received by editor(s) in revised form:
April 4, 2009
Posted:
August 20, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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