Algorithmes pour la factorisation d'entiers et le calcul de logarithme discret

Doctoral thesis French OPEN
Bouvier , Cyril;
(2015)
  • Publisher: HAL CCSD
  • Subject: algorithms | elliptic curve | corps de nombres | filtering step | Algorithmes | [ INFO.INFO-CR ] Computer Science [cs]/Cryptography and Security [cs.CR] | factorization | ECM | étape de filtrage | FFS | polynomial selection | finite field | logarithme discret | corps fini | sélection polynomiale | discrete logarithm | number field | NFS | factorisation | NFS-DL | courbe elliptique

In this thesis, we study the problems of integer factorization and discretelogarithm computation in finite fields. First, we study the ECM algorithmfor integer factorization and present a method to analyze the ellipticcurves used in this algorithm by studying the Galois... View more
  • References (73)
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    [25] C. Bouvier. The filtering step of discrete logarithm and integer factorization algorithms. Preprint, 22 pages. 2013. url : http://hal.inria.fr/hal-00734654 [12] R. Barbulescu, C. Bouvier, J. Detrey, P. Gaudry, H. Jeljeli, E. Thomé, M. Videau et P. Zimmermann. « Discrete logarithm in GFp2809q with FFS ». In : PublicKey Cryptography - PKC 2014. Sous la dir. de H. Krawczyk. T. 8383. Lecture Notes in Computer Science. Springer-Verlag, 2014, p. 221-238. isbn : 978-3-642-54630-3. doi : 10.1007/978-3-642-54631-0_13. url : http://hal.inria.fr/hal-00818124 [27] C. Bouvier et P. Zimmermann. « Division-Free Binary-to-Decimal Conversion ». In : IEEE Transactions on Computers 63.8 (2014), p. 1895-1901. issn : 0018-9340. doi : 10.1109/TC.2014.2315621. url : https://hal.inria.fr/hal-00864293 [5] S. Bai, C. Bouvier, A. Kruppa et P. Zimmermann. « Better polynomials for GNFS ». In : Mathematics of Computation (2015). doi : 10.1090/mcom3048. url : https://hal. inria.fr/hal-01089507

    [3] L. M. Adleman et M.-D. A. Huang. « Function Field Sieve Method for Discrete Logarithms over Finite Fields ». In : Information and Computation 151.1-2 (1999), p. 5-16. issn : 0890-5401. doi : 10.1006/inco.1998.2761. url : http://www.sciencedirect. com/science/article/pii/S0890540198927614 (cité p. 77).

    [4] A. O. L. Atkin et F. Morain. « Finding suitable curves for the elliptic curve method of factorization ». In : Mathematics of Computation 60.201 (1993), p. 399-405 (cité p. 17).

    [5] S. Bai, C. Bouvier, A. Kruppa et P. Zimmermann. « Better polynomials for GNFS ». In : Mathematics of Computation (2015). doi : 10.1090/mcom3048. url : https://hal. inria.fr/hal-01089507 (cité p. 4, 44, 54).

    [6] S. Bai, R. P. Brent et E. Thomé. « Root optimization of polynomials in the number field sieve ». In : Mathematics of Computation (2015). doi : 10.1090/S0025-5718-2015- 02926-3 (cité p. 48).

    [7] S. Bai, E. Thomé et P. Zimmermann. Factorisation of RSA-704 with CADO-NFS. Cryptology ePrint Archive, Report 2012/369. 2012. url : http://eprint.iacr.org/ (cité p. 95).

    [9] R. Barbulescu. « Selecting polynomials for the Function Field Sieve ». Preprint, 23 pages. 2013. url : http://hal.inria.fr/hal-00798386 (cité p. 78, 79).

    [10] R. Barbulescu, J. W. Bos, C. Bouvier, T. Kleinjung et P. L. Montgomery. « Finding ECM-Friendly Curves through a Study of Galois Properties ». In : ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium. Sous la dir. d'E. W. Howe et K. S. Kedlaya. T. 1. Open Book Series. Berkeley : Mathematical Sciences Publishers, 2013, p. 63-86. doi : 10.2140/obs.2013.1.63. url : https://hal.inria.fr/hal00671948 (cité p. 3, 5, 22).

    [11] R. Barbulescu, C. Bouvier, J. Detrey, P. Gaudry, H. Jeljeli, E. Thomé, M. Videau et P. Zimmermann. « The relationship between some guy and cryptography ». ECC2012 rump session talk (humoristic). 2012. url : http://ecc.2012.rump.cr.yp.to/ (cité p. 99).

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