Symmetrized summation polynomials: Using small order torsion points to speed up elliptic curve index calculus

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 8441 LNCS, Issue , 2014, Pages 40-57

  Faugère, Jean Charles ^a,b,c   Huot, Louise ^a,b,c   Joux, Antoine ^b,c,d,e   Renault, Guénaël ^a,b,c   Vitse, Vanessa ^f  

^a INRIA   (France)

^b SORBONNE UNIVERSITÉ   (France)

^c UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^d CryptoExperts ^*  (France)

^e Cryptologie de la Fondation UPMC   (France)

^f UNIV GRENOBLE ALPES   (France)

Author keywords

decomposition method; ECDLP; elliptic curves; index calculus; invariant theory; multivariate polynomial systems; Semaev polynomials

Indexed keywords

CALCULATIONS; CRYPTOGRAPHY; GEOMETRY; TORSIONAL STRESS;

DECOMPOSITION METHODS; ECDLP; ELLIPTIC CURVE; INDEX CALCULUS; INVARIANT THEORY; MULTIVARIATE POLYNOMIAL;

POLYNOMIALS;

EID: 84901675900     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-55220-5_3     Document Type: Conference Paper

References (16)

1. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235-265 (1997);
2. Computational algebra and number theory, London (1993)
3. Diem, C.: On the discrete logarithm problem in elliptic curves. Compos. Math. 147(1), 75-104 (2011)
4. Faugère, J.-C.: FGb: A Library for Computing Gröbner Bases. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 84-87. Springer, Heidelberg (2010)
5. Faugère, J.-C., Gaudry, P., Huot, L., Renault, G.: Using symmetries in the index calculus for elliptic curves discrete logarithm. J. Cryptology, 1-41 (2013), doi:10.1007/s00145-013-9158-5.
6. Faugère, J.-C., Gianni, P., Lazard, D., Mora, T.: Efficient computation of zerodimensional Gröbner bases by change of ordering. J. Symbolic Comput. 16(4), 329-344 (1993)
7. Gaudry, P.: Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem. J. Symbolic Comput. 44(12), 1690-1702 (2008)
8. Gaudry, P., Hess, F., Smart, N.P.: Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptology 15(1), 19-46 (2002)
9. Granger, R., Joux, A., Vitse, V.: New timings for oracle-assisted SDHP on the IPSEC Oakley 'Well Known Group' 3 curve. Announcement on the NBRTHRY Mailing List (July 2010), http://listserv.nodak.edu/archives/nmbrthry.html
10. IETF. The Oakley key determination protocol. IETF RFC 2412 (1998)
11. p6. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 9-26. Springer, Heidelberg (2012)
12. Joux, A., Vitse, V.: Elliptic curve discrete logarithm problem over small degree extension fields. J. Cryptology 26(1), 119-143 (2013)
13. Koblitz, N., Menezes, A.: Another look at non-standard discrete log and Diffie-Hellman problems. J. Math. Cryptol. 2(4), 311-326 (2008)
14. Semaev, I.A.: Summation polynomials and the discrete logarithm problem on elliptic curves. Cryptology ePrint Archive, Report 2004/031 (2004)
15. Serre, J.-P.: Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15(4), 259-331 (1972)
16. Zippel, R.: Interpolating polynomials from their values. Journal of Symbolic Computation 9(3), 375-403 (1990)