Recherche et téléchargement d’archives de revues mathématiques numérisées

  Table des matières de ce fascicule | Article suivant
Frey, Gerhard
On bilinear structures on divisor class groups. Annales mathématiques Blaise Pascal, 16 no. 1 (2009), p. 1-26
Texte intégral djvu | pdf | Analyses MR 2514524 | Zbl 1187.11044
Class. Math.: 11R65, 11R37, 11G20

URL stable:

Voir cet article sur le site de l'éditeur


It is well known that duality theorems are of utmost importance for the arithmetic of local and global fields and that Brauer groups appear in this context unavoidably. The key word here is class field theory. In this paper we want to make evident that these topics play an important role in public key cryptopgraphy, too. Here the key words are Discrete Logarithm systems with bilinear structures. Almost all public key crypto systems used today based on discrete logarithms use the ideal class groups of rings of holomorphic functions of affine curves over finite fields $\mathbf{F}_q$ to generate the underlying groups. We explain in full generality how these groups can be mapped to Brauer groups of local fields via the Lichtenbaum-Tate pairing, and we give an explicit description. Next we discuss under which conditions this pairing can be computed efficiently. If so, the discrete logarithm is transferred to the discrete logarithm in local Brauer groups and hence to computing invariants of cyclic algebras. We shall explain how this leads us in a natural way to the computation of discrete logarithms in finite fields. To end we give an outlook to a globalisation using the Hasse-Brauer-Noether sequence and the duality theorem ot Tate-Poitou which allows to apply index-calculus methods resulting in subexponential algorithms for the computation of discrete logarithms in finite fields as well as for the computation of the Euler totient function (so we have an immediate application to the RSA-problem), and, as application to number theory, a computational method to “describe” cyclic extensions of number fields with restricted ramification.


[1] R. Avanzi, H. Cohen, C. Doche, G. Frey, T. Lange, K. Nguyen and F. Vercauteren, The Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC, 2005  MR 2162716 |  Zbl 1082.94001
[2] P. S. L. M. Barreto, B. Lynn and M. Scott, Constructing elliptic curves with prescribed embedding degrees, Security in Communication Networks – SCN 2002, volume 2576 of Lecture Notes in Comput. Sci., Springer-Verlag, Berlin, 2003  Zbl 1022.94008
[3] P. S. L. M. Barreto and M. Naehrig, Pairing-friendly elliptic curves of prime order, Selected Areas in Cryptography – SAC’2005, Lecture Notes in Comput. Sci. 3897, Springer Verlag, Berlin, 2006  MR 2241646 |  Zbl 1151.94479
[4] D. Boneh and M. Franklin, Identity based encryption from the Weil pairing, SIAM J. Comput., 32(3):586-615, 2003  MR 2001745 |  Zbl 1046.94008
[5] D. Boneh, B. Lynn and H. Shacham, Short signatures from the Weil pairing, Advances in Cryptology – Asiacrypt 2001, Lecture Notes in Comput. Sci. 2248, Springer Verlag Berlin, 2002  MR 1934861 |  Zbl 1064.94554
[6] G. Frey, Applications of arithmetical geometry to cryptographic constructions, Finite fields and applications, Springer, Berlin, 2001  MR 1849086 |  Zbl 1015.94545
[7] G. Frey, On the relation between Brauer groups and discrete logarithms, Tatra Mt. Math. Publ., 33:199-227, 2006  MR 2271447 |  Zbl pre05125150
[8] G. Frey and T. Lange, Mathematical background of public key cryptography, Séminaires et Congrès SMF: AGCT 2003, SMF, 2005  MR 2182837 |  Zbl pre02231488
[9] G. Frey, M. Müller and H. G. Rück, The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems, IEEE Trans. Inform. Theory, 45(5):1717-1719, 1999  MR 1699906 |  Zbl 0957.94025
[10] G. Frey and H. G. Rück, A remark concerning $m$-divisibility and the discrete logarithm problem in the divisor class group of curves, Math. Comp., 62:865-874, 1994  MR 1218343 |  Zbl 0813.14045
[11] M.-D. Huang and W. Raskind, Signature calculus and discrete logarithm problems, Proc. ANTS VII, LNCS 4076, Springer, Berlin, 2006  MR 2282949 |  Zbl 1143.11363
[12] J Neukirch, Algebraic number theory, Springer, 1999  MR 1697859 |  Zbl 0956.11021
[13] A. Joux, A one round protocol for tripartite Diffie–Hellman, Proc. ANTS IV, LNCS 1838, Springer, 2000  MR 1850619 |  Zbl 1029.94026
[14] S. Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math., 7:120-136, 1969  MR 242831 |  Zbl 0186.26402
[15] B. Mazur, Notes on étale cohomology of number fields, Ann. sci. ENS, 64:521-552, 1973
Numdam |  MR 344254 |  Zbl 0282.14004
[16] V.C. Miller, The Weil Pairing, and Its Efficient Calculation, J.Cryptology, 17:235-261, 2004  MR 2090556 |  Zbl 1078.14043
[17] D. Mumford, Abelian Varieties, Oxford University Press, 1970  MR 282985 |  Zbl 0223.14022
[18] K. Nguyen, Explicit Arithmetic of Brauer Groups, Ray Class Fields and Index Calculus, Ph.D. thesis, University of Essen, 2001
[19] J.P. Serre, Groupes algébriques et corps de classes, Hermann, 1959  MR 103191 |  Zbl 0097.35604
[20] J.P. Serre, Corps locaux, Hermann, 1962  MR 354618 |  Zbl 0137.02601
[21] H. Stichtenoth, Algebraic Function Fields and Codes, Springer, 1993  MR 1251961 |  Zbl 0816.14011
[22] J. Tate, $WC$-groups over ${\mathfrak{p}}$-adic fields, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e éd. corrigée, Exposé 156 13, Secrétariat mathématique, 1958
Numdam |  MR 105420 |  Zbl 0091.33701
Copyright Cellule MathDoc 2014 | Crédit | Plan du site