Abderrahmane Nitaj

The abc conjecture

To the abc conjecture home page

Research Interests

Number Theory :

Diophantine Equations, the abc conjecture, Polynomials.

Arithmetic Geometry :

Elliptic curves, Szpiro's conjecture.

Cryptography :

Cryptosystems, The discrete logarithm problem, Elliptic curves.

Implementation of Computer Algebra Algorithms :

SIMATH, APECS, PARI, MAGMA, MAPLE, UBASIC, LIDIA

Fractals :

Fractal pictures, Animations.

Thesis

Conséquences et aspects expérimentaux des conjectures abc et de Szpiro (1994).
[DVI] [Postscript]

Publications

Cryptanalysis of RSA using the ratio of the primes, AFRICACRYPT2009, In Bart Preneel (Ed.): Progress in Cryptology - AFRICACRYPT 2009, Lecture Notes in Computer Science 5580 Springer 2009. Slides
Another generalization of Wiener's attack on RSA, AFRICACRYPT2008, In Serge Vaudenay (Ed.): Progress in Cryptology - AFRICACRYPT 2008, Lecture Notes in Computer Science 5023 Springer 2008. Slides
Application of ECM to a class of RSA keys, Journal of Discrete Mathematical Sciences & Cryptography , To appear
[PS] [PDF]
RSA and a higher degree diophantine equation, Submitted to Journal of Computational Mathematics.
[PS] [PDF]
Cryptanalysis of RSA with constrained keys. International Journal of Number Theory (IJNT), V. 5, p. 311-325 (2009)
[PS] [PDF]
Isogènes des courbes elliptiques définies sur les rationnels, J. Computational Math. 4 (2002), 337-448.
[DVI] [Postscript]
Invariants des courbes de Frey-Hellegouarch et grands groupes de Tate-Shavarevich, Acta Arithmetica 93 (2000), no. 4, pp. 303-327.
[DVI] [Postscript]
(With G. Greaves) Some polynomial identities related to the $abc$- conjecture, Gyoery, Kalman (ed.) et al., Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30--July 9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de Gruyter (1999), 229-236.
[DVI] [Postscript]
Détermination de courbes elliptiques pour la conjecture de Szpiro, Acta Arithmetica 85 (1998), pp. 351-376.
[DVI] [Postscript]
Aspects expérimentaux de la conjecture $abc$, Séminaire de Théorie des Nombres de Paris, London Math. Soc. 235 (1996), pp. 145-156.
[DVI] [Postscript]
La conjecture $abc$, L'Enseignement Mathématique 42 (1996), pp. 3-24.
[DVI] [Postscript]
L'algorithme de Cornacchia, Expositiones Math. 13 (1995), pp. 358-365.
[DVI] [Postscript]
On a conjecture of Erd�s on 3-powerful numbers, Bulletin of the London Mathematical Society 27 (1995), pp. 317-318.
[DVI] [Postscript]
Algorithms for finding good examples for the $abc$ and the Szpiro conjectures, Experimental Mathematics 3 (1993), pp. 223-230.
[Postscript]
An algorithm for finding good $abc$-examples, Comptes Rendus de l'Académie des Sciences de Paris, 317 (1993), pp. 811-815.

Informal Notes

Cryptanalyse de RSA [Course, PDF]
Cryptanalyse de RSA [Maple 12 Worksheet]
Cryptanalyse de RSA
Talk at Ecole CIMPA de Cryptography Oujda (Morocco). [PDF]
Introduction aux courbes elliptiques
Talk at Journées Cryptographie Rabat(Morocco). [PDF]
Clés faibles pour le cryptosystème RSA
Talk at Symposium Cryptographie Et Sécurité de l�Information Oujda (Morocco). [PDF]
Le cryptosystème NTRU, réalités et perspectives
Talk at Symposium Cryptographie Et Sécurité de l�Information Oujda (Morocco). [PDF]
RSA et les équations diophantiennes
Seminar talk at Cryptography Groupe, Caen. [PDF] , [PS]
Cryptanalysis of RSA with constrained keys
Seminar talk at MyCrypt 2005,. [PDF] , [PS]
Cryptanalyse de certains RSA par les fractions continues et l'algorithme LLL.
Seminar talk at Cryptography Groupe, Caen. [PDF] , [PS]
Petites hauteurs des polynômes par l'algorithme LLL
Seminar talk at Cryptography Groupe, Caen. [PDF] , [PS]
A Maple Worksheet for elliptic curves
This Maple Worksheet gives an explicit model for all elliptic curves defined over Q with a non trivial torsion rational point and for all their isogenous curves associated to isogenies with cyclic kernels consisting of rational points.
Le problème du logarithme discret elliptique : Index et Xedni [DVI] , [PS]
Le cryptosystème NTRU. [DVI] , [PS]
Table of all good abc examples [PDF] , [DVI] , [PS]
Table of all good abc-Szpiro examples [PDF] , [DVI] , [PS]