Christophe Doche

Lecturer	Member of Centre for Advanced Computing - Algorithms and Cryptography

How to contact me

Bio

Education

• M.S. in Mathematics, 1995; University of Bordeaux
• Ph.D. in Mathematics, 2000; University of Bordeaux. Title: Mahler measure and real roots of certain family of polynomials (in French)

Areas of Interest

• Cryptography (Elliptic and Hyperelliptic Curve Cryptography)
• Analytic Number Theory (Lehmer's problem, Mahler measure, Moments of Rudin-Shapiro polynomials)
• Algorithmic Number Theory and efficiency aspects

Current Research

• Algebraic and Number Theoretic Methods in Cryptography
• Computation and Algorithms in Finite Fields
• Elliptic and Hyperelliptic curve cryptography

Cours de l'Ecole Doctorale

• Cours du 10 decembre 2007 et le log file PARI
• Cours du 17 decembre 2007, le log file PARI et un fichier maple
• Cours du 07 janvier 2008, le log file PARI et une archive contenant les modules schoof.gp, sea.gp, ecc_ed.gp, et AGM.gp
• Cours du 09 janvier 2008 et le fichier pari pour calculer dans Z[tau]

Honours Projects

• Elliptic Curve Cryptography

  The use of elliptic curves in Public Key Cryptography has been first suggested in 1985. From this date, elliptic curves have become increasingly popular mainly because the only algorithms that we know to solve the hard problem on which cryptography is based, namely the elliptic curve discrete logarithm problem, are of exponential complexity. This is in contrast with the existence of sub-exponential algorithms to solve the factorisation problem on which RSA is based. As a consequence, much shorter keys can be used in elliptic curve cryptography (ECC) to achieve the same level of security provided by RSA. For instance, a key of 160 bits in ECC is believed to be as secure as a 1024-bit RSA key.
  The goal of this project is to research new fast and secure ways to perform arithmetic on elliptic curves and to implement a cryptosystem based on these curves.

• Fair Automarking

  In many areas of computer science, students needs to write programs and receive an appropriate feedback to improve their skills. At the same time, from an instructor point of view, it is clearly impossible to run and evaluate all these programs on an individual basis. A popular solution to this problem is to use an automarking scheme. This is usually a script able to compile each student's program and run it on predefined inputs. The corresponding outputs are then compared with the expected outputs generated by a correct program. This comparison is often done character by character, for instance using diff. Even if some improvements can be made (for instance by ignoring blank characters or letters case), every year a large proportion of students invariably complain that the automarking system is unfair: Despite their program being "mostly correct", they obtained a poor mark in return.
  The aim of this project is to design an automarking system that can be executed by a student. This system should be able to display differences between the output generated by a student's program and the correct output. This will allow students to correct typos and small mistakes on the fly. When required, the system will be asked to generate a file containing enough information to identify a student and compute its mark. This file will later be electronically submitted by the student to a server gathering all the marks.
  Cryptographic techniques will be essential in this project to authenticate students work in order to prevent plagiarism.

Other Recent and Current Activities

• Organiser of the ACAC seminar

Publications

• Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC Press, 2005, with H. Cohen and G. Frey, Editors.
  See also this link for more information, including the Table of Contents, the References, and a sample Chapter.

• Integral geometry and real zeros of Thue-Morse polynomials, with Michel Mendès France, Experiment. Math. 9 no. 3, 339-350, (2000).

• On the spectrum of the Zhang-Zagier height, Math. Comp. 70 no. 233, 419-430, (2001).
  

• Zhang-Zagier heights of perturbed polynomials, Proceedings of the XXIe Journées Arithmétiques. Journal de Théorie des Nombres de Bordeaux 13, 103-110, (2001).
• On the real roots of generalized Thue-Morse polynomials, Acta Arith. XCIX.4 309-319, (2001).
• Real roots! with Michel Mendès France, International Conference in Honour of Jonas Kubilius, Palanga, 2001, Analytic and probabilistic methods in number theory, 42-56, (2002)
• Moments of the Rudin-Shapiro Polynomials, with Laurent Habsieger, J. Fourier Analysis Appl. Vol. 10 Issue 5, 497-505, (2004).
• An exercise on the average number of real zeros of random real polynomials, with Michel Mendès France, Finite and infinite combinatorics, Budapest, 2001. J. Bolyai Mathematical Society, 1-14, (2005).
• Redundant Trinomials for Finite Fields of Characteristic 2, Proceedings of ACISP 05, LNCS 3574, 122-133, (2005).
  A table of redundant polynomials written in gp is also available.
• Even moments of generalized Rudin-Shapiro polynomials, Math. Comp. 74 no. 252, 1923-1935, (2005).
  A gp program to precisely compute the moments of the Rudin-Shapiro polynomial of order q even less than or equal to 32 and the moments of some generalized Rudin-Shapiro polynomials is available here [moments.gp] as well as the tables of minimal polynomials for the recurrences and the first moments [data.gz] and [data32.gz]
• Efficient Scalar Multiplication by Isogeny Decompositions, with Thomas Icart and David Kohel, Proceedings of PKC 06, LNCS 3958, 191-206, (2006).
• Extending Scalar Multiplication using Double Bases, with Roberto Avanzi, Vassil Dimitrov, and Francesco Sica, Proceedings of AsiaCrypt 06, LNCS 4284, 130-144, (2006).
• Extended Double-Base Number System with applications to Elliptic Curve Cryptography, with Laurent Imbert, Proceedings of IndoCrypt 06, LNCS 4329, 335-348, (2006).
  

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