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KryFoVe: Computeralgebra

Project Topic

Applications of advanced algebraic-symbolic methods of computer algebra to crypto analysis, to coding theory and to formal verification of microelectronic systems.

Project Description

The notion of the Groebner basis emerged in the 1960s in the works of Bruno Buchberger, where this notion was introduced along with some algorithmic means to compute it. Soon the notion of the Groebner basis became an important tool for mathematicians, in particular for those working in algebraic geometry and singularity theory. Recently, Groebner bases drew a lot of attention from researchers working in applied mathematics. This is mainly due to applications of Groebner bases in solving systems of polynomial equations. The Groebner bases found their applications in solving systems, where classical numerical methods are unstable or false, also in coding theory, cryptography, control and system theory, biology and other areas. KryFoVe aims at further research of applications of Groebner bases in coding theory and cryptanalysis. We investigate and propose new methods of how the problem of decoding a message, which has been transmitted via a noisy channel, can be reduced to solving a system of polynomial equations over a finite field and can then efficiently be solved. In the frame of this project we also investigate algebraic attacks on block/stream ciphers.

Project Members

Project Chair

Participating Research Groups

Scientific Personnel

PhD Students:

External Cooperation

Project Events and Achievements

Project start: 01.03.2006 with a duration until the middle of 2008

Visits by external researchers:

  • 8. December 2005: Le Van Schroeer (Bundesamt fuer Sicherheit in der Informationstechnik, Bonn)
  • 16. - 20. January 2006: Ruud Pellikaan (Department of Mathematics, Technical University of Eindhoven, Netherlands)
  • 22. - 24. May 2006: Christopher Hillar (Texas A&M and RICAM)
  • 12. - 25. June 2006: Sergiy Ovsienko (Department of Mechanics and Mathematics, Kyiv Shevchenko University)
  • 17. - 18. January 2007: Martin Albrecht (University of Bremen)
  • 17. January - 14. February 2007: Sergiy Ovsienko (Department of Mechanics and Mathematics, Kyiv Shevchenko University)
  • 26. May - 2. June 2007: José Ignacio Farrán (Universidad de Valladolid, Spain)

Talks:

Project Publications

Bounded distance decoding of linear error-correcting codes with Gröbner bases

Stanislav Bulygin, Ruud Pellikaan. In: Journal of Symbolic Computation Special Issue Gröbner Bases Techniques in Cryptography and Coding Theory, to appear. Available online: http://dx.doi.org/10.1016/j.jsc.2007.12.003, 2009

Decoding linear error-correcting codes up to half the minimum distance with Gröbner bases

Stanislav Bulygin, Ruud Pellikaan. In: T. Mora and L. Perret and S. Sakata and M. Sala and C. Traverso ed., Working Title: "Gröbner Bases, Coding, and Cryptography". Proceedings of the Special Semester on Gröbner Bases, to appear, Volume 4, RISC Book Series (Springer, Heidelberg), 2009

Attacking AES via Solving Systems in the Key Variables Only

Michael Brickenstein, Stanislav Bulygin. In: First International Conference on Symbolic Computation and Cryptography, Beijing, China. P. 118--123, April, 2008

Decoding and finding the minimum distance with Gröbner bases: history and new insights

Stanislav Bulygin, Ruud Pellikaan. In: Selected Topics in Information and Coding Theory. World Scientific, 2008

Towards a Better Understanding of the Semigroup Tree

Maria Bras-Amoros, Stanislav Bulygin. In: Discrete Mathematics. Submitted, 2008

Obtaining and solving systems of equations in key variables only for the small variants of AES

Stanislav Bulygin, Michael Brickenstein. In: Mathematics in Computer Science, Special Issue "Symbolic Computation and Cryptography". Available online: http://eprint.iacr.org/2008/435, Submitted, 2008

Decoding error-correcting codes with Groebner bases

Stanislav Bulygin, Ruud Pellikaan. In: Proceedings of the 28th Symposium on Information Theory in the Benelux, Enschede, The Netherlands. P. 3--10, May, 2007

On the structure of generalized toric codes

Diego Ruano. In: Special Issue of JSC EACA'06. 2007

Groebner basis: applications to algebraic coding theory (in Spanish)

E. Martinez-Moro, C. Munuera, Diego Ruano. Instituto Venezolano de Investigaciones Cientificas, 125 pages. ISBN: 978-980-261-087-7, 2007

On the Parameters of r-dimensional Toric Codes

Diego Ruano. In: Finite Fields and their Applications. Volume 13, Number 4, Elsevier, P. 962--976, 2007

Noncommutative Polly Cracker-type cryptosystems and chosen-ciphertext security

T. Rai, Stanislav Bulygin. In: Preprint. Available online: http://eprint.iacr.org/2008/514, 2007

Generalized Hermitian Codes over GF(2^r)

Stanislav Bulygin. In: IEEE Transactions on Information Theory. 2006

Equisingular Calculations for Plane Curve Singularities

A. Campillo, Gert-Martin Greuel, C. Lossen. In: Proceedings of MEGA 2003. 2006

r31 - 01 Dec 2008 - StanislavBulygin

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