Skew polynomials, Gabidulin codes, coding and cryptography

Pierre Loidreau
University of Rennes
France

Abstract:

Skew polynomial rings were originally introduced by Ore in 1933, as a "general non-commutative polynomial theory". The algorithmic applications of these left and right Euclidean rings were first studied in the field of symbolic computation under the denomination of finite difference operator rings. In 2006 a work by Boucher, Geiselmann and Ulmer backed up this field in the coding theory. Under a proper evaluation operator, Gabidulin codes are evaluation codes of skew polynomials, and the so-called rank metric is the natural adapted metric to skew polynomial rings.

In this talk, we will show how to rephrase and generalize the Gabidulin codes theory to more general fields than finite fields, and try to emphasize how the theory of skew polynomial rings increases our insight of the Gabidulin codes theory.

Then we will show how skew polynomials theory and Gabidulin codes can be used to designed bricks or primitive for symmetric and asymmetric cryptography. Finally, it there is time enough, we will show how to design optimal rate/diversity trade off codes for space-time coding from the generalization of Gabidulin codes to number fields.

Biography:

Pierre Loidreau is working in a cryptographic laboratory of the French Department of Defense and he is an associate researcher at the Institute of Mathematical Research at University of Rennes. His main scientific interest includes error-correcting codes and their applications to cryptography and telecommunications.