Topics in Geometry, Coding Theory and Cryptography

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Topics in Geometry, Coding Theory and Cryptography (Algebra and Applications)
Sp--nger | ISBN: 1402053339 | 2006. | 210 pages | PDF | 2 MB

The purpose of this reviewarticle is to serve as an introduction and at the same time, as an invitation to the theory of towers of function fields over finite fields. More specifically, we treat here the case of explicit towers; i.e., towers where the function fields are given by explicit equations. The asymptotic behaviour of the genus and of the number of rational places in towers are important features for applications to coding theory and to cryptography.
The interest in solutions of algebraic equations over finite fields has a long history in mathematics, especially when the equations define a one-dimensional object (a curve or, equivalently, a function field). The major result of this theory is the Hasse-Weil theorem which gives in particular an upper bound for the number of rational points in terms of the genus of the curve and of the cardinality of the finite field.



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