# Algebraic Curves and Finite Fields: Cryptography and Other by Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne

By Harald Niederreiter, Alina Ostafe, Daniel Panario, Arne Winterhof

This publication collects the result of the workshops on purposes of Algebraic Curves and functions of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main famous researchers within the quarter of finite fields and their purposes all over the world, addressing outdated and new difficulties on curves and different facets of finite fields, with emphasis on their different functions to many parts of natural and utilized arithmetic.

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**Extra info for Algebraic Curves and Finite Fields: Cryptography and Other Applications (Radon Series on Computational and Applied Mathematics 16)**

**Sample text**

2, to find the trivariate polynomial ????????2 +????+1 (????, ????, ????). Finding a uniformizing element ????0 of ????(????0 (????2 +????+1)) is somewhat more elaborate. Since such a uniformizing element fills several pages, it is omitted. Below we will state the reduction of ????0 modulo ???? and ???? + 1, so the reader can get an impression of its form. Once ????0 is found, ????0 and ????1 can be expressed in terms of it. In this case we find (????0 + 1)3 (????20 + ????0 + ????2 + ???? + 1) ????0 = ????0 and (????0 + ????2 + ???? + 1)3 (????20 + ????0 + ????2 + ???? + 1) .

24, pp. 235–265, 1997. V. G. Drinfeld and S. G. Vladut, The number of points of an algebraic curve (translated from the Russian paper in Funktsional. Anal. i Prilozhen), Functional Anal. Appl. 17, pp.

The field ???? will play the role of the constant field in the towers we find. From these, towers with a finite field as a constant field can be obtained by reducing the defining equations by a suitably chosen prime element ???? of ???????? [????]. More precisely the constant field of such a reduced tower is ???????? := ???????? [????]/(????). To describe how to obtain (unreduced) towers, we will use the language of Drinfeld modules. Let ???? be a Drinfeld module of rank 2 with ????-invariant ????0 and ???????? be an ????-isogenous Drinfeld module with ????-invariant ????1 .