# 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 e-book 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 in demand researchers within the region of finite fields and their functions world wide, addressing previous and new difficulties on curves and different facets of finite fields, with emphasis on their assorted purposes to many parts of natural and utilized arithmetic.

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**Example text**

As a means of making Condition (ii) more precise, one could restrict to the case that ???? is indecomposable. Another variation is to restrict to the Jacobians of smooth curves. 7]. 2. The original motivation for this nonexistence expectation was the following. Let ????(????????) (????, ????) denote the statement: There exists an Abelian variety ???? of dimension ???? defined over ???? which is not isogenous to the Jacobian of any curve of compact type; (here, this is an isogeny of Abelian varieties without polarization).

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbol. Comput. 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.

Finding a uniformizing element ????0 of ????(????0 (????2 + ????)) and expressing ????0 and ????1 in it, we find ????0 = (????30 + (????2 + ????)????0 + (????2 + ????))3 ????0 (????0 + ????)2 (????0 + ???? + 1)2 and ????1 = (????30 + (????2 + ????)????20 + (????2 + ????)2 )3 . ????40 (????0 + ????)2 (????0 + ???? + 1)2 To find ????????2 +???? (????, ????), we need to factor a bivariate polynomial of ????-degree 9. 2 does not apply, though it still predicts the existence of one factor of ????degree 1. The factors turn out to be ???????? + ????2 + ???? , ????2 ????2 + ????????2 ???? + (????2 + ????)???????? + (????3 + ????2 )???? + ????2 ????2 + ????4 + ????2 , ????2 ????2 + (???? + 1)????2 ???? + (????2 + ????)???????? + (????3 + ????)???? + (????2 + 1)????2 + ????4 + ????2 , and ????4 ????3 + ????4 ????2 + (????2 + ????)????4 ???? + (????2 + ????)????3 ????2 + (????2 + ????)????3 ???? + (????4 + ????2 )????3 + (????2 + ???? + 1)????2 ????3 + (????4 + ????2 )????2 ???? + (????4 + ????2 )????2 + (????2 + ????)????????3 + (????4 + ????)????????2 + (????6 + ????5 + ????4 + ????3 )???? + ????4 .