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

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

ISBN-10: 3110317885

ISBN-13: 9783110317886

This e-book collects the result of the workshops on functions of Algebraic Curves and purposes of Finite Fieldsat the RICAMin 2013. those workshops introduced jointly the main fashionable researchers within the sector of finite fields and their functions worldwide, addressing outdated and new difficulties on curves and different features of finite fields, with emphasis on their diversified functions to many components of natural and utilized arithmetic.

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Doré Subrao, The ????-rank of Artin–Schreier curves, Manuscripta Math. 16 (1975), 169–193. Jacob Tsimerman, The existence of an Abelian variety over ℚ isogenous to no Jacobian, Ann. Math. 176(2) (2012), 637–650. Gerard van der Geer and Marcel van der Vlugt, Reed–Muller codes and supersingular curves. I, Compositio. Math. 84 (1992), 333–367. Kazuhiko Yamaki, Cornalba–Harris equality for semistable hyperelliptic curves in positive characteristic, Asian J. Math. 8 (2004), 409–426. Alp Bassa, Peter Beelen, and Nhut Nguyen Good towers of function fields Abstract: In this chapter, we will give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields.

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 .

The important fact is that we again obtain an irreducible algebraic curve defined over ????2 which determines the isomorphism classes of possible rank 2 Drinfeld modules. This modular curve is known to have genus zero and to be irreducible, but not absolutely irreducible, see [14]. There it is also shown that the number of components is equal to the class number ℎ???? , over which extension field these components are defined and how the Galois group of this extension acts on the components. In our case we obtain that there are five components defined over ????32 and that the Frobenius map of ????32 /????2 acts transitively on these five components.

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


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