Arithmétique des corps de fonctions et ses applications à l'algorithmique et à la cryptologie

Abstract : This thesis is devoted to the study of arithmetic of functional fields and their applications to cryptology. We first present most classical and general results on irreducible polynomials over a finite field $(\bf F)$. To be more specific, we provide existing irreducibility tests and methods to construct such polynomials, either by composition or recursively. Next, we review basic facts on $p$-adic number fields, give the formula of the discriminant of a trinomial and then recall Swan's theorem. We also present an application due to Swan, proving that there is no trinomial of degree $n$ which is irreducible over $(\bf G)$, when $n$ is a multiple of $8$. We then apply these methods to pentanomials. Next, we present basic facts and results on Drinfeld modules over $A=(\bf F)[T]$ and discuss analogies between elliptic curves and Drinfeld modules such as the structure of torsion points, isogenies and the Hasse theorem. We then use elementary methods to determine all possible structures of the $A$-torsion points and a uniform bound for the set of $B$-rational torsion points, where $B$ is a finite integral extension of $A$ and the Drinfeld module $\varphi$ ranges over all Drinfeld modules defined over $A$ and $B$ respectively. Finally, we focus on Drinfeld modules over finite fields and their applications to cryptology.
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Theses
Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2004. French


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Submitted on : Thursday, October 14, 2004 - 8:37:40 AM
Last modification on : Thursday, October 14, 2004 - 8:37:40 AM

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Alexander Gewirtz. Arithmétique des corps de fonctions et ses applications à l'algorithmique et à la cryptologie. Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2004. French. <tel-00007102>

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