Background on Weil descent

G. Frey, T. Lange

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

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Abstract

Weil descent — or, as it is alternatively called — scalar restriction, is a well-known technique in algebraic geometry. It is applicable to all geometric objects like curves, differentials, and Picard groups, if we work over a separable field L of degree d of a ground field K. It relates t-dimensional objects over L to td-dimensional objects over K. As guideline the reader should use the theory of algebraic curves over C, which become surfaces over R. This example, detailed in Section 5.1.2, already shows that the structure of the objects after scalar restriction can be much richer: the surfaces we get from algebraic curves carry the structure of a Riemann surface and so methods from topology and Kähler manifolds can be applied to questions about curves over C. This was the reason to suggest that Weil descent should be studied with respect to (constructive and destructive) applications for DL systems [FRE 1998]. We shall come to such applications in Sections 15.3 and 22.3. In the next two sections we give a short sketch of the mathematical properties of Weil descent. The purpose is to provide a mathematical basis for the descent and show how to construct it. For a thorough discussion in the frame of algebraic geometry and using the language of schemes, we refer to [Die 2001]
Original languageEnglish
Title of host publicationHandbook of Elliptic and Hyperelliptic Curve Cryptography
EditorsH. Cohen, G. Frey
Place of PublicationBoca Raton FL, USA
PublisherChapman & Hall/CRC Press
Chapter7
Pages125-132
ISBN (Print)1-58488-518-1
DOIs
Publication statusPublished - 2006

Publication series

NameDiscrete Mathematics and Its Applications
Volume34

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