Citation
Torreão Dassen, E. (2011, December 20). Basis reduction for layered lattices.
Retrieved from https://hdl.handle.net/1887/18264
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Basis reduction for layered lattices
Proefschrift
ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof. mr. P.F. van der Heijden, volgens besluit van het College voor Promoties te verdedigen op dinsdag 20 December 2011 klokke 16:15 uur.
door
Erwin Lavalli´ere Torre˜ao Dassen,
geboren te Campina Grande, Brazili¨e in 1979.
Overige leden
Prof. Dr. P. Stevenhagen,
Prof. Dr. J. E. Cremona (University of Warwick), Prof. Dr. K. Aardal (TU Delft),
Prof. Dr. R. Cramer (Universiteit Leiden, CWI), Dr. B. de Smit.
Basis reduction for layered lattices
This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/ or send a letter to Cre- ative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
The research leading to this work was supported by Marie-Curie Actions and by NWO.
Typeset in LATEX.
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ISBN/EAN: 978-90-818191-0-7
To my friends and family; to peaceful, happy, and greedless coexistence.
Contents
1 Introduction 9
1.1 Main results . . . . 9
1.2 Review on ordered sets, and on algebra . . . . 14
1.3 Review on complexity theory . . . . 16
1.4 Notation . . . . 17
2 Ordered vector spaces 21 2.1 Ordered rings and fields . . . . 21
2.2 Ordered vector spaces . . . . 23
2.3 Real ordered vector spaces . . . . 27
2.4 Symmetric powers . . . . 28
3 Layered Euclidean spaces 33 3.1 Layered forms . . . . 33
3.2 Orthogonality . . . . 39
3.3 Exterior powers of layered Euclidean spaces . . . . 44
4 Layered lattices 49 4.1 Embedded layered lattices . . . . 49
4.2 Layered lattices . . . . 51
4.3 Exterior powers of layered lattices . . . . 61
4.4 The discriminant . . . . 61
5 The layered Gram-Schmidt procedure 63 5.1 Associated Gram-Schmidt bases . . . . 63
7
6.2 The layered LLL algorithm . . . . 82 6.3 A polynomial-time reduction algorithm . . . . 85
A Two algorithms 91
