Lecture Notes on Lattices, Bases, and the Reduction Problem: Expository Notes (Classic Reprint) - Hardcover

Bud Mishra

9780265271049: Lecture Notes on Lattices, Bases, and the Reduction Problem: Expository Notes (Classic Reprint)

Hardcover

ISBN 10: 0265271045 ISBN 13: 9780265271049

Publisher: Forgotten Books, 2018

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A practical tour of lattices, bases, and reduction methods for numbers and spaces.

This book introduces the geometry of numbers, lattice bases, and the famous LLL basis reduction algorithm with clear definitions, theorems, and proofs.

Geared toward readers with a solid math background, it explains how a lattice can have many bases that relate through unimodular transformations. The text covers determinants as a lattice invariant, dual lattices, and Gram–Schmidt orthogonalization, with Hadamard’s inequality tying geometry to linear algebra. It then moves to algorithmic questions: how to find nice bases and short lattice vectors, and what problems are computationally hard.

What a lattice is, how bases and determinants work, and why unimodular transformations matter.
How Gram–Schmidt orthogonalization connects basis vectors to orthogonal components.
Theoretical bounds and practical algorithms for reducing bases and finding short vectors.
Key results like Minkowski’s theorem, Hermite’s bound, and the LLL reduction procedure.

Ideal for readers of advanced mathematics or theoretical computer science who want solid, applicable insight into lattice reduction and its limits.

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