# LATTICE REDUCTION ALGORITHMS

5-2022

Thesis

## Degree Name

Master of Arts in Mathematics

Mathematics

Meyer, Jeffrey

## Abstract

The purpose of this thesis is to propose and analyze an algorithm that follows
similar steps of Guassian Lattice Reduction Algorithm in two-dimensions and applying
them to three-dimensions. We start off by discussing the importance of cryptography in
our day to day lives. Then we dive into some linear algebra and discuss specific topics that
will later help us in understanding lattice reduction algorithms. We discuss two lattice
problems: the shortest vector problem and the closest vector problem. Then we introduce
two types of lattice reduction algorithms: Guassian Lattice Reduction in two-dimensions
and the LLL Algortihm. We illustrate how both algorithms can provide solutions to the
shortest vector problem and closest vector problem. Afterwards, we use an application
of a cryptosystem, GGH and show how one could break this cryptosystem using lattice
reduction algorithms.
Finally, we propose and analyze a new algorithm that applies the principles of
Gaussian Lattice Reduction Algorithm in two-dimensions. We provide an original proof of
this algorithm outputting a shortest vector in a given lattice L ∈ R3. We state a conjecture
about the termination of the algorithm. Then we analyze this new algorithm and compare
it the LLL Algorithm and Gaussian Lattice reduction Algorithm in two-dimensions. We
provide examples of different three-dimensional lattices. We then summarize the overall
paper and discuss potential research in the near future.

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