LLL Algorithm for Lattice Basis Reduction

TL;DR

This paper provides an accessible introduction to the polynomial-time LLL lattice basis reduction algorithm, explaining its components, correctness, runtime, and applications in solving the shortest vector problem and other mathematical problems.

Contribution

It offers a clear, detailed exposition of the LLL algorithm, including proofs and applications, making it accessible to readers with basic linear algebra knowledge.

Findings

Proves the correctness and polynomial runtime of the LLL algorithm.

Demonstrates the application of LLL to the shortest vector problem.

Explores additional applications of the LLL algorithm in mathematics.

Abstract

This is an expository paper intended to introduce the polynomial time lattice basis reduction algorithm first described by Arjen Lenstra, Hendrik Lenstra, and L\'aszl\'o Lov\'asz in 1982. We begin by introducing the shortest vector problem, which motivates the underlying components of the LLL algorithm. Then, we introduce the details of the algorithm itself, followed by proofs of the correctness and runtime of the algorithm in complete detail, assuming only a basic linear algebra background and an understanding of big O notation. Finally, we apply the LLL algorithm to the shortest vector problem and explore other applications of the algorithm in various mathematical settings.