### A Formalization of the LLL Basis Reduction Algorithm

Jose Divasón, Sebastiaan Joosten, René Thiemann, Akihisa YamadaProceedings of the 9th International Conference on Interactive Theorem Proving, Lecture Notes in Computer Science 10895, pp. 160 – 177, 2018.

### Abstract

The LLL basis reduction algorithm was the first polynomial-time algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It thereby approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has several applications in number theory, computer algebra and cryptography.

In this paper, we develop the first mechanized soundness proof of the LLL algorithm using Isabelle/HOL. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.

### BibTeX

@inproceedings{JDSJRTAY-ITP18, title = {A Formalization of the {LLL} Basis Reduction Algorithm}, author = {Jose Divas\'on and Sebastiaan Joosten and Ren\'e Thiemann and Akihisa Yamada}, year = 2018, booktitle = {Proceedings of the 9th International Conference on Interactive Theorem Proving}, editor = {Jeremy Avigad and Assia Mahboubi}, series = {Lecture Notes in Computer Science}, pages = {160--177}, volume = {10895}, doi = {10.1007/978-3-319-94821-8_10} }