# A Formalization of the LLL Basis Reduction Algorithm

### Authors:

- Divasón, Jose
- Joosten, Sebastiaan J. C.
- Thiemann, René
- Yamada, Akihisa

Open Access

Published in: Interactive Theorem Proving. Avigad, Jeremy Mahboubi, Assia (Eds) pp 160–177

DOI: `10.1007/978-3-319-94821-8_10`

### 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 entry:

```
@inbook{Divason18,
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.},
author = {Jose Divas{\'o}n and Sebastiaan Jozef Christiaan Joosten and Ren{\'e} Thiemann and Akihisa Yamada},
bdsk-url-1 = {https://doi.org/10.1007/978-3-319-94821-8_10},
booktitle = {Interactive Theorem Proving},
day = {4},
doi = {10.1007/978-3-319-94821-8_10},
editor = {Jeremy Avigad and Assia Mahboubi},
isbn = {978-3-319-94820-1},
language = {English},
month = {7},
pages = {160--177},
publisher = {Springer},
series = {Lecture Notes in Computer Science},
title = {A Formalization of the LLL Basis Reduction Algorithm},
year = {2018}
}
```