A Verified Efficient Implementation of the LLL Basis Reduction Algorithm

17 pages•Published: October 23, 2018

Ralph Bottesch, Max W. Haslbeck and René Thiemann

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 approximately solves an NP-hard problem. The algorithm has several applications in number theory, computer algebra and cryptography.
Recently, the first mechanized soundness proof of the LLL algorithm has been developed in Isabelle/HOL. However, this proof did not include a formal statement of the algorithm’s complexity. Furthermore, the resulting implementation was inefficient in practice.
We address both of these shortcomings in this paper. First, we prove the correctness of a more efficient implementation of the LLL algorithm that uses only integer computations. Second, we formally prove statements on the polynomial running-time.

Keyphrases: complexity, isabelle/hol, lattice basis reduction

In: Gilles Barthe, Geoff Sutcliffe and Margus Veanes (editors). LPAR-22. 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 57, pages 164-180.

Links:	https://easychair.org/publications/paper/spJt
	https://doi.org/10.29007/xwwh

BibTeX entry

@inproceedings{LPAR-22:Verified_Efficient_Implementation_LLL,
  author    = {Ralph Bottesch and Max W. Haslbeck and René Thiemann},
  title     = {A Verified Efficient Implementation of the LLL Basis Reduction Algorithm},
  booktitle = {LPAR-22. 22nd International Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Gilles Barthe and Geoff Sutcliffe and Margus Veanes},
  series    = {EPiC Series in Computing},
  volume    = {57},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/spJt},
  doi       = {10.29007/xwwh},
  pages     = {164-180},
  year      = {2018}}

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