Paper Info
Title
Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL.
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 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 applications in number theory, computer algebra and cryptography. In this paper, we provide an implementation of the LLL algorithm. Both its soundness and its polynomial running-time have been verified using Isabelle/HOL. Our implementation is nearly as fast as an implementation in a commercial computer algebra system, and its efficiency can be further increased by connecting it with fast untrusted lattice reduction algorithms and certifying their output. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.
Year
DOI
Venue
2020
10.1007/s10817-020-09552-1
JOURNAL OF AUTOMATED REASONING
Keywords
DocType
Volume
Certified algorithm,Complexity verification,Lattices,Polynomial factorization,Shortest vector problem,Verified LLL implementation
Journal
64.0
Issue
ISSN
Citations 
SP5
0168-7433
0
PageRank 
References 
Authors
0.34
0
6
Name
Order
Citations
PageRank
René Thiemann198469.38
Ralph Bottesch200.34
Jose Divasón3289.96
Max W. Haslbeck400.68
Sebastiaan J. C. Joosten5176.87
Akihisa Yamada 00026347.11