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A verified factorization algorithm for integer polynomials with polynomial complexity
| Title: | A verified factorization algorithm for integer polynomials with polynomial complexity |
| Authors: | Jose Divasón, Sebastiaan Joosten, René Thiemann and Akihisa Yamada (ayamada /at/ trs /dot/ cm /dot/ is /dot/ nagoya-u /dot/ ac /dot/ jp) |
| Submission date: | 2018-02-06 |
| Abstract: | Short vectors in lattices and factors of integer polynomials are related. Each factor of an integer polynomial belongs to a certain lattice. When factoring polynomials, the condition that we are looking for an irreducible polynomial means that we must look for a small element in a lattice, which can be done by a basis reduction algorithm. In this development we formalize this connection and thereby one main application of the LLL basis reduction algorithm: an algorithm to factor square-free integer polynomials which runs in polynomial time. The work is based on our previous Berlekamp–Zassenhaus development, where the exponential reconstruction phase has been replaced by the polynomial-time basis reduction algorithm. Thanks to this formalization we found a serious flaw in a textbook. |
| BibTeX: |
@article{LLL_Factorization-AFP,
author = {Jose Divasón and Sebastiaan Joosten and René Thiemann and Akihisa Yamada},
title = {A verified factorization algorithm for integer polynomials with polynomial complexity},
journal = {Archive of Formal Proofs},
month = feb,
year = 2018,
note = {\url{http://isa-afp.org/entries/LLL_Factorization.html},
Formal proof development},
ISSN = {2150-914x},
}
|
| License: | BSD License |
| Depends on: | LLL_Basis_Reduction, Perron_Frobenius |
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