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 The work is based on our previous Berlekamp–Zassenhaus development, where the exponential reconstruction phase has been replaced by the polynomial-time basis reduction algorithm.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 Divas\'on and Sebastiaan Joosten and Ren\'e 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}
}