### Abstract

Original language | English |
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Number of pages | 79 |

Journal | Archive of Formal Proofs |

Publication status | Published - 6 Feb 2018 |

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### Cite this

*Archive of Formal Proofs*.

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*Archive of Formal Proofs*.

**A verified factorization algorithm for integer polynomials with polynomial complexity.** / Divasón, Jose; Joosten, Sebastiaan; Thiemann, René; Yamada, Akihisa.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - A verified factorization algorithm for integer polynomials with polynomial complexity

AU - Divasón, Jose

AU - Joosten, Sebastiaan

AU - Thiemann, René

AU - Yamada, Akihisa

N1 - Formal proof development

PY - 2018/2/6

Y1 - 2018/2/6

N2 - 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 The work is based on our previous Berlekamp–Zassenhaus development, where the exponential reconstruction phase has been replaced by the polynomial-time basis reduction algorithm.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.

AB - 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 The work is based on our previous Berlekamp–Zassenhaus development, where the exponential reconstruction phase has been replaced by the polynomial-time basis reduction algorithm.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.

M3 - Article

JO - Archive of Formal Proofs

JF - Archive of Formal Proofs

SN - 2150-914x

ER -