### The Factorization Algorithm of Berlekamp and Zassenhaus

Jose Divasón and Sebastiaan Joosten and René Thiemann and Akihisa YamadaArchive of Formal Proofs 2016.

### Abstract

We formalize the Berlekamp-Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials.

The algorithm first performs a factorization in the prime field GF(p) and then performs computations in the integer ring modulo p^{k}, where both p and k are determined at runtime. Since a natural modeling of these structures via dependent types is not possible in Isabelle/HOL, we formalize the whole algorithm using Isabelle’s recent addition of local type definitions.

Through experiments we verify that our algorithm factors polynomials of degree 100 within seconds.

### BibTeX

@article{Berlekamp_Zassenhaus-AFP,
author = {Jose Divasón and Sebastiaan Joosten and René Thiemann and Akihisa Yamada},
title = {The Factorization Algorithm of {B}erlekamp and {Z}assenhaus},
journal = {Archive of Formal Proofs},
month = oct,
year = 2016,
note = {\url{http://isa-afp.org/entries/Berlekamp_Zassenhaus.shtml},
Formal proof development},
ISSN = {2150-914x},

}