# The Factorization Algorithm of Berlekamp and Zassenhaus

 Title: The Factorization Algorithm of Berlekamp and Zassenhaus Authors: Jose Divasón, Sebastiaan Joosten, René Thiemann and Akihisa Yamada Submission date: 2016-10-14 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 Berlekamp and Zassenhaus}, journal = {Archive of Formal Proofs}, month = oct, year = 2016, note = {\url{http://isa-afp.org/entries/Berlekamp_Zassenhaus.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Used by: Algebraic_Numbers, LLL_Basis_Reduction, Smith_Normal_Form