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},
}