A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm
Jose Divasón, Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada · Journal of Automated Reasoning · 2019
We formally verify 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 factorization in the prime field $$\mathrm {GF}(p){}$$ and then performs computations in the ring of integers 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 locales and local type definitions. Through experiments we verify that our algorithm factors polynomials of degree up to 500 within seconds.