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.
- Divasón, J., Joosten, S. J. C., Thiemann, R., & Yamada, A. (2019). A Verified Implementation of the Berlekamp–Zassenhaus Factorization Algorithm. Journal of Automated Reasoning, 64(4), 699–735. https://doi.org/10.1007/s10817-019-09526-y
- von zur Gathen, J., & Gerhard, J. (2013). Modern Computer Algebra. https://doi.org/10.1017/cbo9781139856065