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.
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Related publications
- 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
- Wikipedia
Session Berlekamp_Zassenhaus
- Finite_Field
- Arithmetic_Record_Based
- Finite_Field_Record_Based
- Matrix_Record_Based
- More_Missing_Multiset
- Unique_Factorization
- Unique_Factorization_Poly
- Poly_Mod
- Poly_Mod_Finite_Field
- Karatsuba_Multiplication
- Polynomial_Record_Based
- Poly_Mod_Finite_Field_Record_Based
- Chinese_Remainder_Poly
- Berlekamp_Type_Based
- Distinct_Degree_Factorization
- Finite_Field_Factorization
- Finite_Field_Factorization_Record_Based
- Hensel_Lifting
- Hensel_Lifting_Type_Based
- Berlekamp_Hensel
- Square_Free_Int_To_Square_Free_GFp
- Suitable_Prime
- Degree_Bound
- Mahler_Measure
- Factor_Bound
- Sublist_Iteration
- Reconstruction
- Code_Abort_Gcd
- Berlekamp_Zassenhaus
- Gcd_Finite_Field_Impl
- Square_Free_Factorization_Int
- Factorize_Int_Poly
- Factorize_Rat_Poly