The Factorization Algorithm of Berlekamp and Zassenhaus

Jose Divasón, Sebastiaan J. C. Joosten, René Thiemann and Akihisa Yamada

14 October 2016

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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