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 $$\mathrm {GF}(p){}$$ GF ( p ) and then performs computations in the ring of integers modulo $$p^k$$ 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.

中文翻译：

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Berlekamp-Zassenhaus 分解算法的验证实现

我们正式验证了用于在 Isabelle/HOL 中分解无平方整数多项式的 Berlekamp-Zassenhaus 算法。我们进一步将 Yun 的无平方因式分解算法的现有形式化为整数多项式，从而为任意单变量多项式提供有效且经过认证的因式分解算法。该算法首先在素数域 $$\mathrm {GF}(p){}$$ GF ( p ) 中执行因式分解，然后在以 $$p^k$$ pk 为模的整数环中执行计算，其中 p 和k 在运行时确定。由于在 Isabelle/HOL 中不可能通过依赖类型对这些结构进行自然建模，因此我们使用区域设置和本地类型定义形式化整个算法。通过实验，我们验证了我们的算法可以在几秒钟内分解高达 500 次的多项式。