A formalization of the Berlekamp-Zassenhaus factorization algorithm

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 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 Isabelle's recent addition of local type definitions.Through experiments we verify that our algorithm factors polynomials of degree 100 within seconds.

A Formalization of the LLL Basis Reduction Algorithm

“…In order to factor an integer polynomial f , we may assume a modular factorization of f into several monic factors u i : f ≡ lc f · i u i modulo m where m = p l is some prime power for user-specified l. In Isabelle, we just reuse our verified modular factorization algorithm [1] to obtain the modular factorization of f .…”

“…In order to instantiate Lemma 1, it now suffices to take g as the polynomial corresponding to any short vector in L u,k : u will divide g modulo m by definition of L u,k and moreover degree g < n. The short vector requirement will provide an upper bound to satisfy the assumption ||f 1 …”

“…irreducible g We further combine this algorithm with a pre-processing algorithm of earlier work [1]. This pre-processing splits a polynomial f into c · f…”

“…It reuses most parts of the formalization of the BerlekampZassenhaus factorization algorithm, where the main difference is the replacement of the exponential-time reconstruction phase [1,Sect. 8] by a polynomial-time one based on the LLL algorithm.…”

Comprehending Isabelle/HOL’s Consistency

“…The extensions have already been picked up by Isabelle/HOL power users for translating between different representations of matrices [12,13], for implementing a certified and efficient algorithm for factorization [11], and for tightly integrating invariants in proof rules for a probabilistic programming language [33].…”