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.

Abstract. The LLL basis reduction algorithm was the first polynomialtime algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It thereby approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has several applications in number theory, computer algebra and cryptography.In this paper, we develop the first mechanized soundness proof of the LLL algorithm using Isabelle/HOL. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.

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 squarefree 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 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 locales and local type definitions. Through experiments we verify that our algorithm factors polynomials of degree up to 500 within seconds.

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Matrix interpretations are widely used in automated complexity analysis. Certifying such analyses boils down to determining the growth rate of A n for a fixed non-negative rational matrix A. A direct solution for this task involves the computation of all eigenvalues of A, which often leads to expensive algebraic number computations.In this work we formalize the Perron-Frobenius theorem. We utilize the theorem to avoid most of the algebraic numbers needed for certifying complexity analysis, so that our new algorithm only needs the rational arithmetic when certifying complexity proofs that existing tools can find. To cover the theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of theorems between both libraries. This connection crucially relies on the transfer mechanism in combination with local type definitions, being a non-trivial case study for these Isabelle tools.

In this contribution we present a formalised algorithm in the Isabelle/HOL proof assistant to compute echelon forms, and, as a consequence, characteristic polynomials of matrices. We have proved its correctness over Bézout domains, but its executability is only guaranteed over Euclidean domains, such as the integer ring and the univariate polynomials over a field. This is possible since the algorithm has been parameterised by a (possibly non-computable) operation that returns the Bézout coefficients of a pair of elements of a ring. The echelon form is also used to compute determinants and inverses of matrices. As a by-product, some algebraic structures have been implemented (principal ideal domains, Bézout domains, etc.). In order to improve performance, the algorithm has been refined to immutable arrays inside of Isabelle and code can be generated to functional languages as well.

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In this paper, we present a formalisation in a proof assistant, Isabelle/HOL, of a naive version of the Gauss-Jordan algorithm, with explicit proofs of some of its applications; and, additionally, a process to obtain versions of this algorithm in two different functional languages (SML and Haskell) by means of code generation techniques from the verified algorithm. The aim of this research is not to compete with specialised numerical implementations of Gauss-like algorithms, but to show that formal proofs in this area can be used to generate usable functional programs. The obtained programs show compelling performance in comparison to some other verified and functional versions, and accomplish some challenging tasks, such as the computation of determinants of matrices of big integers and the computation of the homology of matrices representing digital images.