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A verified implementation of the Berlekamp-Zassenhaus factorization algorithm. (English) Zbl 07187044
Summary: 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 $$\text{GF}(p)$$ 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.

##### MSC:
 68V15 Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.)
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##### References:
 [1] Abbott, J., Bounds on factors in $$Z[x]$$, J. Symb. Comput., 50, 532-563 (2013) · Zbl 1295.12010 [2] Axler, SJ, Linear Algebra Done Right. Undergraduate Texts in Mathematics (1997), Berlin: Springer, Berlin · Zbl 0886.15001 [3] Ballarin, C., Locales: a module system for mathematical theories, J. Autom. Reason., 52, 2, 123-153 (2014) · Zbl 1315.68218 [4] Barthe, G., Grégoire, B., Heraud, S., Olmedo, F., Béguelin, S.Z.: Verified indifferentiable hashing into elliptic curves. In: Degano, P., Guttman, J.D. (eds.) Principles of Security and Trust. POST 2012, Volume 7215 of LNCS, pp. 209-228. Springer, Berlin (2012) · Zbl 1354.94021 [5] Berlekamp, ER, Factoring polynomials over finite fields, Bell Syst. Tech. J., 46, 8, 1853-1859 (1967) [6] Blanchette, J.C., Meier, F., Popescu, A., Traytel, D.: Foundational nonuniform (co)datatypes for higher-order logic. In: ACM/IEEE Symposium on Logic in Computer Science, LICS 32, pp. 1-12. IEEE Computer Society (2017). Cross-type induction is explained in Appendix D of the extended report version at http://matryoshka.gforge.inria.fr/pubs/nonuniform_report.pdf · Zbl 06821624 [7] Bottesch, R., Haslbeck, M.W., Thiemann, R.: A verified efficient implementation of the LLL basis reduction algorithm. In: Barthe, G., Sutcliffe, G., Veanes, M. (eds.) Logic for Programming, Artificial Intelligence and Reasoning. LPAR 22, Volume 57 of EPiC Series in Computing, pp. 164-180. EasyChair (2018) · Zbl 1409.68252 [8] Cantor, DG; Zassenhaus, H., A new algorithm for factoring polynomials over finite fields, Math. Comput., 36, 154, 587-592 (1981) · Zbl 0493.12024 [9] Cerlienco, L.; Mignotte, M.; Piras, F., Computing the measure of a polynomial, J. Symb. Comput., 4, 1, 21-33 (1987) · Zbl 0629.12002 [10] Divasón, J., Joosten, S.J.C., Thiemann, R., Yamada, A.: A formalization of the Berlekamp-Zassenhaus factorization algorithm. In: Bertot, Y., Vafeiadis, V. (eds.) Certified Programs and Proofs. CPP 2017, pp. 17-29. ACM (2017) [11] Divasón, J., Joosten, S.J.C., Thiemann, R., Yamada, A.: A formalization of the LLL basis reduction algorithm. In: Avigad, J., Mahboubi, A. (eds.) Interactive Theorem Proving. ITP 2018, Volume 10895 of LNCS, pp. 160-177. Springer, Berlin (2018) · Zbl 06946979 [12] Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) Functional and Logic Programming. FLOPS 2010, Volume 6009 of LNCS, pp. 103-117. Springer, Berlin (2010) · Zbl 1284.68131 [13] Harrison, J., The HOL light theory of Euclidean space, J. Autom. Reason., 50, 2, 173-190 (2013) · Zbl 1260.68373 [14] van Hoeij, M., Factoring polynomials and the knapsack problem, J. Number Theory, 95, 2, 167-189 (2002) · Zbl 1081.11080 [15] Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Certified Programs and Proofs. CPP 2013, Volume 8307 of LNCS, pp. 131-146. Springer, Berlin (2013) · Zbl 1426.68284 [16] Karatsuba, A.; Ofman, Y., Multiplication of multidigit numbers on automata, Sov. Phys. Dokl., 7, 7, 595-596 (1963) [17] Kirkels, B.: Irreducibility certificates for polynomials with integer coefficients. Master’s thesis, Radboud Universiteit Nijmegen (2004) [18] Knuth, DE, The Art of Computer Programming, Volume 2: Seminumerical Algorithms (1998), Reading: Addison-Wesley, Reading [19] Kobayashi, H., Suzuki, H., Ono, Y.: Formalization of Hensel’s lemma. In: Hurd, J., Smith, E., Darbari, A. (eds.) Theorem Proving in Higher Order Logics: Emerging Trends Proceedings, pp. 114-118. Oxford University Computing Laboratory (2005) [20] Krauss, A.: Recursive definitions of monadic functions. In: Bove, A., Komendantskaya, E., Niqui, M. (eds.) Partiality and Recursion in Interactive Theorem Provers. PAR 2010, Volume 43 of EPTCS, pp. 1-13 (2010) [21] Kunčar, O.; Popescu, A., From types to sets by local type definition in higher-order logic, J. Autom. Reason., 62, 2, 237-260 (2019) · Zbl 07024458 [22] Lee, H.: Vector spaces. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/VectorSpace.html (2014) [23] Lenstra, AK; Lenstra, HW; Lovász, L., Factoring polynomials with rational coefficients, Math. Ann., 261, 515-534 (1982) · Zbl 0488.12001 [24] Lochbihler, A.: Fast machine words in Isabelle/HOL. In: Avigad, J., Mahboubi, A. (eds.) Interactive Theorem Proving. ITP 2018, Volume 10895 of LNCS, pp. 388-410. Springer, Berlin (2018) · Zbl 06946992 [25] Maple 2017.3. Maplesoft, a division of Waterloo Maple Inc. Waterloo (2017) [26] Martin-Dorel, É.; Hanrot, G.; Mayero, M.; Théry, L., Formally verified certificate checkers for hardest-to-round computation, J. Autom. Reason., 54, 1, 1-29 (2015) · Zbl 1315.68222 [27] Mignotte, M., An inequality about factors of polynomials, Math. Comput., 28, 128, 1153-1157 (1974) · Zbl 0299.12101 [28] Miola, A.; Yun, DY, Computational aspects of Hensel-type univariate polynomial greatest common divisor algorithms, ACM SIGSAM Bull., 8, 3, 46-54 (1974) [29] Nipkow, T.; Paulson, L.; Wenzel, M., Isabelle/HOL—A Proof Assistant for Higher-Order Logic, Volume 2283 of LNCS (2002), Berlin: Springer, Berlin · Zbl 0994.68131 [30] Thiemann, R.: Computing n-th roots using the Babylonian method. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/Sqrt_Babylonian.html (2013) [31] Thiemann, R., Yamada, A.: Algebraic numbers in Isabelle/HOL. In: Blanchette, J., Merz, S. (eds.) Interactive Theorem Proving. ITP 2016, Volume 9807 of LNCS, pp. 391-408. Springer, Berlin (2016) · Zbl 06644756 [32] Thiemann, R., Yamada, A.: Formalizing Jordan normal forms in Isabelle/HOL. In: Avigad, J., Chlipala, A. (eds.) Certified Programs and Proofs. CPP 2016, pp. 88-99. ACM (2016) [33] von zur Gathen, J.; Gerhard, J., Modern Computer Algebra (2013), Cambridge: Cambridge University Press, Cambridge · Zbl 1277.68002 [34] Mathematica Version 11.2. Wolfram Research, Inc. Champaign (2017) [35] Yun, D.Y.: On square-free decomposition algorithms. In: Symbolic and Algebraic Computation. SYMSAC 1976, pp. 26-35. ACM (1976) [36] Zassenhaus, H., On Hensel factorization, I, J. Number Theory, 1, 3, 291-311 (1969) · Zbl 0188.33703
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