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Univariate polynomial factorization over finite fields with large extension degree

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Abstract

The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with \(d^{1.5 + o (1)}\) for input polynomials of degree d, and with the square of the bit size of the ground field. It relies on a variant of the Cantor–Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen and Shoup, we prove a refinement of this bound when the finite field has a large extension degree over its prime field. We also present fast practical algorithms for the case when the extension degree is smooth.

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  • 19 February 2024

    Copyright right year has been changed from 2021 to 2022

References

  1. Ben-Or, M. Probabilistic algorithms in finite fields. In 22nd Annual Symposium on Foundations of Computer Science (SFCS 1981), pages 394–398. Los Alamitos, CA, USA, 1981. IEEE Computer Society

  2. Berlekamp, E.R.: Factoring polynomials over finite fields. Bell. Syst. Tech. J. 46, 1853–1859 (1967)

    Article  MathSciNet  Google Scholar 

  3. Berlekamp, E.R.: Factoring polynomials over large finite fields. Math. Comput. 24, 713–735 (1970)

    Article  MathSciNet  Google Scholar 

  4. Bostan, A., Chyzak, F., Giusti, M., Lebreton, R., Lecerf, G., Salvy, B. É. Schost.: Algorithmes Efficaces en Calcul Formel. Frédéric Chyzak (self-published), Palaiseau, 2017. Electronic version available from https://hal.archives-ouvertes.fr/AECF

  5. Cantor, D.G., Zassenhaus, H.: A new algorithm for factoring polynomials over finite fields. Math. Comput. 36(154), 587–592 (1981)

    Article  MathSciNet  Google Scholar 

  6. Flajolet, Ph., Gourdon, X., Panario, D.: The complete analysis of a polynomial factorization algorithm over finite fields. J. Algorithm. 40(1), 37–81 (2001)

    Article  MathSciNet  Google Scholar 

  7. Flajolet, P., Steyaert, J.-M.: A branching process arising in dynamic hashing, trie searching and polynomial factorization. In M. Nielsen and E. Schmidt, editors, Automata, Languages and Programming, volume 140 of Lect. Notes Comput. Sci., pages 239–251. Springer–Verlag, 1982

  8. von zur Gathen, J.: Who was who in polynomial factorization. In ISSAC’06: International Symposium on Symbolic and Algebraic Computation, pages 1–2. New York, NY, USA, 2006. ACM

  9. von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, New York (2013)

    Book  Google Scholar 

  10. von zur Gathen, J., Seroussi, G.: Boolean circuits versus arithmetic circuits. Inf. Comput. 91(1), 142–154 (1991)

    Article  MathSciNet  Google Scholar 

  11. von zur Gathen, J., Shoup., V.: Computing Frobenius maps and factoring polynomials. Comput. Complex. 2(3), 187–224 (1992)

    Article  MathSciNet  Google Scholar 

  12. Guo, Z., Narayanan, A.K., Umans, C.: Algebraic problems equivalent to beating exponent 3/2 for polynomial factorization over finite fields. arXiv:1606.04592, 2016

  13. Harvey, D., van der Hoeven, J.: Faster polynomial multiplication over finite fields using cyclotomic coefficient rings. J. Complex. 54, 101404 (2019)

    Article  MathSciNet  Google Scholar 

  14. Harvey, D., van der Hoeven, J.: Polynomial multiplication over finite fields in time \(O (n \log n)\). Technical Report, HAL, 2019. https://hal.archives-ouvertes.fr/hal-02070816. Accepted for publication in JACM

  15. van der Hoeven, J.: The Jolly Writer. Scypress, Your Guide to GNU TeXmacs (2020)

    Google Scholar 

  16. van der Hoeven, J., Lecerf, G.: Modular composition via factorization. J. Complex. 48, 36–68 (2018)

    Article  MathSciNet  Google Scholar 

  17. van der Hoeven, J., Lecerf, G.: Accelerated tower arithmetic. J. Complex. 55, 101402 (2019)

    Article  MathSciNet  Google Scholar 

  18. van der Hoeven, J., Lecerf, G.: Fast multivariate multi-point evaluation revisited. J. Complex. 56, 101405 (2020)

    Article  MathSciNet  Google Scholar 

  19. van der Hoeven, J., Lecerf, G.: Fast amortized multi-point evaluation. J. Complex. 67, 101574 (2021)

    Article  MathSciNet  Google Scholar 

  20. Hopcroft, J., Musinski, J.: Duality applied to the complexity of matrix multiplication and other bilinear forms. SIAM J. Comput. 2(3), 159–173 (1973)

    Article  MathSciNet  Google Scholar 

  21. Kaltofen,E.: Polynomial factorization: a success story. In ISSAC ’03: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pages 3–4. New York, NY, USA, 2003. ACM

  22. Kaltofen, E., Shoup, V.: Subquadratic-time factoring of polynomials over finite fields. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’95, pages 398–406. New York, NY, USA, 1995. ACM

  23. Kaltofen,E., Shoup, V.: Fast polynomial factorization over high algebraic extensions of finite fields. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC ’97, pages 184–188. New York, NY, USA, 1997. ACM

  24. Kaltofen, E., Shoup, V.: Subquadratic-time factoring of polynomials over finite fields. Math. Comput. 67(223), 1179–1197 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  25. Kedlaya, K.S., Umans, C.: Fast modular composition in any characteristic. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), pages 146–155. Los Alamitos, CA, USA, 2008. IEEE Computer Society

  26. Kedlaya, K.S., Umans, C.: Fast polynomial factorization and modular composition. SIAM J. Comput. 40(6), 1767–1802 (2011)

    Article  MathSciNet  Google Scholar 

  27. Le Gall,F., Urrutia, F.: Improved rectangular matrix multiplication using powers of the Coppersmith–Winograd tensor. In A. Czumaj, editor, Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1029–1046. Philadelphia, PA, USA, 2018. SIAM

  28. Lecerf, G.: Fast separable factorization and applications. Appl. Algebra Engrg. Comm. Comput. 19(2), 135–160 (2008)

    Article  MathSciNet  Google Scholar 

  29. Mullen, G.L., Panario. D.: Handbook of Finite Fields. Chapman and Hall/CRC, 2013

  30. Neiger, V., Rosenkilde, J., Solomatov, G.: Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation. In A. Mantzaflaris, editor, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, ISSAC ’20, pages 388–395. New York, NY, USA, 2020. ACM

  31. Paterson, M.S., Stockmeyer, L.J.: On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. Comput. 2(1), 60–66 (1973)

    Article  MathSciNet  Google Scholar 

  32. Poteaux, A., Schost, É.: Modular composition modulo triangular sets and applications. Comput. Complex. 22(3), 463–516 (2013)

    Article  MathSciNet  Google Scholar 

  33. Rabin, M.O.: Probabilistic algorithms in finite fields. SIAM J. Comput. 9(2), 273–280 (1980)

    Article  MathSciNet  Google Scholar 

  34. Shoup, V.: Fast construction of irreducible polynomials over finite fields. J. Symbolic Comput. 17(5), 371–391 (1994)

    Article  MathSciNet  Google Scholar 

  35. Shoup, V.: A new polynomial factorization algorithm and its implementation. J. Symbolic Comput. 20(4), 363–397 (1995)

    Article  MathSciNet  Google Scholar 

  36. Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge University Press, 2nd edition, 2008

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Acknowledgements

We thank the anonymous referees for their useful comments.

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Authors and Affiliations

  1. CNRS, École polytechnique, Institut Polytechnique de Paris, Laboratoire d’informatique de l’École polytechnique (LIX, UMR 7161), Bâtiment Alan Turing, CS35003, 1 rue Honoré d’Estienne d’Orves, 91120, Palaiseau, France

    Joris van der Hoeven & Grégoire Lecerf

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  1. Joris van der Hoeven
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  2. Grégoire Lecerf
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Correspondence to Joris van der Hoeven.

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Hoeven, J.v.d., Lecerf, G. Univariate polynomial factorization over finite fields with large extension degree. AAECC 35, 121–149 (2024). https://doi.org/10.1007/s00200-021-00536-1

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