Subquadratic-Time Algorithms for Normal Bases


For any finite Galois field extension K/F, with Galois group G = Gal (K/F), there exists an element \(\alpha \in \) K whose orbit \(G\cdot\alpha\) forms an F-basis of K. Such an \(\alpha\) is called a normal element, and \(G\cdot\alpha\) is a normal basis. We introduce a probabilistic algorithm for testing whether a given \(\alpha \in\) K is normal, when G is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether \(\alpha\) is normal can be reduced to deciding whether \(\sum_{g \in G} g(\alpha)g \in\) K[G] is invertible; it requires a slightly subquadratic number of operations. Once we know that \(\alpha\) is normal, we show how to perform conversions between the power basis of K/F and the normal basis with the same asymptotic cost.

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  1. D. Augot & P. Camion (1994). A deterministic algorithm for computing a normal basis in a finite field. In Proc. EUROCODE'94, P. Charpin, editor.

  2. E. Bach & J. Shallit (1996). Algorithmic Number Theory, Volume 1: Efficient Algorithms. MIT Press, Cambridge, MA.

  3. A. Bostan, P. Flajolet, B. Salvy & É. Schost (2006). Fast computation of special resultants. J. Symbolic Comput. 41(1), 1–29.

    MathSciNet  Article  Google Scholar 

  4. A. Bostan, C.-P. Jeannerod, C. Mouilleron & É. Schost (2017). On Matrices With Displacement Structure: Generalized Operators and Faster Algorithms. SIAM Journal on Matrix Analysis and Applications 38(3), 733–775.

    MathSciNet  Article  Google Scholar 

  5. R. P. Brent & H. T. Kung (1978). Fast algorithms for manipulating formal power series. Journal of the Association for Computing Machinery 25(4), 581–595.

    MathSciNet  Article  Google Scholar 

  6. Peter Bürgisser, Michael Clausen & M. Amin Shokrollahi (1997). Algebraic complexity theory, volume 315 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. ISBN 3-540-60582-7, xxiv+618 . With the collaboration of Thomas Lickteig.

  7. J. Canny, E. Kaltofen & Y. Lakshman (1989). Solving systems of nonlinear polynomial equations faster. In ISSAC'89, 121–128. ACM.

  8. M. Clausen & M. Müller (2004). Generating fast Fourier transforms of solvable groups. J. Symbolic Comput. 37(2), 137–156. ISSN 0747-7171.

  9. C. Curtis & I. Reiner (1988). Representation theory of finite groups and associative algebras. Wiley Classics Library. John Wiley & Sons Inc, New York, New York. ISBN 0-471-60845-9, xiv+689.

  10. X. Dahan, M. Moreno Maza, É. Schost & Y. Xie (2006). On the complexity of the D5 principle. In Proc. of Transgressive Computing 2006. Granada, Spain.

  11. W. Eberly, M. Giesbrecht, P. Giorgi, A. Storjohann & G. Villard (2007). Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections. In ISSAC '07, 143–150. ACM.

  12. S. Gao, J. Gathen von zur Gathen, D. Panario & V. Shoup (2000). Algorithms for exponentiation in finite fields. Journal of Symbolic Computation 29(6), 879–889.

  13. J. Gathen von zur Gathen & J. Gerhard (2013). Modern Computer Algebra (third edition). Cambridge University Press, Cambridge, U.K. ISBN 9781107039032.

  14. J. Gathen von zur Gathen & M. Giesbrecht (1990). Constructing normal bases in finite fields. J. Symbolic Comput. 10(6), 547–570. ISSN 0747-7171.

  15. J. Gathen von zur Gathen & V. Shoup (1992). Computing Frobenius maps and factoring polynomials. Computational Complexity 2(3), 187–224.

  16. M. Giesbrecht, A. Jamshidpey & É. Schost (2019). Quadratic-Time Algorithms for Normal Elements. In ISSAC '19, 179–186. ACM.

  17. K. Girstmair (1999). An algorithm for the construction of a normal basis. J. Number Theory 78(1), 36–45. ISSN 0022-314X.

  18. D. Holt, B. Eick & E. O'Brien (2005). Handbook of computational group theory. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL. ISBN 1-58488-372-3.

    Google Scholar 

  19. A. Jamshidpey, N. Lemire & É. Schost (2018). Algebraic construction of quasi-split algebraic tori. ArXiv: 1801.09629.

  20. D. L. Johnson (1976). Presentations of Groups. Cambridge University Press, Cambridge-New York-Melbourne, v+204 . London Mathematical Society Lecture Notes Series, No. 22.

  21. E. Kaltofen & V. Shoup (1998). Subquadratic-time factoring of polynomials over finite fields. Math. Comp. 67(223), 1179–1197. ISSN 0025-5718.

  22. M. Kaminski, D.G. Kirkpatrick & N.H. Bshouty (1988). Addition requirements for matrix and transposed matrix products. J. Algorithms 9(3), 354–364.

    MathSciNet  Article  Google Scholar 

  23. K. Kedlaya & C. Umans (2011). Fast polynomial factorization and modular composition. SICOMP 40(6), 1767–1802.

    MathSciNet  Article  Google Scholar 

  24. S. Lang (2002). Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, 3rd edition.

  25. F. Le Gall (2014). Powers of tensors and fast matrix multiplication. In ISSAC'14, 296–303. ACM, Kobe, Japan.

    Google Scholar 

  26. F. Le Gall & F. Urrutia (2018). Improved rectangular matrix multiplication using powers of the Coppersmith-Winograd tensor. In SODA '18, 1029–1046. SIAM, New Orleans, USA.

    Google Scholar 

  27. H. W. Lenstra, Jr. (1991). Finding isomorphisms between finite fields. Math. Comp. 56(193), 329–347. ISSN 0025-5718.

  28. X. Li, M. Moreno Maza & É. Schost (2009). Fast arithmetic for triangular sets: from theory to practice. J. Symb. Comp. 44(7), 891–907.

    MathSciNet  Article  Google Scholar 

  29. G. Lotti & F. Romani (1983). On the asymptotic complexity of rectangular matrix multiplication. Theoretical Computer Science 23(2), 171–185.

    MathSciNet  Article  Google Scholar 

  30. D. Maslen, D. N. Rockmore & S. Wolff (2018). The efficient computation of Fourier transforms on semisimple algebras. J. Fourier Anal. Appl. 24(5), 1377–1400. ISSN 1069-5869.

  31. A. Poli (1994). A deterministic construction for normal bases of abelian extensions. Comm. Algebra 22(12), 4751–4757. ISSN 0092-7872.

  32. H. Schlickewei & S. Stepanov (1993). Algorithms to construct normal bases of cyclic number fields. J. Number Theory 44(1), 30–40. ISSN 0022-314X.

  33. A. Schönhage & V. Strassen (1971). Schnelle Multiplikation großer Zahlen. Computing 7, 281–292.

    MathSciNet  Article  Google Scholar 

  34. I. S. Sergeev (2007). On constructing circuits for transforming the polynomial and normal bases of finite fields from one to the other. Discrete Mathematics and Applications 17(4), 361–373.

    MathSciNet  Article  Google Scholar 

  35. V. Shoup (1995). A new polynomial factorization algorithm and its implementation. J. Symbolic Comput. 20(4), 363–397. ISSN 0747-7171.

  36. C. Giraldo Vergara & F. Brochero Martínez (2002). Wedderburn decomposition of some special rational group algebras. Lect. Mat. 23(2), 99–106. ISSN 0120-1980.

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Giesbrecht, M., Jamshidpey, A. & Schost, É. Subquadratic-Time Algorithms for Normal Bases . comput. complex. 30, 5 (2021).

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  • Normal bases
  • Galois groups
  • Polycyclic groups
  • Metacyclic groups
  • Fast algorithms

Subject classification

  • 12Y05
  • 12F10
  • 11y16
  • 68Q25