Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 437–453 | Cite as

The functional graph of linear maps over finite fields and applications

  • Daniel PanarioEmail author
  • Lucas Reis
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


Let \(\mathbb F_{q}\) be the finite field with q elements and \(n\ge 2\) be a positive integer. We study the functional graph associated to linear maps over finite fields. In particular, we describe the functional graph \(\mathcal {G}_f(\mathbb F_{q^n})\) associated to the map induced by \(L_f\) on \(\mathbb F_{q^n}\), where f is any irreducible divisor of \(x^n-1\) over \(\mathbb F_q\) and \(L_f\) is the q-associate of f. This description derives interesting information on the graph \(\mathcal {G}_f(\mathbb F_{q^n})\), such as the number of cycles and the average of the preperiod length. When \(\gcd (f, x^n-1)=1\), \(L_f\) is a permutation on \(\mathbb F_{q^n}\) and the cycle decomposition of \(\mathcal {G}_f(\mathbb F_{q^n})\) is well known. In this case, we present some applications of this result, such as the construction of linear involutions over odd characteristic and permutations with few fixed points.


Dynamical systems over finite fields Linear maps over finite fields Permutation polynomials Involutions 

Mathematics Subject Classification

12E20 11T06 11T71 



The first author was partially funded by NSERC of Canada. The second author was supported by the Program CAPES-PDSE (process—88881.134747/2016-01) at Carleton University.


  1. 1.
    Blum L., Blum M., Shub M.: A simple unpredictable pseudo-random number generator. SIAM J. Comput. 15(2), 364–383 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Charpin P., Mesnager S., Sarkar S.: Dickson polynomials that are involutions. In: Canteaut A., Effinger G., Huczynska S., Panario D., Storme L. (eds.) Contemporary Developments in Finite Fields and Applications, pp. 22–47. World Scientific, Singapore (2016).CrossRefGoogle Scholar
  3. 3.
    Charpin P., Mesnager S., Sarkar S.: Involutions over the Galois field \({\mathbb{F}}_{2^n}\). IEEE Trans. Inf. Theory 62, 2266–2276 (2016).CrossRefzbMATHGoogle Scholar
  4. 4.
    Chou W., Shparlinski I.: On the cycle structure of repeated exponentiation modulo a prime. J. Number Theory 107(2), 345–356 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gassert T.: Chebyshev action on finite fields. Discret. Math. 315, 83–94 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huczynska S., Mullen G.L., Panario D., Thomson D.: Existence and properties of \(k\)-normal elements over finite fields. Finite Fields Appl. 24, 170–183 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lehmer D.H.: An extended theory of Lucas functions. Ann. Math. 31(3), 419–448 (1930).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics and Its Applications, vol. 20, 2nd edn. Cambridge University Press, Cambridge (1997).Google Scholar
  9. 9.
    Lucas E.: Théorie des fonctions numériques simplement périodiques. Am. J. Math. 1(4), 289–321 (1878).CrossRefGoogle Scholar
  10. 10.
    Mullen G.L., Vaughan T.P.: Cycles of linear permutations over a finite field. Linear Algebra Appl. 108, 63–82 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Peinado A., Montoya F., Muñoz J., Yuste A.: Maximal periods of \(x^2 + c \in {\mathbb{F}}_q\), In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 2227, pp. 219–228 (2001).Google Scholar
  12. 12.
    Pollard J.M.: A Monte Carlo method for factorization. BIT 15(3), 331–334 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pollard J.M.: Monte Carlo methods for index computation (mod \(p\)). Math. Comp. 32(143), 918–924 (1978).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Qureshi C., Panario D.: Rédei actions on finite fields and multiplication map in cyclic groups. SIAM J. Discret. Math. 29(3), 1486–1503 (2015).CrossRefzbMATHGoogle Scholar
  15. 15.
    Qureshi C., Panario D.: The graph structure of the Chebyshev polynomial over finite fields and applications. In: Workshop on Coding and Cryptography (WCC 2017).Google Scholar
  16. 16.
    Qureshi C., Panario D., Martins R.: Cycle structure of iterating Rédei functions. Adv. Math. Commun. 11(2), 397–407 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rogers T.: The graph of the square mapping on the prime fields. Discret. Math. 148, 317–324 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Toledo R.A.H.: Linear finite dynamical systems. Commun. Algebra 33(9), 2977–2989 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Teske E., Williams H.C.: A Note on Shanks’ Chains of Primes. Springer, New York (2000).zbMATHGoogle Scholar
  20. 20.
    Ugolini S.: Graphs associated with the map \(x\mapsto x\,+\,x^{-1}\) in finite fields of characteristic three and five. J. Number Theory 133(4), 1207–1228 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vasiga T., Shallit J.: On the iteration of certain quadratic maps over GF(\(p\)). Discret. Math. 227, 219–240 (2004).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

Personalised recommendations