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Dynamics of polynomial maps over finite fields

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Abstract

Let \(\mathbb {F}_q\) be a finite field with q elements and let n be a positive integer. In this paper, we study the digraph associated to the map \(x\mapsto x^n h(x^{\frac{q-1}{m}})\) over \(\mathbb {F}_q\), where \(h(x)\in \mathbb {F}_q[x].\) We completely determine the associated functional graph of maps that satisfy a certain condition of regularity. In particular, we provide the functional graphs associated to monomial maps. As a consequence of our results, one have the number of connected components, length of the cycles and number of fixed points of these class of maps.

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References

  1. Akbary A., Ghioca D., Wang Q.: On constructing permutations of finite fields. Finite Fields Their Appl. 17, 51–67 (2011).

    Article  MathSciNet  Google Scholar 

  2. Chou W.-S., Shparlinski I.E.: On the cycle structure of repeated exponentiation modulo a prime. J. Number Theory 107, 345–356 (2004).

    Article  MathSciNet  Google Scholar 

  3. Garton D.: Periodic points of polynomials over finite fields. Trans. Am. Math. Soc. 375, 8 (2022).

    Article  MathSciNet  Google Scholar 

  4. Gassert T.A.: Chebyshev action on finite fields. Discret. Math. 315, 83–94 (2014).

    Article  MathSciNet  Google Scholar 

  5. Gómez-Pérez D., Ostafe A., Shparlinski I.E.: On irreducible divisors of iterated polynomials. Revista Matemática Iberoamericana 30(4), 1123–1134 (2014).

    Article  MathSciNet  Google Scholar 

  6. Heath-Brown D.R., Micheli G., David Rodney Heath-Brown and Giacomo Micheli: Irreducible polynomials over finite fields produced by composition of quadratics. Revista Matemática Iberoamericana 35, 847–855 (2019).

    Article  MathSciNet  Google Scholar 

  7. Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory, 84th edn Springer, New York (1982).

    Book  Google Scholar 

  8. Johnson D., Menezes A., Vanstone S.: The elliptic curve digital signature algorithm (ECDSA). Int. J. Inform. Secur. 1(1), 36–63 (2001).

    Article  Google Scholar 

  9. Konyagin S.V., et al.: Functional graphs of polynomials over finite fields. J. Combin. Theory Ser. B 89, 116 (2016).

    MathSciNet  Google Scholar 

  10. Mans B., et al.: On functional graphs of quadratic polynomials. Exp. Math. 28(3), 292–300 (2019).

    Article  MathSciNet  Google Scholar 

  11. Martins R., Panario D., Qureshi C.: A survey on iterations of mappings over finite fields. Combinatorics Finite Fields. De Gruyter 23, 135–172 (2019).

    Article  MathSciNet  Google Scholar 

  12. Panario D., Reis L.: The functional graph of linear maps over finite fields and applications. Des. Codes Cryptogr. 87, 437–453 (2019).

    Article  MathSciNet  Google Scholar 

  13. Peinado A., et al.: Maximal periods of \(x^{2} =c\) in \(\mathbb{F} _{q}\). In: International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes. Springer. (2001), pp. 219–228.

  14. Qureshi C., Panario D.: Rédei actions on finite fields and multiplication map in cyclic group. SIAM J. Discret. Math. 29, 1486–1503 (2015).

    Article  Google Scholar 

  15. Qureshi C., Panario D.: The graph structure of Chebyshev polynomials over finite fields and applications. Des. Codes Cryptogr. 87(2), 393–416 (2019).

    Article  MathSciNet  Google Scholar 

  16. Qureshi C., Reis L.: Dynamics of the a-map over residually finite Dedekind domains and applications. J. Number Theory 204, 134–154 (2019).

    Article  MathSciNet  Google Scholar 

  17. Qureshi C., Reis L.: On the functional graph of the power map over finite groups. In: arXiv preprint arXiv:2107.00584 (2021).

  18. Rogers T.D.: The graph of the square mapping on the prime fields. Discret. Math. 148, 317–324 (1996).

    Article  MathSciNet  Google Scholar 

  19. Ugolini S.: Functional graphs of rational maps induced by endomorphisms of ordinary elliptic curves over finite fields. Periodica Math. Hungarica 77, 237–260 (2018).

    Article  MathSciNet  Google Scholar 

  20. Wang Q.: Polynomials over finite fields: an index approach. Combinatorics Finite Fields De Gruyter 112, 319–348 (2019).

    Article  MathSciNet  Google Scholar 

  21. Wiener MJ., Zuccherato RJ.: Faster attacks on elliptic curve cryptosystems. In: International workshop on selected areas in cryptography. Springer. pp. 190–200 (1998).

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Acknowledgements

The first author was supported by FAPESP, Brazil, under grant 2021/13712-5. The second author was supported by FAPEMIG, Brazil, under grant APQ-02973-17.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Lavras, UFLA, Lavras, MG, 37200-900, Brazil

    José Alves Oliveira

  2. Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Belo Horizonte, MG, 30123-970, Brazil

    F. E. Brochero Martínez

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  1. José Alves Oliveira
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  2. F. E. Brochero Martínez
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Correspondence to José Alves Oliveira.

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Communicated by D. Panario.

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Oliveira, J.A., Brochero Martínez, F.E. Dynamics of polynomial maps over finite fields. Des. Codes Cryptogr. 92, 1113–1125 (2024). https://doi.org/10.1007/s10623-023-01332-3

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  • DOI: https://doi.org/10.1007/s10623-023-01332-3

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