Skip to main content
Springer Nature Link
Log in

Fractional Jumps: Complete Characterisation and an Explicit Infinite Family

  • Conference paper
  • First Online:
Arithmetic of Finite Fields (WAIFI 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11321))

Included in the following conference series:

Abstract

In this paper we provide a complete characterisation of transitive fractional jumps. In particular, we prove that they can only arise from transitive projective automorphisms apart from a couple of degenerate cases which we entirely classify. Furthermore, we prove that such construction is feasible for arbitrarily large dimension by exhibiting an infinite class of projectively primitive polynomials whose companion matrix can be used to define a full orbit sequence over an affine space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Amadio Guidi, F., Lindqvist, S., Micheli, G.: Full orbit sequences in affine spaces via fractional jumps and pseudorandom number generation. arXiv preprint arXiv:1712.05258v2 (2017). (To appear in Mathematics of Computation)

  2. Brandstätter, N., Winterhof, A.: Some notes on the two-prime generator of order 2. IEEE Trans. Inf. Theory 51(10), 3654–3657 (2005)

    Article  MathSciNet  Google Scholar 

  3. Cao, X.: On the order of the polynomial \(x^p-x-a\). Cryptology ePrint Archive, Report 2010/034 (2010). https://eprint.iacr.org/2010/034.pdf

  4. Chou, W.-S.: On inversive maximal period polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 6(4), 245–250 (1995)

    Article  MathSciNet  Google Scholar 

  5. Eichenauer-Herrmann, J.: Inversive congruential pseudorandom numbers avoid the planes. Math. Comp. 56(193), 297–301 (1991)

    Article  MathSciNet  Google Scholar 

  6. El-Mahassni, E.D., Gomez, D.: On the distribution of nonlinear congruential pseudorandom numbers of higher orders in residue rings. In: Bras-Amorós, M., Høholdt, T. (eds.) AAECC 2009. LNCS, vol. 5527, pp. 195–203. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02181-7_21

    Chapter  Google Scholar 

  7. Ferraguti, A., Micheli, G., Schnyder, R.: On sets of irreducible polynomials closed by composition. In: Duquesne, S., Petkova-Nikova, S. (eds.) WAIFI 2016. LNCS, vol. 10064, pp. 77–83. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-55227-9_6

    Chapter  MATH  Google Scholar 

  8. Gómez-Pérez, D., Ostafe, A., Shparlinski, I.E.: Algebraic entropy, automorphisms and sparsity of algebraic dynamical systems and pseudorandom number generators. Math. Comput. 83(287), 1535–1550 (2014)

    Article  MathSciNet  Google Scholar 

  9. Gutierrez, J., Shparlinski, I.E., Winterhof, A.: On the linear and nonlinear complexity profile of nonlinear pseudorandom number generators. IEEE Trans. Inf. Theory 49(1), 60–64 (2003)

    Article  MathSciNet  Google Scholar 

  10. Heath-Brown, D.R., Micheli, G.: Irreducible polynomials over finite fields produced by composition of quadratics. arXiv preprint arXiv:1701.05031 (2017)

  11. Lang, S.: Algebra - Revised Third Edition. Graduate Texts in Mathematics, vol. 211. Springer, New York (2002). https://doi.org/10.1007/978-1-4613-0041-0

    Book  Google Scholar 

  12. Niederreiter, H., Shparlinski, I.E.: Recent advances in the theory of nonlinear pseudorandom number generators. In: Fang, K.T., Niederreiter, H., Hickernell, F.J. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 86–102. Springer, Heidelberg (2002). https://doi.org/10.1007/978-3-642-56046-0_6

    Chapter  Google Scholar 

  13. Niederreiter, H., Shparlinski, I.E.: Dynamical systems generated by rational functions. In: Fossorier, M., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 6–17. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-44828-4_2

    Chapter  Google Scholar 

  14. Ostafe, A.: Pseudorandom vector sequences derived from triangular polynomial systems with constant multipliers. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 62–72. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13797-6_5

    Chapter  Google Scholar 

  15. Ostafe, A., Pelican, E., Shparlinski, I.E.: On pseudorandom numbers from multivariate polynomial systems. Finite Fields Appl. 16(5), 320–328 (2010)

    Article  MathSciNet  Google Scholar 

  16. Ostafe, A., Shparlinski, I.E.: On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators. Math. Comput. 79(269), 501–511 (2010)

    Article  MathSciNet  Google Scholar 

  17. Ostafe, A., Shparlinski, I.E.: On the length of critical orbits of stable quadratic polynomials. Proc. Am. Math. Soc. 138(8), 2653–2656 (2010)

    Article  MathSciNet  Google Scholar 

  18. Topuzoğlu, A., Winterhof, A.: Pseudorandom sequences. In: Garcia, A., Stichtenoth, H. (eds.) Topics in Geometry, Coding Theory and Cryptography. AA, vol. 6, pp. 135–166. Springer, Dordrecht (2006). https://doi.org/10.1007/1-4020-5334-4_4

    Chapter  Google Scholar 

  19. Winterhof, A.: Recent results on recursive nonlinear pseudorandom number generators. In: Carlet, C., Pott, A. (eds.) SETA 2010. LNCS, vol. 6338, pp. 113–124. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15874-2_9

    Chapter  Google Scholar 

Download references

Acknowledgment

The authors are grateful to Andrea Ferraguti for preliminary reading of this manuscript, and for useful discussions and suggestions. The second author is thankful to the Swiss National Science Foundation grant number 171248.

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, UK

    Federico Amadio Guidi & Giacomo Micheli

Authors
  1. Federico Amadio Guidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giacomo Micheli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giacomo Micheli .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Lilya Budaghyan

  2. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico, Mexico

    Francisco Rodríguez-Henríquez

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Amadio Guidi, F., Micheli, G. (2018). Fractional Jumps: Complete Characterisation and an Explicit Infinite Family. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-05153-2_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05152-5

  • Online ISBN: 978-3-030-05153-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics