Fractional Jumps: Complete Characterisation and an Explicit Infinite Family

  • Federico Amadio Guidi
  • Giacomo MicheliEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11321)


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.



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.


  1. 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. 2.
    Brandstätter, N., Winterhof, A.: Some notes on the two-prime generator of order 2. IEEE Trans. Inf. Theory 51(10), 3654–3657 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cao, X.: On the order of the polynomial \(x^p-x-a\). Cryptology ePrint Archive, Report 2010/034 (2010).
  4. 4.
    Chou, W.-S.: On inversive maximal period polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 6(4), 245–250 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Eichenauer-Herrmann, J.: Inversive congruential pseudorandom numbers avoid the planes. Math. Comp. 56(193), 297–301 (1991)MathSciNetCrossRefGoogle Scholar
  6. 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). Scholar
  7. 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). Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Heath-Brown, D.R., Micheli, G.: Irreducible polynomials over finite fields produced by composition of quadratics. arXiv preprint arXiv:1701.05031 (2017)
  11. 11.
    Lang, S.: Algebra - Revised Third Edition. Graduate Texts in Mathematics, vol. 211. Springer, New York (2002). Scholar
  12. 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). Scholar
  13. 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). Scholar
  14. 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). Scholar
  15. 15.
    Ostafe, A., Pelican, E., Shparlinski, I.E.: On pseudorandom numbers from multivariate polynomial systems. Finite Fields Appl. 16(5), 320–328 (2010)MathSciNetCrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 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)MathSciNetCrossRefGoogle Scholar
  18. 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). Scholar
  19. 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). Scholar

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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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