Multiplicative Character Sums with Counter-Dependent Nonlinear Congruential Pseudorandom Number Generators

  • Domingo Gomez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)


Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper we present a bound for multiplicative character sums for nonlinear sequences generated by counter-dependent generators.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Shamir, A., Tsaban, B.: Guaranteeing the diversity of number generators. Inform. and Comput. 171(2), 350–363 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Niederreiter, H., Shparlinski, I.E.: On the distribution of power residues and primitive elements in some nonlinear recurring sequences. Bull. London Math. Soc. 35(4), 522–528 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Gutierrez, J., Shparlinski, I.E., Winterhof, A.: On the linear and nonlinear complexity profile of nonlinear pseudorandom number-generators. IEEE Trans. Inform. Theory 49(1), 60–64 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Niederreiter, H., Shparlinski, I.E.: On the distribution and lattice structure of nonlinear congruential pseudorandom numbers. Finite Fields Appl. 5(3), 246–253 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Niederreiter, H., Winterhof, A.: Exponential sums for nonlinear recurring sequences. Finite Fields Appl. 14(1), 59–64 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Topuzoğlu, A., Winterhof, A.: Pseudorandom sequences. In: Topics in Geometry, Coding Theory and Cryptography. Algebr. Appl., vol. 6, pp. 135–166. Springer, Dordrecht (2007)CrossRefGoogle Scholar
  7. 7.
    Winterhof, A.: Recent results on recursive nonlinear pseudorandom number generators. In: Sequences and their Applications SETA 2010. LNCS. Springer, Heidelberg (2010)Google Scholar
  8. 8.
    Niederreiter, H., Winterhof, A.: Multiplicative character sums for nonlinear recurring sequences. Acta Arith. 111(3), 299–305 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    El-Mahassni, E., Winterhof, A.: On the distribution and linear complexity of counter-dependent nonlinear congruential pseudorandom number generators. Journal of Algebra, Number Theory and Applications 2, 1–6 (2006)MathSciNetGoogle Scholar
  10. 10.
    Griffin, F., Niederreiter, H., Shparlinski, I.E.: On the distribution of nonlinear recursive congruential pseudorandom numbers of higher orders. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 87–93. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Gutierrez, J., Gomez-Perez, D.: Iterations of multivariate polynomials and discrepancy of pseudorandom numbers. In: Bozta, S., Sphparlinski, I. (eds.) AAECC 2001. LNCS, vol. 2227, pp. 192–199. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Ostafe, A., Pelican, E., Shparlinski, I.E.: On pseudorandom numbers from multivariate polynomial systems. Finite Fields and their Applications (to appear 2010)Google Scholar
  13. 13.
    Topuzoğlu, A., Winterhof, A.: On the linear complexity profile of nonlinear congruential pseudorandom number generators of higher orders. Appl. Algebra Engrg. Comm. Comput. 16(4), 219–228 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ostafe, A.: Multivariate permutation polynomial systems and nonlinear pseudorandom number generators. Finite Fields and Their Applications 16(3), 144–154 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ostafe, A., Shparlinski, I.E.: On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators. Math. Comp. 79(269), 501–511 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ostafe, A., Shparlinski, I.E., Winterhof, A.: Multiplicative character sums of a class of nonlinear recurrence vector sequences (2010) (Preprint)Google Scholar
  17. 17.
    Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms, 3rd edn. Undergraduate Texts in Mathematics. Springer, New York (2007); An introduction to computational algebraic geometry and commutative algebra zbMATHGoogle Scholar
  18. 18.
    Hong, H.: Subresultants under composition. J. Symbolic Comput. 23(4), 355–365 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lidl, R., Niederreiter, H.: Finite fields. In: Encyclopedia of Mathematics and its Applications, 2nd edn., vol. 20. Cambridge University Press, Cambridge (1997), With a foreword by P. M. Cohn Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Domingo Gomez
    • 1
  1. 1.Faculty of SciencesUniversity of CantabriaSantanderSpain

Personalised recommendations