Skip to main content
SpringerLink
Log in

Recent Results on Recursive Nonlinear Pseudorandom Number Generators

(Invited Paper)

  • Conference paper
Sequences and Their Applications – SETA 2010 (SETA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6338))

Included in the following conference series:

Abstract

This survey article collects recent results on recursive nonlinear pseudorandom number generators and sketches some important proof techniques. We mention upper bounds on additive character sums which imply uniform distribution results. Moreover, we present lower bounds on the linear complexity profile and closely related lattice tests and thus results on the suitability in cryptography. Finally, we give bounds on multiplicative character sums from which one can derive results on the distribution of powers and primitive elements.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aly, H., Winterhof, A.: On the linear complexity profile of nonlinear congruential pseudorandom number generators with Dickson polynomials. Des. Codes Cryptogr. 39, 155–162 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourgain, J.: Mordell’s exponential sum estimate revisited. J. Amer. Math. Soc. 18, 477–499 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Çeşmelioğlu, A., Winterhof, A.: On the average distribution of power residues and primitive elements in inversive and nonlinear recurring sequences. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 60–70. Springer, Heidelberg (2008)

    Google Scholar 

  4. Chen, Z., Ostafe, A., Winterhof, A.: Structure of pseudorandom numbers derived from Fermat quotients. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 73–85. Springer, Heidelberg (2010)

    Google Scholar 

  5. Chou, W.-S.: The period lengths of inversive congruential recursions. Acta Arith. 73, 325–341 (1995)

    MATH  MathSciNet  Google Scholar 

  6. Chou, W.-S.: The period lengths of inversive pseudorandom vector generations. Finite Fields Appl. 1, 126–132 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dorfer, G.: Lattice profile and linear complexity profile of pseudorandom number sequences. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) Fq7 2003. LNCS, vol. 2948, pp. 69–78. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  8. Dorfer, G., Meidl, W., Winterhof, A.: Counting functions and expected values for the lattice profile at n. Finite Fields Appl. 10, 636–652 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dorfer, G., Winterhof, A.: Lattice structure and linear complexity profile of nonlinear pseudorandom number generators. Appl. Algebra Engrg. Comm. Comput. 13, 499–508 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dorfer, G., Winterhof, A.: Lattice structure of nonlinear pseudorandom number generators in parts of the period. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 199–211. Springer, Berlin (2004)

    Google Scholar 

  11. Drmota, M., Tichy, R.F.: Sequences, discrepancies and applications. LNM, vol. 1651. Springer, Berlin (1997)

    MATH  Google Scholar 

  12. El-Mahassni, E.D.: On the distribution of the power generator modulo a prime power for parts of the period. Bol. Soc. Mat. Mexicana 13(3), 7–13 (2007)

    MATH  MathSciNet  Google Scholar 

  13. El-Mahassni, E.D.: On the distribution of the power generator over a residue ring for parts of the period. Rev. Mat. Complut. 21, 319–325 (2008)

    MATH  MathSciNet  Google Scholar 

  14. El-Mahassni, E.D.: Exponential sums for nonlinear recurring sequences in residue rings. Albanian J. Math. (to appear)

    Google Scholar 

  15. 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-18. LNCS, vol. 5527, pp. 195–203. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  16. El-Mahassni, E.D., Shparlinski, I.E., Winterhof, A.: Distribution of nonlinear congruential pseudorandom numbers modulo almost squarefree integers. Monatsh. Math. 148, 297–307 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. El-Mahassni, E.D., Winterhof, A.: On the distribution of nonlinear congruential pseudorandom numbers in residue rings. Int. J. Number Theory 2, 163–168 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Friedlander, J.B., Hansen, J., Shparlinski, I.E.: Character sums with exponential functions. Mathematika 47, 75–85 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Friedlander, J.B., Shparlinski, I.E.: On the distribution of the power generator. Math. Comp. 70, 1575–1589 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fu, F.W., Niederreiter, H.: On the counting function of the lattice profile of periodic sequences. J. Complexity 23, 423–435 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E.: Exponential sums with Dickson polynomials. Finite Fields Appl. 12, 16–25 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gomez, D., Winterhof, A.: Character sums for sequences of iterations of Dickson polynomials. Finite fields and applications. Contemp. Math. 461, 147–151 (2008)

    MathSciNet  Google Scholar 

  23. Gomez, D., Winterhof, A.: Multiplicative character sums of recurring sequences with Redéi functions. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 175–181. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  24. Griffin, F., Shparlinski, I.E.: On the linear complexity profile of the power generator. IEEE Trans. Inform. Theory 46, 2159–2162 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  25. 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)

    Chapter  Google Scholar 

  26. 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)

    Chapter  Google Scholar 

  27. Gutierrez, J., Niederreiter, H., Shparlinski, I.E.: On the multidimensional distribution of inversive congruential pseudorandom numbers in parts of the period. Monatsh. Math. 129, 31–36 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Gutierrez, J., Winterhof, A.: Exponential sums of nonlinear congruential pseudorandom number generators with Rédei functions. Finite Fields Appl. 14, 410–416 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ibeas, A., Winterhof, A.: Exponential sums and linear complexity of nonlinear pseudorandom number generators with polynomials of small p-weight degree. Unif. Distrib. Theory 5, 79–93 (2010)

    Google Scholar 

  31. Kurlberg, P., Pomerance, C.: On the periods of the linear congruential and power generators. Acta Arith. 119, 149–169 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lidl, R., Mullen, G.L.: Cycle structure of Dickson permutation polynomials. Math. J. Okayama Univ. 33, 1–11 (1991)

    MATH  MathSciNet  Google Scholar 

  33. Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications, Revision of the 1986 first edition. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  34. Marsaglia, G.: The structure of linear congruential sequences. In: Zaremba, S.K. (ed.) Applications of Number Theory to Numerical Analysis, pp. 249–285. Academic Press, New York (1972)

    Google Scholar 

  35. Meidl, W., Winterhof, A.: On the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions. Finite Fields Appl. 13, 628–634 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Niederreiter, H.: Random number generation and quasi-Monte Carlo methods. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)

    Google Scholar 

  37. Niederreiter, H.: Design and analysis of nonlinear pseudorandom number generators. In: Monte Carlo Simulation, pp. 3–9. A.A. Balkema Publishers (2001)

    Google Scholar 

  38. Niederreiter, H.: Linear complexity and related complexity measures for sequences. In: Johansson, T., Maitra, S. (eds.) INDOCRYPT 2003. LNCS, vol. 2904, pp. 1–17. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  39. Niederreiter, H., Rivat, J.: On the correlation of pseudorandom numbers generated by inversive methods. Monatsh. Math. 153, 251–264 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Niederreiter, H., Shparlinski, I.E.: On the distribution and lattice structure of nonlinear congruential pseudorandom numbers. Finite Fields Appl. 5, 246–253 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  41. Niederreiter, H., Shparlinski, I.E.: On the distribution of pseudorandom numbers and vectors generated by inversive methods. Appl. Algebra Engrg. Comm. Comput. 10, 189–202 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Niederreiter, H., Shparlinski, I.E.: Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus. Acta Arith. 92, 89–98 (2000)

    MATH  MathSciNet  Google Scholar 

  43. Niederreiter, H., Shparlinski, I.E.: On the distribution of inversive congruential pseudorandom numbers in parts of the period. Math. Comp. 70, 1569–1574 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  44. Niederreiter, H., Shparlinski, I.E.: Recent advances in the theory of nonlinear pseudorandom number generators. In: Monte Carlo and quasi-Monte Carlo methods, 2000 (Hong Kong), pp. 86–102. Springer, Berlin (2002)

    Google Scholar 

  45. Niederreiter, H., Shparlinski, I.E.: On the average distribution of inversive pseudorandom numbers. Finite Fields Appl. 8, 491–503 (2002)

    MATH  MathSciNet  Google Scholar 

  46. Niederreiter, H., Shparlinski, I.E.: On the distribution of power residues and primitive elements in some nonlinear recurring sequences. Bull. London Math. Soc. 35, 522–528 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  47. Niederreiter, H., Shparlinski, I.E.: Dynamical systems generated by rational functions. In: Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 6–17. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  48. Niederreiter, H., Winterhof, A.: On the distribution of compound inversive congruential pseudorandom numbers. Monatsh. Math. 132, 35–48 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  49. Niederreiter, H., Winterhof, A.: Lattice structure and linear complexity of nonlinear pseudorandom numbers. Appl. Algebra Engrg. Comm. Comput. 13, 319–326 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  50. Niederreiter, H., Winterhof, A.: Multiplicative character sums for nonlinear recurring sequences. Acta Arith. 111, 299–305 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  51. Niederreiter, H., Winterhof, A.: Exponential sums and the distribution of inversive congruential pseudorandom numbers with power of two modulus. Int. J. Number Theory 1, 431–438 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  52. Niederreiter, H., Winterhof, A.: Exponential sums for nonlinear recurring sequences. Finite Fields Appl. 14, 59–64 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  53. Niederreiter, H., Winterhof, A.: On the structure of inversive pseudorandom number generators. In: Boztaş, S., Lu, H.-F(F.) (eds.) AAECC 2007. LNCS, vol. 4851, pp. 208–216. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  54. Ostafe, A.: Multivariate permutation polynomial systems and pseudorandom number generators. Finite Fields Appl. 16, 144–154 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  55. Ostafe, A.: Pseudorandom vector sequences derived from triangular polynomial systems with constant multipliers. In: Anwar Hasan, M. (ed.) WAIFI 2010. LNCS, vol. 6087, pp. 62–72. Springer, Heidelberg (2010)

    Google Scholar 

  56. Ostafe, A., Pelican, E., Shparlinski, I.E.: On pseudorandom numbers from multivariate polynomial systems. Finite Fields Appl. (to appear)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  58. Ostafe, A., Shparlinski, I.E.: Pseudorandom numbers and hash functions from iterations of multivariate polynomials. Cryptography and Communications 2, 49–67 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  59. Ostafe, A., Shparlinski, I.E., Winterhof, A.: On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences. Adv. Math. Commun. 4, 369–379 (2010)

    Article  Google Scholar 

  60. Ostafe, A., Shparlinski, I.E., Winterhof, A.: Multiplicative character sums of a class of nonlinear recurrence vector sequences (preprint)

    Google Scholar 

  61. Pirsic, G., Winterhof, A.: On the structure of digital explicit nonlinear and inversive pseudorandom number generators. J. Complexity 26, 43–50 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  62. Shparlinski, I.E.: On the linear complexity of the power generator. Des. Codes Cryptogr. 23, 5–10 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  63. Shparlinski, I.E.: On some dynamical systems in finite fields and residue rings. Discrete Contin. Dyn. Syst. 17, 901–917 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  64. Shparlinski, I.E.: On the average distribution of pseudorandom numbers generated by nonlinear permutations. Math. Comp. (to appear)

    Google Scholar 

  65. Topuzoğlu, A., Winterhof, A.: On the linear complexity profile of nonlinear congruential pseudorandom number generators of higher orders. Appl. Algebra Engrg. Comm. Comput. 16, 219–228 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  66. Topuzoğlu, A., Winterhof, A.: Pseudorandom sequences. In: Topics in Geometry, Coding Theory and Cryptography. Algebr. Appl., vol. 6, pp. 135–166. Springer, Dordrecht (2007)

    Chapter  Google Scholar 

  67. Wang, L.-P., Niederreiter, H.: Successive minima profile, lattice profile, and joint linear complexity profile of pseudorandom multisequences. J. Complexity 24, 144–153 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  68. Winterhof, A.: Linear complexity and related complexity measures. In: Selected Topics in Information and Coding Theory, pp. 3–40. World Scientific, Singapore (2010)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, 4040, Linz, Austria

    Arne Winterhof

Authors
  1. Arne Winterhof
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LAGA,Universities of Paris 8 and Paris 13 and CNRS, France

    Claude Carlet

  2. Institute for Algebra and Geometry, Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, D-39016,, Magdeburg, Germany

    Alexander Pott

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Winterhof, A. (2010). Recent Results on Recursive Nonlinear Pseudorandom Number Generators. In: Carlet, C., Pott, A. (eds) Sequences and Their Applications – SETA 2010. SETA 2010. Lecture Notes in Computer Science, vol 6338. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15874-2_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15874-2_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15873-5

  • Online ISBN: 978-3-642-15874-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation