Advertisement

Recent Results on Recursive Nonlinear Pseudorandom Number Generators

(Invited Paper)
  • Arne Winterhof
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)

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.

Keywords

Linear Complexity Pseudorandom Number Polynomial System Nonlinear Generator Pseudorandom Number Generator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bourgain, J.: Mordell’s exponential sum estimate revisited. J. Amer. Math. Soc. 18, 477–499 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 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. 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. 5.
    Chou, W.-S.: The period lengths of inversive congruential recursions. Acta Arith. 73, 325–341 (1995)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Chou, W.-S.: The period lengths of inversive pseudorandom vector generations. Finite Fields Appl. 1, 126–132 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 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. 11.
    Drmota, M., Tichy, R.F.: Sequences, discrepancies and applications. LNM, vol. 1651. Springer, Berlin (1997)zbMATHGoogle Scholar
  12. 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)zbMATHMathSciNetGoogle Scholar
  13. 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)zbMATHMathSciNetGoogle Scholar
  14. 14.
    El-Mahassni, E.D.: Exponential sums for nonlinear recurring sequences in residue rings. Albanian J. Math. (to appear)Google Scholar
  15. 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)CrossRefGoogle Scholar
  16. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Friedlander, J.B., Hansen, J., Shparlinski, I.E.: Character sums with exponential functions. Mathematika 47, 75–85 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Friedlander, J.B., Shparlinski, I.E.: On the distribution of the power generator. Math. Comp. 70, 1575–1589 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fu, F.W., Niederreiter, H.: On the counting function of the lattice profile of periodic sequences. J. Complexity 23, 423–435 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gomez-Perez, D., Gutierrez, J., Shparlinski, I.E.: Exponential sums with Dickson polynomials. Finite Fields Appl. 12, 16–25 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Gomez, D., Winterhof, A.: Character sums for sequences of iterations of Dickson polynomials. Finite fields and applications. Contemp. Math. 461, 147–151 (2008)MathSciNetGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 24.
    Griffin, F., Shparlinski, I.E.: On the linear complexity profile of the power generator. IEEE Trans. Inform. Theory 46, 2159–2162 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Gutierrez, J., Winterhof, A.: Exponential sums of nonlinear congruential pseudorandom number generators with Rédei functions. Finite Fields Appl. 14, 410–416 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 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. 31.
    Kurlberg, P., Pomerance, C.: On the periods of the linear congruential and power generators. Acta Arith. 119, 149–169 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Lidl, R., Mullen, G.L.: Cycle structure of Dickson permutation polynomials. Math. J. Okayama Univ. 33, 1–11 (1991)zbMATHMathSciNetGoogle Scholar
  33. 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. 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. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 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. 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. 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)CrossRefGoogle Scholar
  39. 39.
    Niederreiter, H., Rivat, J.: On the correlation of pseudorandom numbers generated by inversive methods. Monatsh. Math. 153, 251–264 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Niederreiter, H., Shparlinski, I.E.: On the distribution and lattice structure of nonlinear congruential pseudorandom numbers. Finite Fields Appl. 5, 246–253 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 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)zbMATHMathSciNetGoogle Scholar
  43. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 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. 45.
    Niederreiter, H., Shparlinski, I.E.: On the average distribution of inversive pseudorandom numbers. Finite Fields Appl. 8, 491–503 (2002)zbMATHMathSciNetGoogle Scholar
  46. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  47. 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)CrossRefGoogle Scholar
  48. 48.
    Niederreiter, H., Winterhof, A.: On the distribution of compound inversive congruential pseudorandom numbers. Monatsh. Math. 132, 35–48 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Niederreiter, H., Winterhof, A.: Lattice structure and linear complexity of nonlinear pseudorandom numbers. Appl. Algebra Engrg. Comm. Comput. 13, 319–326 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Niederreiter, H., Winterhof, A.: Multiplicative character sums for nonlinear recurring sequences. Acta Arith. 111, 299–305 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Niederreiter, H., Winterhof, A.: Exponential sums for nonlinear recurring sequences. Finite Fields Appl. 14, 59–64 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 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)CrossRefGoogle Scholar
  54. 54.
    Ostafe, A.: Multivariate permutation polynomial systems and pseudorandom number generators. Finite Fields Appl. 16, 144–154 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 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. 56.
    Ostafe, A., Pelican, E., Shparlinski, I.E.: On pseudorandom numbers from multivariate polynomial systems. Finite Fields Appl. (to appear)Google Scholar
  57. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Ostafe, A., Shparlinski, I.E.: Pseudorandom numbers and hash functions from iterations of multivariate polynomials. Cryptography and Communications 2, 49–67 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  59. 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)CrossRefGoogle Scholar
  60. 60.
    Ostafe, A., Shparlinski, I.E., Winterhof, A.: Multiplicative character sums of a class of nonlinear recurrence vector sequences (preprint)Google Scholar
  61. 61.
    Pirsic, G., Winterhof, A.: On the structure of digital explicit nonlinear and inversive pseudorandom number generators. J. Complexity 26, 43–50 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Shparlinski, I.E.: On the linear complexity of the power generator. Des. Codes Cryptogr. 23, 5–10 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Shparlinski, I.E.: On some dynamical systems in finite fields and residue rings. Discrete Contin. Dyn. Syst. 17, 901–917 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Shparlinski, I.E.: On the average distribution of pseudorandom numbers generated by nonlinear permutations. Math. Comp. (to appear)Google Scholar
  65. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  66. 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)CrossRefGoogle Scholar
  67. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Winterhof, A.: Linear complexity and related complexity measures. In: Selected Topics in Information and Coding Theory, pp. 3–40. World Scientific, Singapore (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Arne Winterhof
    • 1
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

Personalised recommendations