, Volume 40, Issue 1, pp 115–120 | Cite as

The lattice structure of nonlinear congruential pseudorandom numbers

  • Jürgen Eichenauer-Herrmann


Several known deficiencies of the classical linear congruential method for generating uniform pseudorandom numbers led to the development of nonlinear congruential pseudorandom number generators. In the present paper a general class of nonlinear congruential methods with prime power modulus is considered. It is proved that these generators show certain undesirable linear structures, too, which stem from the composite nature of the modulus.


Lattice Structure Pseudorandom Number Congruential Sequence Congruential Method Congruential Pseudorandom Number 
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  1. Eichenauer J, Lehn J (1987) On the structure of quadratic congruential sequences, manuscripta math. 58:129–140zbMATHCrossRefMathSciNetGoogle Scholar
  2. Eichenauer J, Grothe H, Lehn J (1988a) Marsaglia’s lattice test and non-linear congruential pseudo random number generators, Metrika 35:241–250CrossRefzbMATHGoogle Scholar
  3. Eichenauer J, Lehn J, Topuzoğlu A (1988b) A nonlinear congruential pseudorandom number generator with power of two modulus, Math. Comp. 51:757–759zbMATHCrossRefMathSciNetGoogle Scholar
  4. Eichenauer-Herrmann J (1991) Inversive congruential pseudorandom numbers avoid the planes, Math. Comp. 56:297–301zbMATHCrossRefMathSciNetGoogle Scholar
  5. Eichenauer-Herrmann J (1992a) Construction of inversive congruential pseudorandom number generators with maximal period length, J. Comp. Appl. Math. 40:345–349zbMATHCrossRefMathSciNetGoogle Scholar
  6. Eichenauer-Herrmann J (1992b) Inversive congruential pseudorandom numbers: a tutorial, Int. Statist. Rev. 60:167–176zbMATHCrossRefGoogle Scholar
  7. Eichenauer-Herrmann J, Topuzoğlu A (1990) On the period length of congruential pseudorandom number sequences generated by inversions, J. Comp. Appl. Math. 31:87–96zbMATHCrossRefGoogle Scholar
  8. Eichenauer-Herrmann J, Grothe H, Niederreiter H, Topuzoğlu A (1990) On the lattice structure of a nonlinear generator with modulus 2α, J. Comp. Appl. Math. 31:81–85zbMATHCrossRefGoogle Scholar
  9. Knuth DE (1981) The art of computer programming, vol. 2, Addison-Wesley, Reading, Mass., 2nd edzbMATHGoogle Scholar
  10. Niederreiter H (1988) Remarks on nonlinear congruential pseudorandom numbers, Metrika 35:321–328zbMATHCrossRefMathSciNetGoogle Scholar
  11. Niederreiter H (1991) Recent trends in random number and random vector generation, Ann. Operations Res. 31:323–346zbMATHCrossRefMathSciNetGoogle Scholar
  12. Niederreiter H (1992) Random number generation and quasi-Monte Carlo methods, SIAM, PhiladelphiazbMATHGoogle Scholar

Copyright information

© Physica-Verlag GmbH 1993

Authors and Affiliations

  • Jürgen Eichenauer-Herrmann
    • 1
  1. 1.Fachbereich Mathematik, Technische Hochschule DarmstadtDarmstadtGermany

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