The serial test for congruential pseudorandom numbers generated by inversions

Author:
Harald Niederreiter

Journal:
Math. Comp. **52** (1989), 135-144

MSC:
Primary 65C10; Secondary 11K45

DOI:
https://doi.org/10.1090/S0025-5718-1989-0971407-2

MathSciNet review:
971407

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Abstract: Two types of congruential pseudorandom number generators based on inversions were introduced recently. We analyze the statistical independence properties of these pseudorandom numbers by means of the serial test. The results show that these pseudorandom numbers perform satisfactorily under the serial test. The methods of proof rely heavily on bounds for character sums such as the Weil-Stepanov bound for character sums over finite fields.

**[1]**J. Eichenauer, H. Grothe & J. Lehn, "Marsaglia's lattice test and non-linear congruential pseudo random number generators,"*Metrika*, v. 35, 1988, pp. 241-250.**[2]**Jürgen Eichenauer and Jürgen Lehn,*A nonlinear congruential pseudorandom number generator*, Statist. Hefte**27**(1986), no. 4, 315–326. MR**877295****[3]**Jürgen Eichenauer, Jürgen Lehn, and Alev Topuzoğlu,*A nonlinear congruential pseudorandom number generator with power of two modulus*, Math. Comp.**51**(1988), no. 184, 757–759. MR**958641**, https://doi.org/10.1090/S0025-5718-1988-0958641-1**[4]**Donald E. Knuth,*The art of computer programming. Vol. 2*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**633878****[5]**Rudolf Lidl and Harald Niederreiter,*Finite fields*, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR**746963****[6]**Harald Niederreiter,*Pseudo-random numbers and optimal coefficients*, Advances in Math.**26**(1977), no. 2, 99–181. MR**0476679**, https://doi.org/10.1016/0001-8708(77)90028-7**[7]**Harald Niederreiter,*Quasi-Monte Carlo methods and pseudo-random numbers*, Bull. Amer. Math. Soc.**84**(1978), no. 6, 957–1041. MR**508447**, https://doi.org/10.1090/S0002-9904-1978-14532-7**[8]**H. Niederreiter, "Number-theoretic problems in pseudorandom number generation," in*Proc. Sympos. on Applications of Number Theory to Numerical Analysis*, Lecture Notes No. 537, Research Inst. of Math. Sciences, Kyoto, 1984, pp. 18-28.**[9]**Harald Niederreiter,*The serial test for pseudorandom numbers generated by the linear congruential method*, Numer. Math.**46**(1985), no. 1, 51–68. MR**777824**, https://doi.org/10.1007/BF01400255**[10]**Harald Niederreiter,*Statistical independence of nonlinear congruential pseudorandom numbers*, Monatsh. Math.**106**(1988), no. 2, 149–159. MR**968332**, https://doi.org/10.1007/BF01298835**[11]**H. Niederreiter,*Remarks on nonlinear congruential pseudorandom numbers*, Metrika**35**(1988), no. 6, 321–328. MR**980847**, https://doi.org/10.1007/BF02613320**[12]**H. Salié, "Über die Kloostermanschen Summen ,"*Math. Z.*, v. 34, 1932, pp. 91-109.**[13]**S. A. Stepanov,*The estimation of rational trigonometric sums with prime denominator*, Trudy Mat. Inst. Steklov.**112**(1971), 346–371, 389. (errata insert) (Russian). Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday. I. MR**0318158****[14]**André Weil,*On some exponential sums*, Proc. Nat. Acad. Sci. U. S. A.**34**(1948), 204–207. MR**0027006**

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0971407-2

Article copyright:
© Copyright 1989
American Mathematical Society