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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**J. Eichenauer & J. Lehn, "A non-linear congruential pseudo random number generator,"*Statist. Papers*, v. 27, 1986, pp. 315-326. MR**877295 (88i:65014)****[3]**J. Eichenauer, J. Lehn & A. Topuzoglu, "A nonlinear congruential pseudorandom number generator with power of two modulus,"*Math. Comp.*, v. 51, 1988, pp. 757-759. MR**958641 (89i:65007)****[4]**D. E. Knuth,*The Art of Computer Programming*, vol. 2:*Seminumerical Algorithms*, 2nd ed., Addison-Wesley, Reading, Mass., 1981. MR**633878 (83i:68003)****[5]**R. Lidl & H. Niederreiter,*Finite Fields*, Addison-Wesley, Reading, Mass., 1983. MR**746963 (86c:11106)****[6]**H. Niederreiter, "Pseudo-random numbers and optimal coefficients,"*Adv. in Math.*, v. 26, 1977, pp. 99-181. MR**0476679 (57:16238)****[7]**H. Niederreiter, "Quasi-Monte Carlo methods and pseudo-random numbers,"*Bull. Amer. Math. Soc.*, v. 84, 1978, pp. 957-1041. MR**508447 (80d:65016)****[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]**H. Niederreiter, "The serial test for pseudo-random numbers generated by the linear congruential method,"*Numer. Math.*, v. 46, 1985, pp. 51-68. MR**777824 (86i:65010)****[10]**H. Niederreiter, "Statistical independence of nonlinear congruential pseudorandom numbers,"*Monatsh. Math.*(To appear.) MR**968332 (89j:11079)****[11]**H. Niederreiter, "Remarks on nonlinear congruential pseudorandom numbers,"*Metrika*(To appear.) MR**980847 (90e:11119)****[12]**H. Salié, "Über die Kloostermanschen Summen ,"*Math. Z.*, v. 34, 1932, pp. 91-109.**[13]**S. A. Stepanov, "On estimating rational trigonometric sums with prime denominator,"*Trudy Mat. Inst. Steklov.*, v. 112, 1971, pp. 346-371. (Russian) MR**0318158 (47:6706)****[14]**A. Weil, "On some exponential sums,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 34, 1948, pp. 204-207. MR**0027006 (10:234e)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65C10,
11K45

Retrieve articles in all journals with MSC: 65C10, 11K45

Additional Information

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

Article copyright:
© Copyright 1989
American Mathematical Society