[11] J.V. Uspensky, Introduction to Mathematical Probability (McGraw Hill, New

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1 [11] J.V. Uspensky, Introduction to Mathematica Probabiity (McGraw Hi, New York, 1937) 77{84. [12] Wiiam G. Cochran, The 2 Test of Goodness of Fit (John Hopkins University, Department of Biostatistics, Paper No. 282). [13] David M. Arnow, DP: A Library for Buiding Portabe, Reiabe, Distributed Appications, in: Proceedings of the USENIX Winter '95 Technica Conference (Usenix Association, New Oreans,1995). [14] Scott Kirkpatrick and Erich P. Sto, A Very Fast Shift-Register Sequence Random Number Generator, J. Comput. Phys. 40 (1981) 517{526. [15] W.L. Maier, A Fast Pseudorandom Number Generator, Dr. Dobb's Journa 176 (1991). [16] Aan M. Ferrenberg and D.P. Landau, Monte Caro Simuations: Hidden Errors from "Good" Random Number Generators, Phys. Rev. Lett. 69 (1992) 3382{ [17] H. Wey, Uber die Geichvereiung von Zahan mod. Eins, Math. Ann. 77 (1916) 313{352. [18] Brad Lee Hoian, Ora E. Percus, Tony T. Warnock and Paua A. Whitock, Pseudorandom Number Generator for Massive Parae Moecuar-Dynamics Simuations, Phys. Rev. E 50 (1994) 1607{1615. [19] J. Eichenauer-Herrmann, Statistica independence of a new cass of inversive congruentia pseudorandom numbers, Math. Comp. 60 (1993) 375{384. [20] Otmar Lend, prng2.2, A ibrary for the generation of pseudorandom numbers, obtained January 27, 1997, [21] Pierre L'Ecuyer and Serge C^ote, Impementing a Random Number Package with Spitting Faciities, ACM Trans. on Math. Software 17 (1991) 98{111. [22] Pierre L'Ecuyer and S. Tezuka, Structura properties for two casses of combined random number generators, Math. of Compu. 57 (1991) 735{746. [23] SunOS 5.6 manua pages, "man srand48", 22 Jan [24] G. Marsagia, Random numbers fa mainy in the panes, Proc. Nat. Acad. Sci. USA 60 (1968) 25{28. [25] P. L'Ecuyer, R. Simard and S. Wegenkitt, Sparse seria tests of uniformity for random number generators, preprint,

2 Acknowedgments This work was supported in part by NSF Grant No. ASC and by PSC/ CUNY Grant No P.A.W. is aso supported by ONR Grant No. N References [1] D.E. Knuth, The Art of Computer Programming, Vo. 2, Seminumerica Agorithms, 2 n d edition (Addison-Wesey, Reading, MA, 1981). [2] A. De Matteis and S. Pagnutti, A Cass of Parae Random Number Generators, Parae Comput. 13 (1990) 193{198. [3] Jian Chen and Paua A. Whitock, Impementation of A Distributed Pseudorandom Number Generator, in: H. Niederreiter and P. J-S. Shiue, eds., Monte Caro and Quasi-Monte Caro Methods in Scientic Computing, Lecture Notes in Statistics 106 (Springer-Verag, Berin, 1995) 168{185. [4] Mark J. Durst, Using Linear Congruentia Generators for Parae Random Number Generation, in: Edward A. MacNair, Kenneth J. Musseman, Phiip Heideberger, eds., Proceedings of the 1989 Winter Simuation Conference (Society for Computer Simuation, 1989) 462{466. [5] Harad Niederreiter, New Deveopments in Pseudorandom Number and Vector Generation, in: H. Niederreiter and P. J-S. Shiue, eds., Monte Caro and Quasi- Monte Caro Methods in Scientic Computing, Lecture Notes in Statistics 106 (Springer-Verag, Berin, 1995) 87{120. [6] O.E. Percus and M.H. Kaos, Random Number Generators for MIMD Parae Processors, J. Parae and Dist. Comput. 6 (1989) 477{497. [7] P. Heekaek, Inversive pseudorandom number generators: concepts, resuts, and inks, in: C. Aexopouos, K. Kang, W.R. Liegdon and D. Godsman, eds., Proceedings of the 1995 Winter Simuation Conference (Society of Computer Simuation, 1995) 255{262. [8] W.F. Eddy, Random number generators for parae processors, J. Comp. App. Math. 31 (1990) 63{71. [9] Michae Mascagni, Steven A. Cuccaro, Danie V. Pryor and M.L. Robinson, A Fast, High Quaity, and Reproducibe Parae Lagged-Fibonacci Pseudorandom Generator, J. Comp. Phys. 119 (1995) 211{219. [10] Ora E. Percus, Testing for Correations Between Independent Parae Random Number Generators, New York University (1995). 13

3 Generators Spit Combination Srand48 run q bits K + q K? q q bits K + q K? q (1) 150 0x x3C (2) 60 0x x (3) 60 0x Tabe 4 The parae sequence test resuts for the Spit Mutipicative Linear Congruentia Combination geneator and Srand48 other consists of the even terms fx 2 (k) = srand48(2k) k = 1; 2; g. Each stream is further spit into sequences of ength of 1,000,000. One sequence from each stream in the corresponding order is chosen to form a pair of sequences. Two experiments were performed and described in Tabe 4. One experiment examined bits 26?29 and in the other, bits 27?30 were tested. The F K (K q ) in each experiment for q = 60 were outside the interva [10%; 90%]. Hence we have 90% of condence to reect it. The generator srand48 was not caimed to be a PRNG [23]; so it is not surprising that it fais the empirica test. 6 Concusions We present a new empirica test for PRNG's based on Percus' theory. It is a generic test that can be used to study both bit stream generators and fu word size generators. The test was impemented on a distributed network and used to study severa pubished PRNG. The resuts conrmed the expectation that the nested Wey generator and the C ibrary function, srand48, woud fai the test. Both of these generators have we-documented shortcomings, even when used as seria pseudorandom number generators and aso have parae correations that the parae sequence test detects. The resuts of the appying the parae sequence test to R250 is aso consistent with pubished reports. Researchers have reported both successes and faiures using it in parae simuations. The faiure of the shued, nested Wey generator and the expicit inversive congruentia method were not anticipated and are under further investigation. Finay, mutipicative congruentia generators are known to have many types of seria correations [24], [25]. However, when used as parae sequences, the most signicant bits do not exhibit across sequence correations as tested by the parae sequence test. The parae sequence test appears to predict correations that are important in some casses of simuations and shoud be a usefu too in examining new parae pseudo-random number generators. 12

4 run a 0 bits tested q K + q K? q (1) x (3) x (4) x Tabe 3 The parae sequence test resuts for the Expicit Inversive Congruentia Generator The version of the generator used is based on the impementation by Otmar Lend [20] with p = = 2 31?1. Three experiments were performed. Tabe 3 ists the vaues of the mutipiers a 0 (a 1 = a 0 ) used in each experiment and the number of repications. The additives c 0 and c 1 were (5 m) and (5 m + 91) respectivey for the m th pair of sequences. The 2 most signicant bits (bits 28 and bit 29) in 3 experiments and the 3 most signicant bits (bits 27-29) in the other were tested. In a experiments, the F K (K q ) have the trend to go out of the range of [6%; 87%]. We have 87% condence to reect it as a PRNG. 5.5 Spit Mutipicative Linear Congruentia Generator The generator is based on L'Ecuyer's impementation [21] of a combination generator using two dierent inear congruentia generators: X +1 = ax + c mod p 0 (23) The (a 1 ; a 2 ; c 1 ; c 2 ; p) are chosen to yied the maximum period. Then the period of the sequence is divided up into N subsequences whose starting points are far apart. Each process using the generator is assigned a non-overapping supsequence with a period of = p=n. The structure of this generator was anayzed in [22]. Three experiments were performed. One of the experiments tested the 3 most signicant bits (mask 0x ) and the others tested the 4 most signicant bits (mask 0x ). The resuts of each experiment are shown in Tabe 4. The F K (K q ) are in the range of [40%; 60%]. We can say this generator passes our test. 5.6 SUN Workstation C ibrary function, srand48 Two pseudorandom number streams were formed from srand48 [23], one consists of a of its odd terms fx 1 (k) = srand48(2k? 1) k = 1; 2; g and the 11

5 5.3 Shued Nested Wey Sequence This generator was an attempt to create a PRNG with better properties than the nested Wey sequence [18]. An integer M >> 1 is seected and a sequence of processor numbers n is given as n = Mfnfngg (18) The shued nested Wey sequence is dened by x n = f n f n gg. To create mutipe sequences, the k th sequence is dened by x (k) n = fkf nf n ggg (19) and (17) is modied to X (k) n k = 2 s 0 x (k) n = b2 s 0 fkf n f n gggc (20) In this case, s 0 was 4, = p 2 and M = A bits in the whoe number part are tested. Two experiments were done with q = 15, Tabe 2. The F K (K q ) in each experiment are outside of [1%; 99%]. Hence we have 99% of condence to reect the PRNG. 5.4 Expicit Inversive Congruentia Method When used as an individua RNG, the Expicit Inversive Congruentia Method [5,7,19] is dened as X n = an + c mod p for n 0 (21) where p is a prime and x = x?1 ed F p. is the inverse of the eement x in the nite As a PRNG, a dierent pair of a and c is assigned for each dierent pseudorandom sequence. So the generator for the k th process is given by X (k) n = a k n + c k mod p for n 0 (22) The pairs of a k and c k shoud be seected propery such that c 0 a 0 ; c 1 a 1 ; ; c N?1a N?1 2 F p are distinct. 10

6 Generators R250 Nested Wey SNWS q run K + q K? q K + q K? q K + q K? q (1) (2) (3) (4) Tabe 2 The quantity >ff K (K q )g? 0:5> obtained from running the parae sequence test for the pseudo-random number generators R250, the nested Wey sequence and the shued nested Wey sequence (SNWS). R250. In practice, R250 was found to introduce correations into some casses of parae simuations [16]. 5.2 Nested Wey Sequence The nested Wey sequence (NWS) is a natura extension of the Wey sequence [17] and is dened as x n = fnfngg, where fg indicates the fractiona part of the encosed number, within the precision of the computer used. When used as a PRNG, the k th sequence is dened by x (k) n = fkfnfnggg (16) Because our empirica test ony accepts integers, the above formua was repaced by X (k) n k = 2 s 0 x (k) n = b2 s 0 fkfnfngggc (17) where byc is the oor of y, and s 0 is an integer such that x (k) n are converted to integers in [0; 2 s 0 ). In our tests, = p 2, s 0 was 4 and a bits in the whoe number part were tested. Two experiments were done and for q = 10 (arger vaues of q showed no improvement), Tabe 2, the F K (K q ) in each experiment are outside of the range [1%; 99%]. Hence we have 99% of condence to reect it. This resut was not unexpected since previous work had found the generator correated [18]. 9

7 4.2 The Test Modue The test modue uses the DP ibrary to distribute processes to carry out the testing across a network of workstations. A primary process on the initia host begins the execution of test modue processes on the other network hosts. It coects the ongest run statistics, cacuates the 2 vaues and performs the Komogorov-Smirnov test. Each host on the network executes two processes, one that provides the needed pseudo-random variabes and another that carries out the detais of the test. That is, the distributed test modue process obtains the t sequences of random variabes, produces the new binary sequence and cacuates the ongest run of the new sequence. The DP message passing faciities are used to communicate with the primary process. 4.3 The Conguration Modue The conguration modue is used to set the parameters needed by the test and the provider modues. The parameters incude the PRNG to be tested (mutipe PRNG's can be contained in the provider modue), the bit ocations to be tested, the ength of a sequence, the number of ongest runs for a chi-square test L, the tota number of 2 vaues to be observed for the Komogorov-Smirnov test, L ks, and the number of sequences in a group t. The conguration modue can be modied any time before performing the test without having to recreate the executabe programs. 5 Resuts of Appying the Parae Sequence test to Specic Generators 5.1 R250 The R250 pseudo-random number generator [14] as impemented by W. L. Maier [15] was used. Since 16 R250 streams can be obtained from the generator at the same time, two of them were randomy chosen for the test. The test was repeated four times and the resuts are shown in Tabe 2, for q = 150 repications. Ony one of the experiments exhibited acceptabe behavior. The other 3 experiments showed poorer performance; the F K (K q ) were out of the range [15%; 85%]. According to (15), we have a 85% condence to reect 8

8 1 q L ks, which are uniformy distributed on [0,1]. For any integer q 0 much smaer than L ks, e.g. q 0 < 1 2 L ks, if q > q 0 such that >ff K (K q )g? 0:5 > > d% (15) for any positive number d < 50, we have (50+d) percent of condence to reect the hypothesis that any t sequences of pseudo-random variabes generated by the PRNG are independent. 4 Impementation of the Empirica Test A prototype of the parae sequence test was impemented and run on a custer of workstations distributed across a network. The test was impemented in the C anguage on a UNIX operating system environment with the message passing faciity, DP [13], instaed. DP is a ibrary of process management and communication toos for faciitating writing portabe distributed programs on MIMD systems. It supports dynamic process creation and message passing with a variety of semantics. To perform the test upon a PRNG, three modues were created, the test modue, the provider modue and the conguration modue. The test modue is a executabe program consisting of a of the maor functionaity for testing a particuar PRNG. The provider modue is another executabe program providing the test modue with pseudo-random variabes generated by the specic PRNG to be tested. The conguration modue is a text e with the parameters for a test. 4.1 The Provider Modue The provider modue consists of one or more PRNG's to be tested and the interface between the PRNG's and the test modue. The interface transfers pseudo-random variabes from the PRNG specied in the conguration modue to the test modue. It is hence caed the provider whie the test modue acts as a consumer of pseudo-random variabes. The program begins execution when a process running the test modue requests pseudo- random variabes. The provider modue can be maintained by users of the testing software without the need to know the detais of the test modue or the DP faciity. 7

9 are N c mutua excusive casses [n i?1; n i ), (1 i N c ). The probabiity that the ongest run, r (k), fas in the mutuay excusive cass, [n i?1; n i ) can be obtained from P (r; ), (4) - (6), the probabiity that a ongest run exceeds ength r. That is, P fr (k) 2 [n i?1; n i )g = P (n i?1; )? P (n i ; ) (10) The expected number of ongest runs faing into an interva, P fr (k) 2 [n i?1; n i )g 5, to yied a good approximation to the asymptotic chi square distribution. Let Z (k) i that n i?1 r (k) n i, then be the number of r (k) V k = 2 = XN c i=1, (1 L ); (1 k L ks ) such (Z (k) i? e i ) 2 e i ; (11) where e i = P fr (k) 2 [n i?1; n i )g. V k is asymptoticay chi-square distributed with N c? 1 degrees of freedom, V k 2 N c?1, see [12]. 3.3 Performing the Komogorov-Smirnov Test Each time a new vaue of 2 is cacuated, it is added to the set, V 1 ; ; V q, obtained from (11). The V i are sorted in ascending order and denoted by V 1 Vq. A Komogorov-Smirnov test is performed on the vaues and the quantities K q + and K? q are formed: K + q K? q = p q max 1q = p q max 1qek! q? F ( 2 Nc?1 V?1) F 2 Nc?1 ( V?1)? q! (12) (13) When q is arge, the distribution of K + q and K? q is given by F K q (t) = 1? e 2t2 1? 2t 3 p q! ; t 0; (14) The distribution functions (14) can be used to transform the sequences of reaizations of K + q and K? q into sequences of numbers ff K +(K + q )g and ff K?(K? q )g, 6

10 Longest Number of Bits Run Tabe 1 The distributions of engths of ongest runs in binary sequences of ength 1,000,000 with the number of bits considered varying from 1 to 6. 5

11 and the anaysis continues as in (3). 3 The Parae Sequence Test The basis of the empirica test is to consider many groups of pseudo-random number sequences derived from a particuar PRNG. The number of sequences, t, tested in each group can be two or more. Each group of sequences is converted to a new binary sequence with (2), if t = 2 or (7), if t 3. The ongest run of 1's in each new binary sequence is determined. A chi-square statistic, dened beow, is cacuated for each group. The vaues of the chi-square statistic are used in a Komogorov-Smirnov test to udge whether the PRNG produces sequences whose properties mimic those of a truy random sequence. 3.1 Cacuate Longest Runs The set of bit ocations, I, is chosen prior to the test and are xed throughout a singe repication of the test. The number of casses, N c, representing the distribution of possibe ongest runs depends upon the number of bits considered. Tabe 1 iustrates the observed distributions of probabiities, p, for a ongest run of a random binary sequence when t = 2,, the ength of the sequence = 1,000,000 and 1 s 6 bit ocations are examined. It can be seen that N c becomes smaer as the number of bits to be tested increases. To be meaningfu, s shoud be chosen so that p is greater than 5. From each group of sequences using the s bits, a new binary sequence Y (;k) 1 ; Y (;k) 2 ; ; Y (;k) (1 L 1 k L ks ) (9) is obtained from (2) if t = 2 or (7) if t 3. Each binary sequence is formed from independent, non-overapping sequences. L is the number of groups used to form the 2 statistic and L ks is the maximum number of 2 vaues used in the Komogorov-Smirnov test. The ongest run of 1's in each sequence of (9) is determined and is denoted by r (k). 3.2 Forming the Chi-square Statistic Longest runs exceeding a ength of 100 vaues are unikey and are grouped together in the statistica anaysis. In genera, we have 0 r (k) 100. Let n i be integers where (1 i N c ), and n 0 = 0, n Nc < 101, n i?1 < n i. There 4

12 P fength of ongest run rg is [11] P (r; ) = 1? r+1 X =0 (?1) (qp r )? r! + p r?r r+1 X =0 (?1) (qp r )? ( + 1)r! :(3) It is chaenging to compute P (r; ) with (3). Aternativey, et = p r (1? p) (4) It can be proved [10] that the equation? 1? r+1 = 0 (5) has at east one positive rea root. Denote the smaest one as 0. Then P (r; ) has the foowing estimate P (r; ) = 1? 1? p 0?1 0 r + (1? p) 0 r + 1? r 0 1? p p+2 (6) where < 1. The root 0 can be found either by direct soution of the trinomia equation foowing Gauss' method, or by appication of Lagrange's series. Percus's anaysis can be extended to the case of t parae sequences X (1) 1 ; X (1) 2 ; ; X (1) X (2) 1 ; X (2) 2 ; ; X (2) X (t) 1 ; X (t) 2 ; ; X (t) : A binary sequence fy 1 g can be dened such that Y = 8 >< >: 1 if at east w of the fx (k) g t k=1 are equa for a i 1 ; ; i s 2 I 0 otherwise (7) where (max(1; t? 2 s ) w t). Then the probabiity p is p = P fy = 1g = tx =w! t 1 2 (1? 1 s(?1) 2 s )t? (8) 3

13 in actua simuations. As the performance in theoretica tests is no guarantee, but ony an indicator of what we may expect in practice, empirica testing of generators is a necessity [7]. Some theoretica tests have been deveoped for specic PRNG's [6,9]. Our goa was to deveop and impement a genera empirica test that can be appied to a wide range of PRNG's. The empirica test described here is based upon the theoretica investigations of Percus [10]. Section 2 briey reviews Percus's theory on the properties of parae sequences of truy random numbers. The transation of the theory into an empirica test, the parae sequence test, is described in Section 3. An outine of the impementation of the test as a distributed cacuation is given in Section 4. Finay, Section 5 discusses the appication of the parae sequence test to severa pubished PRNG's. We concude that the test is usefu in predicting correations that aect a distributed or parae simuation. 2 Theoretica Basis of the Parae Sequence Test Consider 2 sequences of random variabes X (1) 1 ; X (1) 2 ; ; X (1) X (2) 1 ; X (2) 2 ; ; X (2) ; (1) where the X (k) are integers represented by an arbitrary, but xed number of bits and is the ength of the sequence. Consider the set I = fi 1 ; :::; i s g of indices of bit ocations in each integer, where the tota number of bit ocations of interest is s. These bit ocations do not have to be consecutive. A new binary sequence fy 1 g is created such that Y = 8 >< >: 1 if X (1) equas X (2) for i 1 ; ; i s 2 I 0 otherwise (2) Assume each bit of X (k) has the probabiity of 1=2 to be 0 or 1. If the two sequences of (1) are independent, fy g woud be a random number sequence with the probabiity P fy = 1g = 2?s. Let q = 1? p, and P (r; ) be the probabiity of having r ones in succession in fy g, where (1 r ). Percus [10] derived that the probabiity, 2

14 A New Empirica Test for Parae Pseudorandom Number Generators Yufeng Liang and P.A. Whitock Department of Computer and Information Science, Brookyn Coege, Brookyn, NY Abstract Recenty, Percus derived probabiities and distributions for pare, i.i.d. random sequences of integers. This was accompished by considering s given bit ocations in each random variabe (represented as a predetermined number of bits) in each sequence. These s bits were used to create a new binary sequence whose expected behavior can be anayzed. Based upon Percus' work, an empirica test for parae pseudo-random number generators has been devised. For each generator, parae sequences of various engths are considered and anayzed as proposed by Percus and the resuts are statisticay compared to the expected behavior for truy random sequences. A variety of parae pseudo-random number generators from the iterature are studied and the usefuness of the new empirica test is discussed. 1 Introduction Computer simuations use random variabes to mode the probabiity distribution functions important to the cacuations. The random variabes are repaced by vaues, pseudo-random numbers, provided by deterministic agorithms whose properties mimic those of a truy random sequence [1]. A parae pseudo-random number generator (PRNG) is used to produce mutipe sequences of pseudo-random numbers for parae simuations [2{8]. To avoid introducing unwanted correations, sequences of pseudo-random numbers must mimic the properties of the truy random sequences that are most important to the simuations. This is a we-studied topic for singe sequences of random variabes[1]. However, in the case of parae sequences ess is understood about the type of correations that coud ead to poor resuts in use. Theoretica tests can be deveoped which consider the properties of the whoe period of a specic generator [6]. This does not necessariy predict the behavior of subsets, i.e. sampes of smaer size than the whoe period, that are used Preprint submitted to Esevier Preprint 27 January 2000