Marsaglia G. and Zaman A. (1991) `A new class of random number generators', The Annals of Applied Probability, 1:462--480.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2) Typically the RNG is a member of a family of similar generators with di erent parameters and one hopes that parameters and seeds may be easily chosen so as to guarantee properties (1) 2) 3) and (4) There is no known family of RNG with all four properties (see, for example, M1] 1.2. In [MZ], Marsaglia and Zaman showed that their add with carry (AWC) generators satisfy condition (1) By giving up on (4) and using an appropriate base b, they achieve good distribution properties of d tuples for values d which are less than the lag. It has been shown [TE] that these generators fail ....

....(2.1.2) and (2.1.3) might be optimized by appropriate choice of coecients a i : In the next few paragraphs we show that these generators have many desirable properties. The proofs for most of these facts follow from standard techniques which may be distilled from the literature ( C1] C2] [MZ], TE] However, short and ecient proofs may be obtained by an algebraic technique, which is a simple but not entirely obvious modi cation of the method of [KG3] so we include them at the end for completeness. 2.2. Throughout the rest of this section we x a modulus b and consider the MWC ....

G. Marsaglia and A. Zaman, A new class of random number generators. Ann. Appl. Probabl. 1 (1991), pp. 462-480.

....# sequences may be generated using feedback with carry shift registers as described in [11, 12] where their role in stream ciphers was investigated. See also [5] and [14] This method of generating # sequences (and their mod p generalizations) was discovered independently by Marsaglia and Zaman [16] in special cases and by Couture and L Ecuyer [4] in general, who proposed using them as pseudo random number generators for Monte Carlo simulations. These # sequences exhibit important randomness properties. In [1] it was shown that they have perfect distribution properties: for any d log 2 (q) ....

G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Applied Probability. 1 (1991) pp. 462-480. 14

....stream cipher known as the summation combiner [14] It was further shown that various desirable statistical properties hold for FCSR sequences (see below) in most cases paralleling properties of LFSR sequences. FCSR sequences were discovered independently in a slightly di#erent context by [12] [13] who showed they were useful as (pseudo ) random number generators for simulations. The simplest p ary FCSR sequences may be described by a i = Ap i (modN) modp) # Z (p) 1) where p and N are prime numbers for which p is a primitive root (cf 9) modulo N,whereA # Z (N)andwhere (modN) modp) ....

G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Applied Probability 1 (1991) pp. 462-480.

....a power of 2) Typically the RNG is a member of a family ofsimilar generrxI withdi#erq tparU xIEU and one hopes that parKq qxI and seeds may be easily chosen so as toguarF tee pr er E (1) 2) 3) and (4) Ther is no known family of RNG with all four pr er KS (see,for example, M1] 1.2. In [MZ], Mar aglia and Zaman showed that their add with carc (AWC) gener ator satisfy condition (1) By giving up on (4) and using an appr FxIE base b, they achieve good distrxSKEKx pr er Kq of d tuplesfor values d which a r less than the lag. It has been shown [TE] that thesegenerxI U fail the ....

....computations (2.1.2) and (2.1.3) might be optimized byapprKxI UF choice of coe#cients a i . In the ne t fewparEBxI K we show that these gener ator have many desirq U prq erq s. The pr ofs for most of these facts follow fr standar techniques which may be distilled frs the literF xI ( C1] C2] [MZ], TE] Howevershor and e#cient pr ofs may be obtained by analgebr x technique, which is a simple but not entirUE obvious modification of the method of [KG3] so we include them at the endfor completeness. 2.2. ThrB SxI ther est of this section we fi a modulus b and consider the MWC gener ator ....

G. Marsaglia and A. Zaman, A new class of random number generators. Ann. Appl. Probabl. 1 (1991), pp. 462--480.

....when 26 2 m . One with modulus equal to 48 2 failed when 28 2 m . Two shift register generators [Golomb 1982] failed when 23 2 m . The additive generators generally passed but those with lags less than 40 failed. Similar results were obtained for the subtract with borrow generators [Marsaglia 1991]. The least significant bits of words generated by Super Duper, a combination generator, failed as early as 23 2 = m . The Mersenne Twister [Matsumoto 1998] passed alright, so did the KISS generator [Marsaglia 1999] 4 One major difficulty we encountered in our investigation was computing the ....

Marsaglia, G., and Zaman, A., 1991, A new class of random number generators, The Annals of Applied Probability 1, No. 3, 462-480.

....of the basic steps of the simulations: random calls, log calls, Discrete choices and insertion. To perform fast random choices, it is compulsory to have an efficient random generator. The generator we use is called FSU ULTRA and has been proposed by researchers at Florida State University 3 [9]. 3 It can be obtained by contacting Arif Zaman (arif stat.fsu.edu) or George Marsaglia (geo stat.fsu.edu) 5 The table below gives the time spent in each function call for our implementation on an Ultra Sparc 2 with 512 Megabytes of memory. In the table 1 dis(n) indicates a discrete choice on ....

G. Marsaglia and A. Zaman. A new class of random number generators. Annals of Applied Probability, 1:462-480, 1991.

....stream cipher known as the summation combiner [14] It was further shown that various desirable statistical properties hold for FCSR sequences (see below) in most cases paralleling properties of LFSR sequences. FCSR sequences were discovered independently in a slightly di erent context by [12] [13] who showed they were useful as (pseudo ) random number generators for simulations. The simplest p ary FCSR sequences may be described by a i = Ap i (modN) mod p) 2 Z= p) 1) where p and N are prime numbers for which p is a primitive root (cf x9) modulo N , where A 2 Z= N) and where (modN) ....

G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Applied Probability 1 (1991) pp. 462-480.

....has a primitive characteristic polynomial. However, there is a compromise to be made in terms of implementation speed. If the B j s do not have a special structure that can be exploited, the generator will be too slow. 3.9. ADD WITH CARRY AND SUBTRACT WITH BORROW GENERATORS Marsaglia and Zaman [62] propose two types of random number generators, called addwith carry (AWC) and subtract with borrow (SWB) which are slight modifications of the lagged Fibonacci generators with the and Gamma operations, respectively. The AWC generator is based on the recurrence x j = x j Gammar x j ....

....k = 5 and a 1 = 107374182; a 5 = 104480; a 2 = a 3 = a 4 = 0 Generators G1 and G2 are used in various software packages [10, 47] and recommended by some authors [79, 47] Fishman and Moore [33] recommend G3. G7 is proposed by Matsumoto and Kurita [66] while G8 is proposed by Marsaglia and Zaman [62] and further recommended by James [41] The results were that besides G5, G6, and G9, all other generators failed spectacularly at least one of the tests. Moreover, each of G1 to G4 failed spectacularly at least 6 tests out of 10. In more than half of the fail cases, the descriptive level ffi 2 ....

G. Marsaglia and A. Zaman, A New Class of Random Number Generators, The Annals of Applied Probability 1 (1991) 462--480.

....; un k ) n 0, lie in only two planes in the three dimensional unit cube (see L Ecuyer 1994a) so this author believes that they should be avoided. Slight variations of additive and subtractive generators, called add with carry (AWC) and subtractwith borrow (SWB) were proposed recently by Marsaglia and Zaman (1991). The modification is that a carry (or borrow) bit is maintained with the recurrence (6) This permits a period length of up to M Gamma 1, where M = m k Sigma m r Sigma 1 (depending on the variant) It is a tremendous increase. Unfortunately, as shown by Tezuka, L Ecuyer, and Couture (1994) ....

Marsaglia, G., and A. Zaman. 1991. A new class of random number generators. The Annals of Applied Probability, 1:462--480.

....understood. In this paper we introduce a new but very simple feedback architecture for shift register generation of pseudorandom sequences, which we call feedback with carry shift registers (FCSR) These are the shift register analogues of the Marsaglia Zaman pseudorandom number generators [36]. It turns out that sequences generated by an FCSR share many of the important properties enjoyed by linear recurrence sequences. However, the analysis of FCSR sequences involves a completely different mathematical toolkit: instead of arithmetic in finite fields, we use arithmetic in the 2 adic ....

....in x17.4 and x17.5 of [45] where various architectures for combining them with other shift register sequences are suggested. See also [10] to appear) We also wish to draw the reader s attention to the subsequent developments [22] and [27] The closely related paper of Marsaglia and Zaman [36] (cf [35] was recently brought to our attention: their random number generator may be described as an FCSR with two taps, whose cells contain integers modulo b (rather than modulo 2) Here, b is some large integer. Thus there is some overlap between their analysis and ours. In particular, ....

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G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Applied Probability. vol. 1, 1991 pp. 462-480.

.... noise ) and can also be analyzed with the Beyer and spectral tests. Other classes of combined generators are not (yet) well understood theoretically. See L Ecuyer (1990) and the references given there. The highly efficient add with carry and subtract withborrow generators, recently proposed by Marsaglia and Zaman (1991), also turn out to be equivalent to LCG s with huge moduli, with a truncated fractional expansion (see Tezuka and L Ecuyer 1992) So, again, those generators have an approximate lattice structure (in high dimensions) Tausworthe and GFSR generators also have a lattice structure. Tezuka and ....

....tests defined in Table 1, and applied them to the 8 generators described in Table 2. The generators G1 to G3 are recommended respectively by Bratley, Fox, and Schrage (1987) Law and Kelton (1991) and Fishman and Moore (1986) G7 is proposed by Matsumoto and Kurita (1992) while G8 is proposed by Marsaglia and Zaman (1991) and further recommended by James (1990) Table 3 gives the descriptive level ffi 2 for each test generator pair. For each generator, the different tests were Table 3: Descriptive Levels ffi 2 G1 G2 G3 G4 G5 G6 G7 G8 T1 .3072 .7752 .2208 .5122 .1317 .0051 .1719 .0038 T2 .4650 .9565 .9763 .0750 ....

Marsaglia, G. and A. Zaman. 1991. A New Class of Random Number Generators. The Annals of Applied Probability , 1, 3:462--480.

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Marsaglia G. and Zaman A. (1991) `A new class of random number generators', The Annals of Applied Probability, 1:462--480.

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Marsaglia, G., and Zaman, A., 1991, A new class of random number generators, The Annals of Applied Probability 1, No. 3, 462-480.

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G. Marsaglia and A. Zaman, "A new class of random number generators", The Annals of Applied Probability 1 (1991), 462-480. 15

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G. Marsaglia and A. Zaman, A new class of random number generators. The Annals of Applied Probability, 1:462--480, 1991.

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G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Appl. Prob. 1 (1991) pp. 462-480.

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G. Marsaglia and A. Zaman, A new class of random number generators, Ann. of Appl. Prob. 1 (1991), 462-480.

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G. Marsaglia and A. Zaman, "A new class of random number generators", The Annals of Applied Probability 1 (1991), 462-480.

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G. Marsaglia and A. Zaman, "A new class of random number generators", The Annals of Applied Probability 1 (1991), 462-480.

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G. Marsaglia and A. Zaman, A new class of random number generators, Ann. Appl. Probab., vol. 1, 1991, pp. 462--480.

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G. Marsaglia and A. Zaman, A new class of random number generators. The Annals of Applied Probability, 1:462--480, 1991.

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G. Marsaglia and A. Zaman, A new class of random number generators, The Annals of Applied Probability, 1 (1991), pp. 462--480.

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Marsaglia, G. and A. Zaman (1991). A new class of random number generators. The Annals of Applied Probability, Vol. 1, pp. 462--480.

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G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Appl. Probab., 1(1991), pp. 462480.

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G. Marsaglia and A. Zaman, A new class of random number generators, Annals of Appl. Prob. 1 (1991) pp. 462-480.

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