# Does It Beat the Minimal Standard?

Suppose that you've written (or just seen someone else announce) a brand new PRNG. Cool! It's nice to have new things. But here's the question you should ask yourself; “Is it better than a reasonable ‘minimal standard’? Does it beat methods devised more than sixty years ago?”

### Wisdom from the Past

Back in 1987, more than thirty years ago now, Steve Park and Keith Miller wrote the paper Random Number Generators: Good Ones Are Hard to Find, which lamented the poor quality of many of the PRNGs in use at the time and proposed a bare mininum for a “good enough” PRNG (the paper appeared in CACM early the following year). In that paper, they wrote

In retrospect it is evident that a generally satisfactory algorithm for random number generation was proposed by D. H. Lehmer 36 years ago [26]. This parametric multiplicative linear congruential algorithm has withstood the test of time. It can be implemented efficiently [27, 31, 37, 41], numerous empirical tests of the randomness of its output have been published [8, 15, 27, 28, 37], and its important theoretical properties have been analyzed [9, 14, 18, 30]. The conclusion to be drawn from all this research is now clear. Although Lehmer’s algorithm has some statistical defects, if the algorithm parameters are chosen properly and if the software implementation of the algorithm is correct, the resulting generator can produce a virtually infinite sequence of numbers that will satisfy almost any statistical test of randomness. In other words, with properly chosen parameters, Lehmer’s algorithm, correctly implemented, represents a good minimal standard generator against which all other random number generators can—and should—be judged.

Thirty years later, some specifics may have changed (i.e., the necessary parameters for what we might consider a reasonable LCG), but the fundamental idea—that any proponent of a new PRNG should be asked to explain how their technique is better than an appropriate “minimal standard” LCG—remains as relevant today is it was then.

If you can't show that your PRNG technique beats an idea from over sixty years ago, you haven't proven its worth.

### Today's 32-bit Minimal Standard: A 96-bit Truncated LCG or MCG

The code below represents a reasonable minimal-standard MCG:

``````uint96_t state = 1;   // can be seeded to any odd number

uint32_t next()
{
state *= 0xdc87976860b11728995deb95;
return state >> 64;
}
``````

And this version is a reasonable minimal-standard LCG:

``````uint96_t state;   // can be seeded to any number

uint32_t next()
{
state *= MULTIPLIER;
state += MULTIPLIER;
// Any odd odd number will do, the multiplier is odd and
// conveniently to hand
return state >> 64;
}
``````

What we know about these two PRNGs is that

• The code is incredibly simple (although it does presume the existence of a 96-bit (or larger) integer type).
• On 64-bit machines, the necessary multiplication is not difficult to perform. In fact, on modern x86 machines, the entire generation process can take place in less than a nanosecond.
• The generators are known to have long periods and no short cycles. The LCG has period 296, which is maximal, and no bad initializations—any initial seed is okay. The MCG provides two separate cycles of size 262 and only requires that the seed be odd.
• The underlying theory ensures that the generators are uniform (each output occurs the same number of times).
• Because the output function is truncation, each output occurs a vast number of times (262 and 264, respectively), meaning that producing one output does not significantly reduce the chance of it recurring.
• The underlying theory is very well understood. The multiplicative constants were chosen to have excellent “spectral test” values (for the MCG, normalized LLL-spectral test values are M8 = 0.70279, M16 = 0.65345, M24 = 0.58118 and for the LCG, M8 = 0.73201, M16 = 0.64438, M24 = 0.53151, as computed by software by Karl Entacher).
• The output passes stringent statistical tests, such as PractRand up to 32 TB and TestU01's BigCrush test.

In fact, not only does this 96-bit PRNG pass statistical tests, it passes them well. Testing with PractRand, we can plot the failure point for different sized 32-bit generators as follows:

This graph shows us that we leave clear-fail territory at 78 bits for LCGs and 80 bits for MCGs. Thus a 96-bit LCG or MCG has considerable headroom available to allow it to pass larger statistical tests.

Extrapolating the line from this graph shows that we'll get to the clear failure point for a 96-bit MCG at 128 petabytes of data, and for the LCG at the half exabyte point. If it takes PractRand a week of testing to test 32 TB, half an exabyte will require more than three hundred years of testing to detect statistical flaws.

### Today's 64-bit Minimal Standard: A 128-bit Truncated LCG or MCG

Similarly, if you want 64-bit output, a 128-bit truncated LCG or MCG represents a reasonable minimal standard. In fact, on 64-bit hardware, these generators are no more costly to compute that the 32-bit output versions given above.

``````uint128_t state = 1;   // can be seeded to any odd number

uint64_t next()
{
state *= 0x0fc94e3bf4e9ab32866458cd56f5e605;
// Spectral test: M8 = 0.71005, M16 = 0.66094, M24 = 0.61455
return state >> 64;
}
``````
``````uint128_t state;   // can be seeded to any number

uint64_t next()
{
const uint128_t MULTIPLIER = 0x2d99787926d46932a4c1f32680f70c55;
state *= MULTIPLIER;
// Spectral test: M8 = 0.70420, M16 = 0.65412, M24 = 0.60209
state += MULTIPLIER;
// Any odd odd number will do, the multiplier is odd and
// conveniently to hand
return state >> 64;
}
``````

Essentially the same observations we made for the 32-bit generators apply here as well. Once again, we can plot the failure point for different sized 64-bit generators as follows:

This graph shows us that we leave clear-fail territory at 109 bits for LCGs and 111 bits for MCGs. Thus a 128-bit LCG or MCG has considerable headroom available to allow it to pass larger statistical tests.

As before, we can extrapolate the line from this graph, this time finding that we reach the clear failure point for a 128-bit MCG at 256 petabytes of data, and for the LCG at the 1 exabyte point. If it takes PractRand a week of testing to test 32 TB, an exabyte will require more than six hundred years of testing to detect statistical flaws.

### Conclusion

Linear congruential generators are frequently cited as being a poor choice for random-number generation, and at small sizes that remains true, but as we have seen, at larger sizes these generators do very well in statistical tests. Forty years ago, these larger versions might have been unwieldy, but that has not been the case for a long time. Today modern 64-bit architectures have little difficulty performing 128-bit math quickly, and since 2016, even a \$35 Raspberry Pi has a 64-bit quad-core CPU. And older 32-bit machines can perform 96-bit math acceptably fast—it's worth realizing that the longstanding `drand48` generator present in all Unix systems was designed in the age of 16-bit machines and can be trivially adapted to use 32-bit data, creating `drand96`.

Any new PRNG with 64-bit output needs to answer the question, “How is it better than the minimal standard, a 128-bit LCG?” and likewise any PRNG with 32-bit output needs to answer the question “How is it better than the minimal standard, a 96-bit LCG?”. It is not enough just to be fairly fast, because these minimal standards are fast. It is not enough to pass stringent statistical tests, because these minimal standard generators pass those same tests, and pass them easily.

Moreover, the graphs we have seen show that it is possible to take a scientific approach to testing PRNGs. The trend lines we have seen allow us to make good predictions about tests that that might be prohibitively expensive to run, so we see another useful tip for designers of PRNGs. Don't just provide a single data point, provide many and do some science.

### Appendix

To allow others to recreate the above graphs, below are multiplicative constants for MCGs and LCGs of different sizes. The constants were found by using a random search to find constants that had good normalized LLL-spectral test results; each constant is intended to be “very good” rather than truly outstanding. Each table shows the M8 result for the normalized LLL-spectral test. The M16 values are all above 0.625.

#### MCG Constants

Bits Multiplicative Constant Spectral Test
32 2739110765 0.716768631223
33 6416791565 0.741609517211
34 2110734621 0.730357331437
35 9409920061 0.727139444967
36 8124055821 0.726338729537
37 85397059285 0.745062524229
38 268237653253 0.742041038480
39 220322046269 0.729472249491
40 533248573853 0.741393982264
41 1159580999389 0.721303065856
42 4207420190797 0.727290948336
43 1162000279733 0.717554748268
44 14291355528493 0.721569546451
45 32316819487421 0.722852594721
46 27424978670797 0.723975492235
47 112357400097837 0.725660187612
48 203797216008469 0.728474675519
49 14838598453045 0.736512565883
50 406663326808437 0.755377478054
51 942227593642253 0.727499556775
52 276168391955373 0.738214029774
53 7207664573232381 0.747016695380
54 1678565460045845 0.727249278946
55 15485172535814365 0.710832966834
56 65157942103163701 0.729099254104
57 41826564414174485 0.737033978607
58 265278118812494821 0.720996827858
59 435341156619262861 0.713519610623
60 222915332783142117 0.737161497144
61 2042033352170702485 0.730916689500
62 3852213006194032149 0.717693049282
63 2681400331702771381 0.730222321935
64 14647171131086947261 0.738375341778
65 23586512187791901165 0.749719574465
66 4021396209304885645 0.737035361383
67 71338817522127457661 0.749586645884
68 23537159970439144981 0.741383430613
69 574409946355343590093 0.728499816481
70 847531188765324779717 0.735818093856
71 1540150647070481776589 0.736156422148
72 3629512676251493547269 0.733279877031
73 5691505500811439494453 0.731779832097
74 2687589216551525493141 0.717594710954
75 5944024556493776365117 0.731218718355
76 2540004110552927093773 0.741476230780
77 9614823434306722085205 0.716421804008
78 256445958763511920394925 0.733008689483
79 601712692362582751632037 0.744911691987
80 39606201209357049487589 0.725258169835
81 1290131957331746079373301 0.734219557372
82 3977675389778782791388165 0.756010134150
83 7621526099150505215659221 0.725543053961
84 9877239282961482572937501 0.725928715061
85 25236437150407555247393029 0.749180805625
86 8171031101133229061033037 0.740169374705
87 49362540671389698004118317 0.724803286548
88 2577223908231812491115693 0.719764670518
89 249522469266726746228387597 0.749826167287
90 839446671699896991292676421 0.727750604637
91 958856983424445520621212381 0.720868271564
92 936151086994606743394269365 0.717081973076
93 4398551187470736665765410061 0.737375655009
94 16575717516719574569518905037 0.733588588969
95 16924451041812820016938051413 0.738072389915
96 63684207872218969504639112949 0.722889364342
97 123612427885422382889677805949 0.720047707563
98 13564477857739332917956337261 0.723224899486
99 567192134585768783122181505413 0.727178102380
100 605302915592620321688005792373 0.745754947466
101 1940675479040113117551892359605 0.739161068408
102 3658916473478134582167183652629 0.753704464419
103 1571359329199485794141792737933 0.730697664731
104 16904445017912738760767735186589 0.742586816074
105 22485855570768166567205990626413 0.718485657687
106 74313279853458239997575461406493 0.720008226330
107 69789568092498811246130593789565 0.731015334466
108 199185148331384386512624391509277 0.753423145342
109 445995822870053631298584088971085 0.737911420945
110 522038943738820526113376981570765 0.728651914451
111 1790166223466728928894031445659221 0.745853280704
112 2376232594419346738793516431034285 0.739509229331
113 1739365516208071203047989829164373 0.752781474861
114 13159937790046761463548017391558589 0.735305543510
115 26771634451446385007601761491488901 0.724847025658
116 14592594772439033291729959800277197 0.744686357317
117 60966792259065248069366924264146173 0.741555898957
118 68836090842512112509636367404547397 0.746365192374
119 524186352600080584049930567274522237 0.739180239317
120 610775152132035755583590899855195013 0.747582831613
121 900387334203823311229546540549529981 0.745007443886
122 2306725953348125746190075336418508061 0.727909253640
123 1800413948600614978725384679020807301 0.723551474912
124 18484242130631853726820903993747978157 0.718440060620
125 26907765095583321268647989660234740333 0.722561248710
126 25076288795215231685469951584245457885 0.724645916794
127 34218442865726806866117399643452229733 0.719956739979
128 63788880824840432877499191278319602189 0.731606096275

#### LCG Constants

Bits Multiplicative Constant Spectral Test
32 1019135901 0.726561516888
33 6527674237 0.718126733013
34 15029734005 0.723277276010
35 3660796997 0.742645750618
36 29510867981 0.719429027076
37 63514268605 0.732551422652
38 116624537293 0.724558743348
39 194195883829 0.722415400434
40 568512975829 0.730711283118
41 1322742409629 0.738030287624
42 126882788757 0.728710926727
43 1391190336909 0.729854488924
44 9272727027237 0.722758310108
45 12790945375773 0.737588630205
46 15105438078773 0.739295363228
47 50521972618661 0.723112105229
48 249419361368573 0.730241902102
49 404273197363917 0.750413967945
50 178519384041629 0.731476180651
51 435569983484149 0.734296368387
52 672969485740877 0.748370625927
53 1997786961644461 0.721800591403
54 7946058339844821 0.741940429724
55 34210420404670109 0.735305419562
56 49682230424721469 0.751246366335
57 41552818007412365 0.732767196482
58 208768124490119261 0.720192561204
59 534186703036545445 0.732258727961
60 718198937956463965 0.717132171609
61 1492761048249620805 0.732951920801
62 285762664057093061 0.720297424345
63 5562408657945804781 0.740322612872
64 9199940308585234877 0.740159647448
65 11160153871882754749 0.722374537852
66 30609332369975655877 0.750132565315
67 102504574710574106733 0.727072404636
68 289008808635447097333 0.737066005448
69 200965074862812859469 0.727446643227
70 995216606425775766885 0.742508065267
71 1383446250119146840917 0.735902660153
72 1277974755392284889725 0.745059476563
73 755621007724937721413 0.723731561436
74 1403601537669505748437 0.735159325664
75 34430510360273607538717 0.739570300200
76 56021341046004321357677 0.732060732087
77 57428549995224081070613 0.737418286180
78 161279981175413397505341 0.736420906775
79 601009572732792860513077 0.744752347985
80 812978456606438465717861 0.725653623399
81 2230578935970333198265389 0.726202750322
82 3110954275176992599611333 0.722803877503
83 2549771676202439997119837 0.728988412935
84 17241119348271108561932573 0.733198806677
85 26457731553295723861284917 0.751942767791
86 59090791729463201494191589 0.726268118857
87 15271881663998384886817965 0.734997793995
88 70622416032282250170082701 0.730360551193
89 426541151666756081629040421 0.732673387845
90 1033056376707427530835730733 0.730196493664
91 396024176885822028546435213 0.730113412997
92 2388182321528562871077577165 0.737154295148
93 8010225565907072943071535853 0.728882371264
94 13261206629595072908252835605 0.725329755925
95 16902422804113592611251823813 0.750861377225
96 61124247442928732736190063229 0.732014535166
97 27911782745385710563574645949 0.734864728131
98 29938634752950390286186539269 0.723675443983
99 26229225708957686851411456901 0.735866461121
100 819029723418697416441104865757 0.716956721057
101 2322146411595516589823065344981 0.741946609263
102 1784183314298914792221437953997 0.730507073217
103 3235104345222420208314737169685 0.726129037214
104 67188141926644620722638870253 0.747754151762
105 2436249846286644479688629522261 0.723418570653
106 1825670733282477966908844273549 0.760448328874
107 3246539704836717185167544760429 0.732216405411
108 303587630294941123502336685825389 0.719120482824
109 11392648997474423235502837719821 0.751921546570
110 964268383221670159557101452712557 0.734010450204
111 639338642199625092864608394898789 0.735970283015
112 1906368178963804336466096107548453 0.747352401943
113 6749730758631022595149452958693405 0.730041084931
114 16707654375351138070845852572447229 0.723933957671
115 17207620364045872019394706534426421 0.740743821760
116 15623495417764251607984624356818501 0.731135297999
117 131745201739913730912678565034201349 0.737765053219
118 34201110710269651177692545734465877 0.726921739220
119 144754161049240217053622458007212997 0.719557456180
120 1139027868931473050531506703519921533 0.723535881344
121 2612106425891462620739196525511584853 0.734994863425
122 2009517146465439999957517982454009829 0.744269096369
123 8342579080939882558302521411171977885 0.745363344384
124 17774964191369474761588846250681198813 0.726741230435
125 18418224551009769296825399045980769997 0.724759980253
126 56967995961931505441914624497778363077 0.735443492446
127 92155423410782672437200603327585209317 0.742100869722
128 199967246047888932297834045878657099405 0.723283410960