NOTE Most of the tests in DIEHARD return a p-value, which should be uniform on [0,1) if the input file contains truly independent random bits. Those p-values are obtained by p=F(X), where F is the assumed distribution of the sample random variable X---often normal. But that assumed F is often just an approximation, for which the fit will likely be worst in the tails. Thus you should not be surprised with occasion- al p-values near 0 or 1, such as .0012 or .9983. When a bit stream really FAILS BIG, you will get p`s of 0 or 1 to six or more places. By all means, do not, as a Statistician might, think that a p < .025 or p> .975 means that the RNG has "failed the test at the .05 level". Such p`s happen among the hundreds that DIEHARD produces, even with good RNGs. So keep in mind that "p happens" Enter the name of the file to be tested. This must be a form="unformatted",access="direct" binary file of about 10-12 million bytes. Enter file name: Which tests do you want to perform? For all tests, enter 17 1's: 11111111111111111: To choose, say, tests 1, 3, 7, 14 enter: 10100010000001000: HERE ARE YOUR CHOICES: 1 Birthday Spacings 2 GCD 3 Gorilla 4 Overlapping Permutations 5 Ranks of 31x31 and 32x32 matrices 6 Ranks of 6x8 Matrices 7 Monkey Tests on 20-bit Words 8 Monkey Tests OPSO,OQSO,DNA 9 Count the 1`s in a Stream of Bytes 10 Count the 1`s in Specific Bytes 11 Parking Lot Test 12 Minimum Distance Test 13 Random Spheres Test 14 The Sqeeze Test 15 Overlapping Sums Test 16 Runs Up and Down Test 17 The Craps Test Enter your choices, 1's yes, 0's no using 17 columns: 12345678901234567 |-------------------------------------------------------------| | This is the BIRTHDAY SPACINGS TEST | |Choose m birthdays in a "year" of n days. List the spacings | |between the birthdays. Let j be the number of values that | |occur more than once in that list, then j is asymptotically | |Poisson distributed with mean m^3/(4n). Experience shows n | |must be quite large, say n>=2^18, for comparing the results | |to the Poisson distribution with that mean. This test uses | |n=2^24 and m=2^10, so that the underlying distribution for j | |is taken to be Poisson with lambda=2^30/(2^26)=16. A sample | |of 200 j's is taken, and a chi-square goodness of fit test | |provides a p value. The first test uses bits 1-24 (counting | |from the left) from 32-bit integers in the specified file. | |The file is closed and reopened, then bits 2-25 of the same | |integers are used to provide birthdays, and so on to bits | |9-32. Each set of bits provides a p-value, and the nine p- | |values provide a sample for a KSTEST. | |------------------------------------------------------------ | RESULTS OF BIRTHDAY SPACINGS FOR urandom.bin (no_bdays=1024, no_days/yr=2^24, lambda=16.00, sample size=500) Bits used mean chisqr p-value 1 to 24 69.57 21905.9731 1.000000 2 to 25 68.94 21905.9731 1.000000 3 to 26 55.48 21905.9731 1.000000 4 to 27 52.94 21905.9731 1.000000 5 to 28 53.68 21905.9731 1.000000 6 to 29 54.15 21905.9731 1.000000 7 to 30 57.90 21905.9731 1.000000 8 to 31 63.21 21905.9731 1.000000 9 to 32 68.99 21905.9731 1.000000 Chisquare degrees of freedom: 17 --------------------------------------------------------------- p-value for KStest on those 9 p-values: 0.000000 |-------------------------------------------------------------| | This is the "tough" BIRTHDAY SPACINGS TEST | |Choose 4096 birthdays in a "year" of 2^32 days. Thus each | |birthday is a 32-bit integer and the test uses 2^12 of them, | |so that j, the number of duplicate spacings, is asympotically| |Poisson distributed with lambda=4 . Generators that pass the| |earlier tests for m=1024 and n=2^24 often fail this test, yet| |those that pass this test seem to pass the "weaker" test. | |Each set of 4096 birthdays provide a Poisson variate j, and | |500 such j's lead to a chisquare test to see if the result | |is consistent with the Poisson distribution with lambda=16. | |------------------------------------------------------------ | Tough bday spacings test for urandom.bin: 4096 birthdays, year=2^32 days Table of Expected vs. Observed counts: Duplicates 0 1 2 3 4 5 6 7 8 9 >=10 Expected 9.2 36.6 73.3 97.7 97.7 78.1 52.1 29.8 14.9 6.6 4.1 Observed 0 0 0 0 0 0 0 0 0 0 500 (O-E)^2/E 9.2 36.6 73.3 97.7 97.7 78.1 52.1 29.8 14.9 6.660487.7 Birthday Spacings: Sum(O-E)^2/E=60983.660, p= 1.000 |-----------------------------------------------------------| |This is the GCD TEST. Let the (32-bit) RNG produce two | |successive integers u,v. Use Euclids algorithm to find the| |gcd, say x, of u and v. Let k be the number of steps needed| |to get x. Then k is approximately binomial with p=.376 | |and n=50, while the distribution of x is very close to | | Pr(x=i)=c/i^2, with c=6/pi^2. The gcd test uses ten | |million such pairs u,v to see if the resulting frequencies | |of k's and x's are consistent with the above distributions.| |Congruential RNG's---even those with prime modulus---fail | |this test for the distribution of k, the number of steps, | |and often for the distribution of gcd values x as well. | |-----------------------------------------------------------| RESULTS OF GCD FOR urandom.bin Not enough random numbers for this test. Minimum is 20000000. The test is skipped. |-----------------------------------------------------------| |This is the GORILLA test, a strong version of the monkey | |tests that I developed in the 70's. It concerns strings | |formed from specified bits in 32-bit integers from the RNG.| |We specify the bit position to be studied, from 0 to 31, | |say bit 3. Then we generate 67,108,889 (2^26+25) numbers | |from the generator and form a string of 2^26+25 bits by | |taking bit 3 from each of those numbers. In that string of | |2^26+25 bits we count the number of 26-bit segments that | |do not appear. That count should be approximately normal | |with mean 24687971 and std. deviation 4170. This leads to | |a normal z-score and hence to a p-value. The test is | |applied for each bit position 0 (leftmost) to 31. | |(Some older tests use Fortran's 1-32 for most- to least- | |significant bits. Gorilla and newer tests use C's 0 to 31.)| |-----------------------------------------------------------| Gorilla test for 2^26 bits, positions 0 to 31 for urandom.bin: Note: lengthy test---for example, ~20 minutes for 850MHz PC Not enough random numbers for this test. Minimum is 67108889. The test is skipped. |-------------------------------------------------------------| | THE OVERLAPPING 5-PERMUTATION TEST | |This is the OPERM5 test. It looks at a sequence of ten mill-| |ion 32-bit random integers. Each set of five consecutive | |integers can be in one of 120 states, for the 5! possible or-| |derings of five numbers. Thus the 5th, 6th, 7th,...numbers | |each provide a state. As many thousands of state transitions | |are observed, cumulative counts are made of the number of | |occurences of each state. Then the quadratic form in the | |weak inverse of the 120x120 covariance matrix yields a test | |that the 120 cellcounts came from the specified (asymptotic) | |distribution with the specified means and 120x120 covariance.| |-------------------------------------------------------------| The OPERM5 test for 10 million (overlapping) 5-tuples for urandom.bin, p-values for 5 runs: Not enough random numbers for this test. Minimum is 50000025. The test is skipped. |-------------------------------------------------------------| |This is the BINARY RANK TEST for 31x31 matrices. The leftmost| |31 bits of 31 random integers from the test sequence are used| |to form a 31x31 binary matrix over the field {0,1}. The rank | |is determined. That rank can be from 0 to 31, but ranks< 28 | |are rare, and their counts are pooled with those for rank 28.| |Ranks are found for 40,000 such random matrices and a chisqu-| |are test is performed on counts for ranks 31,30,28 and <=28. | | (The 31x31 choice is based on the unjustified popularity of | | the proposed "industry standard" generator | | x(n) = 16807*x(n-1) mod 2^31-1, not a very good one.) | |-------------------------------------------------------------| Rank test for binary matrices (31x31) for urandom.bin RANK OBSERVED EXPECTED (O-E)^2/E SUM r<=28 37949 211.4 6736063.347 6736063.347 r= 29 1947 5134.0 1978.382 6738041.730 r= 30 102 23103.0 22899.498 6760941.227 r= 31 2 11551.5 11547.524 6772488.752 chi-square = 6772488.752 with df = 3; p-value = 1.000 -------------------------------------------------------------- |-------------------------------------------------------------| |This is the BINARY RANK TEST for 32x32 matrices. A random 32x| |32 binary matrix is formed, each row a 32-bit random integer.| |The rank is determined. That rank can be from 0 to 32. Ranks | |less than 29 are rare, and their counts are pooled with those| |for rank 29. Ranks are found for 40,000 such random matrices| |and a chisquare test is performed on counts for ranks 32,31,| |30 and <=29. | |-------------------------------------------------------------| Rank test for binary matrices (32x32) for urandom.bin RANK OBSERVED EXPECTED (O-E)^2/E SUM r<=29 37843 211.4 6698275.027 6698275.027 r= 30 2025 5134.0 1882.728 6700157.755 r= 31 129 23103.0 22845.768 6723003.523 r= 32 3 11551.5 11545.525 6734549.048 chi-square = 6734549.048 with df = 3; p-value = 1.000 -------------------------------------------------------------- |-------------------------------------------------------------| |This is the BINARY RANK TEST for 6x8 matrices. From each of | |six random 32-bit integers from the generator under test, a | |specified byte is chosen, and the resulting six bytes form a | |6x8 binary matrix whose rank is determined. That rank can be| |from 0 to 6, but ranks 0,1,2,3 are rare; their counts are | |pooled with those for rank 4. Ranks are found for 100,000 | |random matrices, and a chi-square test is performed on | |counts for ranks <=4, 5 and 6. | |-------------------------------------------------------------| Rank test for binary matrices (6x8) for urandom.bin b-rank test for bits 1 to 8, p=1.00000 b-rank test for bits 2 to 9, p=1.00000 b-rank test for bits 3 to 10, p=1.00000 b-rank test for bits 4 to 11, p=1.00000 b-rank test for bits 5 to 12, p=1.00000 b-rank test for bits 6 to 13, p=1.00000 b-rank test for bits 7 to 14, p=1.00000 b-rank test for bits 8 to 15, p=1.00000 b-rank test for bits 9 to 16, p=1.00000 b-rank test for bits 10 to 17, p=1.00000 b-rank test for bits 11 to 18, p=1.00000 b-rank test for bits 12 to 19, p=1.00000 b-rank test for bits 13 to 20, p=1.00000 b-rank test for bits 14 to 21, p=1.00000 b-rank test for bits 15 to 22, p=1.00000 b-rank test for bits 16 to 23, p=1.00000 b-rank test for bits 17 to 24, p=1.00000 b-rank test for bits 18 to 25, p=1.00000 b-rank test for bits 19 to 26, p=1.00000 b-rank test for bits 20 to 27, p=1.00000 b-rank test for bits 21 to 28, p=1.00000 b-rank test for bits 22 to 29, p=1.00000 b-rank test for bits 23 to 30, p=1.00000 b-rank test for bits 24 to 31, p=1.00000 b-rank test for bits 25 to 32, p=1.00000 TEST SUMMARY, 25 tests, each on 100,000 random 6x8 matrices The above should be 25 uniform [0,1] random variables: The KS test for those 25 supposed UNI's yields p = 0.000000 |-------------------------------------------------------------| | THE BITSTREAM TEST | |The file under test is viewed as a stream of bits. Call them | |b1,b2,... . Consider an alphabet with two "letters", 0 and 1| |and think of the stream of bits as a succession of 20-letter | |"words", overlapping. Thus the first word is b1b2...b20, the| |second is b2b3...b21, and so on. The bitstream test counts | |the number of missing 20-letter (20-bit) words in a string of| |2^21 overlapping 20-letter words. There are 2^20 possible 20| |letter words. For a truly random string of 2^21+19 bits, the| |number of missing words j should be (very close to) normally | |distributed with mean 141,909 and sigma 428. Thus | | (j-141909)/428 should be a standard normal variate (z score)| |that leads to a uniform [0,1) p value. The test is repeated | |twenty times. | |-------------------------------------------------------------| THE OVERLAPPING 20-TUPLES BITSTREAM TEST for urandom.bin (20 bits/word, 2097152 words 20 bitstreams. No. missing words should average 141909.33 with sigma=428.00.) ---------------------------------------------------------------- BITSTREAM test results. Bitstream No. missing words z-score p-value 1 331719 443.48 1.000000 2 331066 441.95 1.000000 3 331775 443.61 1.000000 4 332808 446.02 1.000000 5 331337 442.59 1.000000 6 332472 445.24 1.000000 7 330704 441.11 1.000000 8 331201 442.27 1.000000 9 330214 439.96 1.000000 10 330820 441.38 1.000000 11 332314 444.87 1.000000 12 331405 442.75 1.000000 13 331855 443.80 1.000000 14 332006 444.15 1.000000 15 331248 442.38 1.000000 16 331019 441.85 1.000000 17 331285 442.47 1.000000 18 330756 441.23 1.000000 19 331906 443.92 1.000000 20 331628 443.27 1.000000 ---------------------------------------------------------------- |-------------------------------------------------------------| | OPSO means Overlapping-Pairs-Sparse-Occupancy | |The OPSO test considers 2-letter words from an alphabet of | |1024 letters. Each letter is determined by a specified ten | |bits from a 32-bit integer in the sequence to be tested. OPSO| |generates 2^21 (overlapping) 2-letter words (from 2^21+1 | |"keystrokes") and counts the number of missing words---that | |is,2-letter words which do not appear in the entire sequence.| |That count should be very close to normally distributed with | |mean 141,909, sigma 290. Thus (missingwrds-141909)/290 should| |be a standard normal variable. The OPSO test takes 32 bits at| |a time from the test file and uses a designated set of ten | |consecutive bits. It then restarts the file for the next de- | |signated 10 bits, and so on. | |------------------------------------------------------------ | OPSO test for urandom.bin Bits used No. missing words z-score p-value 23 to 32 782551 2209.1092 1.000000 22 to 31 779480 2198.5196 1.000000 21 to 30 776242 2187.3540 1.000000 20 to 29 773756 2178.7816 1.000000 19 to 28 772183 2173.3575 1.000000 18 to 27 771347 2170.4747 1.000000 17 to 26 967704 2847.5678 1.000000 16 to 25 966798 2844.4437 1.000000 15 to 24 782561 2209.1437 1.000000 14 to 23 779399 2198.2402 1.000000 13 to 22 776141 2187.0058 1.000000 12 to 21 773540 2178.0368 1.000000 11 to 20 771883 2172.3230 1.000000 10 to 19 770980 2169.2092 1.000000 9 to 18 967357 2846.3713 1.000000 8 to 17 966452 2843.2506 1.000000 7 to 16 782542 2209.0782 1.000000 6 to 15 779383 2198.1851 1.000000 5 to 14 776110 2186.8989 1.000000 4 to 13 773493 2177.8747 1.000000 3 to 12 771796 2172.0230 1.000000 2 to 11 770904 2168.9471 1.000000 1 to 10 967323 2846.2540 1.000000 ----------------------------------------------------------------- |------------------------------------------------------------ | | OQSO means Overlapping-Quadruples-Sparse-Occupancy | | The test OQSO is similar, except that it considers 4-letter| |words from an alphabet of 32 letters, each letter determined | |by a designated string of 5 consecutive bits from the test | |file, elements of which are assumed 32-bit random integers. | |The mean number of missing words in a sequence of 2^21 four- | |letter words, (2^21+3 "keystrokes"), is again 141909, with | |sigma = 295. The mean is based on theory; sigma comes from | |extensive simulation. | |------------------------------------------------------------ | OQSO test for urandom.bin Bits used No. missing words z-score p-value 28 to 32 141740 -0.5740 0.282984 27 to 31 141998 0.3006 0.618131 26 to 30 142039 0.4396 0.669872 25 to 29 968837 2803.1446 1.000000 24 to 28 962445 2781.4768 1.000000 23 to 27 957586 2765.0057 1.000000 22 to 26 954482 2754.4836 1.000000 21 to 25 952775 2748.6972 1.000000 20 to 24 142034 0.4226 0.663710 19 to 23 141905 -0.0147 0.494145 18 to 22 141003 -3.0723 0.001062 17 to 21 968719 2802.7446 1.000000 16 to 20 962110 2780.3413 1.000000 15 to 19 957024 2763.1006 1.000000 14 to 18 953898 2752.5040 1.000000 13 to 17 952057 2746.2633 1.000000 12 to 16 141525 -1.3028 0.096319 11 to 15 141978 0.2328 0.592034 10 to 14 141968 0.1989 0.578822 9 to 13 968691 2802.6497 1.000000 8 to 12 962051 2780.1413 1.000000 7 to 11 956952 2762.8565 1.000000 6 to 10 953586 2751.4463 1.000000 5 to 9 951805 2745.4091 1.000000 4 to 8 141453 -1.5469 0.060946 3 to 7 142210 1.0192 0.845951 2 to 6 142143 0.7921 0.785849 1 to 5 968750 2802.8497 1.000000 ----------------------------------------------------------------- |------------------------------------------------------------ | | The DNA test considers an alphabet of 4 letters: C,G,A,T,| |determined by two designated bits in the sequence of random | |integers being tested. It considers 10-letter words, so that| |as in OPSO and OQSO, there are 2^20 possible words, and the | |mean number of missing words from a string of 2^21 (over- | |lapping) 10-letter words (2^21+9 "keystrokes") is 141909. | |The standard deviation sigma=339 was determined as for OQSO | |by simulation. (Sigma for OPSO, 290, is the true value (to | |three places), not determined by simulation. | |------------------------------------------------------------ | DNA test for urandom.bin Bits used No. missing words z-score p-value 31 to 32 141644 -0.7827 0.216906 30 to 31 141784 -0.3697 0.355801 29 to 30 142333 1.2498 0.894307 28 to 29 141250 -1.9449 0.025892 27 to 28 141874 -0.1042 0.458498 26 to 27 141503 -1.1986 0.115339 25 to 26 1041026 2652.2616 1.000000 24 to 25 1035843 2636.9725 1.000000 23 to 24 142136 0.6686 0.748138 22 to 23 141565 -1.0157 0.154881 21 to 22 142338 1.2645 0.896977 20 to 21 141843 -0.1957 0.422437 19 to 20 142371 1.3619 0.913379 18 to 19 142398 1.4415 0.925279 17 to 18 1041095 2652.4651 1.000000 16 to 17 1035922 2637.2055 1.000000 15 to 16 142102 0.5683 0.715101 14 to 15 141813 -0.2842 0.388144 13 to 14 141582 -0.9656 0.167128 12 to 13 142330 1.2409 0.892681 11 to 12 142003 0.2763 0.608846 10 to 11 141709 -0.5909 0.277279 9 to 10 1041087 2652.4415 1.000000 8 to 9 1035903 2637.1495 1.000000 7 to 8 141427 -1.4228 0.077397 6 to 7 142447 1.5860 0.943636 5 to 6 141912 0.0079 0.503142 4 to 5 142093 0.5418 0.706022 3 to 4 141290 -1.8269 0.033855 2 to 3 141497 -1.2163 0.111933 1 to 2 1041176 2652.7040 1.000000 ----------------------------------------------------------------- |-------------------------------------------------------------| | This is the COUNT-THE-1's TEST on a stream of bytes. | |Consider the file under test as a stream of bytes (four per | |32 bit integer). Each byte can contain from 0 to 8 1's | |with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let | |the stream of bytes provide a string of overlapping 5-letter| |words, each "letter" taking values A,B,C,D,E. The letters are| |determined by the number of 1's in a byte: 0,1,or 2 yield A | |3 yields B, 4 yields C, 5 yields D and 6,7 or 8 yield E. Thus| |we have a monkey at a typewriter hitting five keys with vari-| |ous probabilities (37,56,70,56,37 over 256). There are 5^5 | |possible 5-letter words, and from a string of 256,000 (over- | |lapping) 5-letter words, counts are made on the frequencies | |for each word. The quadratic form in the weak inverse of | |the covariance matrix of the cell counts provides a chisquare| |test: Q5-Q4, the difference of the naive Pearson sums of | |(OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts. | |-------------------------------------------------------------| Test result COUNT-THE-1's in bytes for urandom.bin (Degrees of freedom: 5^4-5^3=2500; sample size: 2560000) chisquare z-score p-value 489020.14 6880.434 1.000000 |-------------------------------------------------------------| | This is the COUNT-THE-1's TEST for specific bytes. | |Consider the file under test as a stream of 32-bit integers. | |From each integer, a specific byte is chosen , say the left- | |most: bits 1 to 8. Each byte can contain from 0 to 8 1's, | |with probabilitie 1,8,28,56,70,56,28,8,1 over 256. Now let | |the specified bytes from successive integers provide a string| |of (overlapping) 5-letter words, each "letter" taking values| |A,B,C,D,E. The letters are determined by the number of 1's, | |in that byte: 0,1,or 2 ---> A, 3 ---> B, 4 ---> C, 5 ---> D, | |and 6,7 or 8 ---> E. Thus we have a monkey at a typewriter | |hitting five keys with with various probabilities: 37,56,70, | |56,37 over 256. There are 5^5 possible 5-letter words, and | |from a string of 256,000 (overlapping) 5-letter words, counts| |are made on the frequencies for each word. The quadratic form| |in the weak inverse of the covariance matrix of the cell | |counts provides a chisquare test: Q5-Q4, the difference of | |the naive Pearson sums of (OBS-EXP)^2/EXP on counts for 5- | |and 4-letter cell counts. | |-------------------------------------------------------------| Test results for specific bytes for urandom.bin (Degrees of freedom: 5^4-5^3=2500; sample size: 256000) bits used chisquare z-score p-value 1 to 8 50139.38 673.723 1.000000 2 to 9 49012.64 657.788 1.000000 3 to 10 50039.96 672.317 1.000000 4 to 11 50477.42 678.503 1.000000 5 to 12 51328.08 690.533 1.000000 6 to 13 50942.96 685.087 1.000000 7 to 14 51159.82 688.154 1.000000 8 to 15 51461.14 692.415 1.000000 9 to 16 51319.51 690.412 1.000000 10 to 17 50256.26 675.376 1.000000 11 to 18 50677.69 681.335 1.000000 12 to 19 51329.76 690.557 1.000000 13 to 20 50756.97 682.457 1.000000 14 to 21 51173.42 688.346 1.000000 15 to 22 51938.16 699.161 1.000000 16 to 23 52790.63 711.217 1.000000 17 to 24 52278.46 703.974 1.000000 18 to 25 51208.38 688.840 1.000000 19 to 26 50328.49 676.397 1.000000 20 to 27 50271.32 675.588 1.000000 21 to 28 50807.55 683.172 1.000000 22 to 29 51241.84 689.314 1.000000 23 to 30 51098.71 687.290 1.000000 24 to 31 51362.88 691.025 1.000000 25 to 32 51231.67 689.170 1.000000 |-------------------------------------------------------------| | THIS IS A PARKING LOT TEST | |In a square of side 100, randomly "park" a car---a circle of | |radius 1. Then try to park a 2nd, a 3rd, and so on, each | |time parking "by ear". That is, if an attempt to park a car | |causes a crash with one already parked, try again at a new | |random location. (To avoid path problems, consider parking | |helicopters rather than cars.) Each attempt leads to either| |a crash or a success, the latter followed by an increment to | |the list of cars already parked. If we plot n: the number of | |attempts, versus k: the number successfully parked, we get a | |curve that should be similar to those provided by a perfect | |random number generator. Theory for the behavior of such a | |random curve seems beyond reach, and as graphics displays are| |not available for this battery of tests, a simple characteriz| |ation of the random experiment is used: k, the number of cars| |successfully parked after n=12,000 attempts. Simulation shows| |that k should average 3523 with sigma 21.9 and be approximate| |to normally distributed. Thus (k-3523)/21.9 should serve as | |a standard normal variable, which, converted to a p uniform | |in [0,1), provides input to a KSTEST based on a sample of 10.| |-------------------------------------------------------------| CDPARK for urandom.bin: result of 10 tests (Of 12000 tries, the average no. of successes should be 3523.0 with sigma=21.9) No. succeses z-score p-value 1178 -107.0776 0.000000 1186 -106.7123 0.000000 1173 -107.3059 0.000000 1187 -106.6667 0.000000 1197 -106.2100 0.000000 1178 -107.0776 0.000000 1195 -106.3014 0.000000 1186 -106.7123 0.000000 1163 -107.7626 0.000000 1184 -106.8037 0.000000 Square side=100, avg. no. parked=1182.70 sample std.=9.61 p-value of the KSTEST for those 10 p-values: 0.000000 |-------------------------------------------------------------| | THE MINIMUM DISTANCE TEST | |It does this ten times: choose n=8000 random points in a | |square of side 10000. Find d, the minimum distance between | |the (n^2-n)/2 pairs of points. If the points are truly inde-| |pendent uniform, then d^2, the square of the minimum distance| |should be (very close to) exponentially distributed with mean| |.995 . Thus 1-exp(-d^2/.995) should provide a p-value and a| |KSTEST on the resulting 10 values serves as a test of uni- | |formity for those samples of 8000 random points in a square. | |-------------------------------------------------------------| Results for the MINIMUM DISTANCE test for urandom.bin 0.0803,0.0480,0.1170,0.0949,0.0512,0.0022,0.0153,0.0013,0.1701,0.1019, The KS test for those 10 p-values: 0.000000 |-------------------------------------------------------------| | THE 3DSPHERES TEST | |Choose 4000 random points in a cube of edge 1000. At each | |point, center a sphere large enough to reach the next closest| |point. Then the volume of the smallest such sphere is (very | |close to) exponentially distributed with mean 120pi/3. Thus | |the radius cubed is exponential with mean 30. (The mean is | |obtained by extensive simulation). The 3DSPHERES test gener-| |ates 4000 such spheres 20 times. Each min radius cubed leads| |to a uniform variable by means of 1-exp(-r^3/30.), then a | | KSTEST is done on the 20 p-values. | |-------------------------------------------------------------| The 3DSPHERES test for urandom.bin sample no r^3 equiv. uni. 1 0.963 0.031597 2 0.118 0.003925 3 0.869 0.028544 4 5.058 0.155142 5 0.216 0.007183 6 5.770 0.174970 7 0.100 0.003314 8 0.184 0.006110 9 0.695 0.022914 10 0.680 0.022418 11 0.487 0.016110 12 1.311 0.042757 13 0.487 0.016099 14 0.788 0.025917 15 0.735 0.024213 16 0.019 0.000621 17 1.948 0.062871 18 0.935 0.030687 19 14.768 0.388763 20 1.460 0.047488 -------------------------------------------------------------- p-value for KS test on those 20 p-values: 0.000000 |-------------------------------------------------------------| | This is the SQUEEZE test | | Random integers are floated to get uniforms on [0,1). Start-| | ing with k=2^31=2147483647, the test finds j, the number of | | iterations necessary to reduce k to 1, using the reduction | | k=ceiling(k*U), with U provided by floating integers from | | the file being tested. Such j's are found 100,000 times, | | then counts for the number of times j was <=6,7,...,47,>=48 | | are used to provide a chi-square test for cell frequencies. | |-------------------------------------------------------------| RESULTS OF SQUEEZE TEST FOR urandom.bin Table of standardized frequency counts (obs-exp)^2/exp for j=(1,..,6), 7,...,47,(48,...) 28.9 70.8 145.9 267.2 414.2 576.1 646.0 603.6 461.7 285.3 123.4 16.0 -43.8 -72.2 -83.3 -88.5 -90.5 -90.5 -88.6 -85.0 -80.0 -74.0 -67.2 -60.1 -52.9 -45.9 -39.2 -33.1 -27.5 -22.6 -18.4 -14.8 -11.7 -9.2 -7.2 -5.5 -4.2 -3.2 -2.4 -1.8 -1.3 -1.0 -1.1 Chi-square with 42 degrees of freedom:1776573.502178 z-score=193835.478172, p-value=1.000000 _____________________________________________________________ |-------------------------------------------------------------| | The OVERLAPPING SUMS test | |Integers are floated to get a sequence U(1),U(2),... of uni- | |form [0,1) variables. Then overlapping sums, | | S(1)=U(1)+...+U(100), S2=U(2)+...+U(101),... are formed. | |The S's are virtually normal with a certain covariance mat- | |rix. A linear transformation of the S's converts them to a | |sequence of independent standard normals, which are converted| |to uniform variables for a KSTEST. | |-------------------------------------------------------------| Results of the OSUM test for urandom.bin Test no p-value 1 0.000000 2 0.000000 3 0.000000 4 0.000000 5 0.000000 6 0.000000 7 0.000000 8 0.000000 9 0.000000 10 0.000000 _____________________________________________________________ p-value for 10 kstests on 100 sums: 0.000000 |----------------------------------------------------------| |This is the UP-DOWN RUNS test. An up-run of length n has | |x_1<...x_(n+1), while a down-run of length n | |has x_1>...>x_n and x_n=2, the prob. of a run of length k is 2*k/(k+1)!) Length Expected UpRuns (O-E)^2/E DownRuns (O-E)^2/E 2 66666.67 66621 0.03 66892 0.76 3 25000.00 24993 0.00 24880 0.58 4 6666.67 6706 0.23 6547 2.15 5 1388.89 1400 0.09 1399 0.07 6 238.10 243 0.10 238 0.00 7 34.72 35 0.00 36 0.05 8 4.96 1 3.16 7 0.84 p=0.27193 p=0.38339 Number of rngs required: 687880, p-value: 0.273 |-------------------------------------------------------------| |This the CRAPS TEST. It plays 200,000 games of craps, counts| |the number of wins and the number of throws necessary to end | |each game. The number of wins should be (very close to) a | |normal with mean 200000p and variance 200000p(1-p), and | |p=244/495. Throws necessary to complete the game can vary | |from 1 to infinity, but counts for all>21 are lumped with 21.| |A chi-square test is made on the no.-of-throws cell counts. | |Each 32-bit integer from the test file provides the value for| |the throw of a die, by floating to [0,1), multiplying by 6 | |and taking 1 plus the integer part of the result. | |-------------------------------------------------------------| RESULTS OF CRAPS TEST for urandom.bin No. of wins: Observed Expected 132432 98585.9 z-score=151.380, pvalue=1.00000 Analysis of Throws-per-Game: Throws Observed Expected Chisq Sum of (O-E)^2/E 1 66611 66666.7 0.046 0.046 2 35010 37654.3 185.701 185.747 3 24816 26954.7 169.698 355.446 4 17790 19313.5 120.172 475.618 5 12984 13851.4 54.321 529.938 6 9556 9943.5 15.104 545.042 7 6985 7145.0 3.584 548.627 8 5358 5139.1 9.326 557.953 9 4045 3699.9 32.195 590.148 10 3154 2666.3 89.208 679.356 11 2519 1923.3 184.484 863.840 12 1924 1388.7 206.304 1070.144 13 1513 1003.7 258.411 1328.555 14 1221 726.1 337.243 1665.798 15 992 525.8 413.264 2079.062 16 837 381.2 545.188 2624.250 17 686 276.5 606.271 3230.521 18 540 200.8 572.805 3803.326 19 498 146.0 848.824 4652.150 20 409 106.2 863.140 5515.290 21 2552 287.1 17866.452 23381.742 Chisq=23381.74 for 20 degrees of freedom, p= 1.00000 SUMMARY of craptest p-value for no. of wins: 1.000000 p-value for throws/game: 1.000000 _____________________________________________________________ |-------------------------------------------------------------| |This is the CRAPS TEST with different dice. Each die value is| |determined by the rightmost three bits of the 32-bit random | |integer; values 1 to 6 are accepted, others rejected. As in | |the first test, 200,000 games of craps are played, counting | |the number of wins and the number of throws necessary to end | |each game. The number of wins should be (very close to) a | |normal with mean 200000p and variance 200000p(1-p), and | |p=244/495. Throws necessary to complete the game can vary | |from 1 to infinity, but counts for all>21 are lumped with 21.| |A chi-square test is made on the no.-of-throws cell counts. | |-------------------------------------------------------------| RESULTS OF CRAPS TEST2 for urandom.bin No. of wins: Observed Expected 98516 98585.9 z-score=-0.312, pvalue=0.37735 Analysis of Throws-per-Game: Throws Observed Expected Chisq Sum of (O-E)^2/E 1 66874 66666.7 0.645 0.645 2 37536 37654.3 0.372 1.017 3 26921 26954.7 0.042 1.059 4 19248 19313.5 0.222 1.281 5 13892 13851.4 0.119 1.400 6 9958 9943.5 0.021 1.421 7 7170 7145.0 0.087 1.508 8 5167 5139.1 0.152 1.660 9 3694 3699.9 0.009 1.669 10 2596 2666.3 1.853 3.522 11 1938 1923.3 0.112 3.634 12 1386 1388.7 0.005 3.640 13 1023 1003.7 0.371 4.010 14 698 726.1 1.091 5.101 15 525 525.8 0.001 5.102 16 380 381.2 0.003 5.106 17 271 276.5 0.111 5.217 18 199 200.8 0.017 5.233 19 134 146.0 0.984 6.217 20 101 106.2 0.256 6.473 21 289 287.1 0.012 6.486 Chisq= 6.49 for 20 degrees of freedom, p= 0.00194 SUMMARY of craptest p-value for no. of wins: 0.377350 p-value for throws/game: 0.001938 _____________________________________________________________ ***** TEST SUMMARY ***** All p-values: 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,0.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,0.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,0.2830,0.6181,0.6699,1.0000,1.0000,1.0000,1.0000,1.0000, 0.6637,0.4941,0.0011,1.0000,1.0000,1.0000,1.0000,1.0000,0.0963,0.5920, 0.5788,1.0000,1.0000,1.0000,1.0000,1.0000,0.0609,0.8460,0.7858,1.0000, 0.2169,0.3558,0.8943,0.0259,0.4585,0.1153,1.0000,1.0000,0.7481,0.1549, 0.8970,0.4224,0.9134,0.9253,1.0000,1.0000,0.7151,0.3881,0.1671,0.8927, 0.6088,0.2773,1.0000,1.0000,0.0774,0.9436,0.5031,0.7060,0.0339,0.1119, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000, 1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,1.0000,0.0000,0.0000,0.0000, 0.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0803,0.0480, 0.1170,0.0949,0.0512,0.0022,0.0153,0.0013,0.1701,0.1019,0.0000,0.0316, 0.0039,0.0285,0.1551,0.0072,0.1750,0.0033,0.0061,0.0229,0.0224,0.0161, 0.0428,0.0161,0.0259,0.0242,0.0006,0.0629,0.0307,0.3888,0.0475,0.0000, 1.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000,0.0000, 0.0000,0.0000,0.2719,0.3834,0.2732,1.0000,1.0000,0.3773,0.0019, Overall p-value after applying KStest on 229 p-values = 0.000000 In response to requests, we have provided a list of all the p-values produced by the tests you have chosen for this run. The individual p-values are supposed to be uniform in [0,1), but they are not necessarily independent. So even though we have applied a KSTEST to the accumulated p-values, the result is not necessarily---even if your file contains truly random bits---uniform in [0,1). But it is probably pretty close, so take that last p-value with a grain of salt. In particular, there may be some values so close to 0 or 1 that the tests they came from should be applied several more times, or new, related tests should be undertaken.