# HotBits Statistical Testing

## Introduction

While it's essential that the theory behind a random number generator be well understood, and that its hardware and software realisation be carefully verified to implement the theoretical design, there is no substitute for detailed statistical testing of the actual output of the generator against the expectation for genuinely random data. Although a generator's design may be simple, its implementation on a complex computer and software environment makes it vulnerable to a multitude of potential problems ranging from simple programming errors to subtle bias introduced by the behaviour of instruction and data caches in the microprocessor, interference from interrupts (if not prevented), and metastability in the logic gates which receive events from the generator.

A large data set produced by the HotBits generator has been subjected to the scrutiny of three different randomness test suites, whose results are presented below. You can download this large data set (which took almost two days to create with the HotBits generator running continuously) and subject it to your own analyses, if you are so inclined.

There are many different ways to test for randomness, but all of them, in essence, boil down to computing a mathematical metric from the data stream being tested and comparing the result with the expectation value for an infinite sequence of genuinely random data. For a truly random sequence, any value is equally probable. The sequence of bits “0000000000000000” is just as likely to occur in a random data stream as “1100100100001111”, and is no less “random”. (The latter sequence is, in fact, the first sixteen bits of the mathematical constant π in binary, whose algorithmic complexity is only modestly greater than the all zero sequence!)

Randomness can be defined only statistically over a long sequence, which is why it is essential to test a large data set. Data can fail to be random in many ways. For example, one of the most obvious tests one can apply to a sequence of binary data is to count the number of ones and zeroes: as the length of the sequence increases, the difference in these values can be used to calculate the probability the sequence is random. But this test, used in isolation, would consider a sequence of alternating zero and one bits (“0101010101010101…”) perfectly random. Hence, it must be used as part of a test battery, including other measures which are sensitive to repeating patterns, improbably long runs of zeroes and ones, and other, more subtle, deviations from randomness.

Speaking as a programmer and not a mathematician or statistician, the two widely-used randomness test batteries: Diehard and the NIST SP 800-22 Statistical Test Suite, whose results are reported below, are quite messy and fragile programs. When using them, it is wise to use data sets of the same size as those employed in the examples supplied with the programs, and to select test parameters identical to those of the examples. In my experience, deviating from the domain in which the programs are known to have been tested may yield surprising and dismaying results. Also, before testing your own data with one of these test batteries, be sure to re-run the examples in the documentation and verify that you're able to reproduce the published results; changes in compilers and libraries, file formats, and operating system compatibility issues may have to be resolved before you can obtain reliable results from these tests.

## Tests with the Fourmilab ENT Utility

The Fourmilab ENT program is a public domain utility which tests binary data sequences, either as a series of 8 bit bytes, or as a bit stream, with five standard tests for randomness which are described in the document linked to above. These are all straightforward mathematical metrics, and while they identify major departures from randomness, may miss subtle forms of bias identified by the more comprehensive test suites. The following are the results of an ENT test of the same 11,468,800 data set used for the Diehard test battery in the next section.

```Entropy = 7.999975 bits per byte.

Optimum compression would reduce the size
of this 11468800 byte file by 0 percent.

Chi square distribution for 11468800 samples is 402.53, and randomly
would exceed this value 0.01 percent of the times.

Arithmetic mean value of data bytes is 127.5423 (127.5 = random).
Monte Carlo value for Pi is 3.141486168 (error 0.00 percent).
Serial correlation coefficient is -0.000053 (totally uncorrelated = 0.0).
```

## Tests with the Marsaglia Diehard Battery of Tests of Randomness

Professor George Marsaglia of Florida State University published the “Diehard Battery of Tests of Randomness” in 1995, as part of the Marsaglia Random Number CDROM; these programs may now be downloaded from the link above. The Diehard tests are rather “quirky” measurements of randomness compared to the mathematical properties tested by ENT. Diehard tests include items such as a spacings between birthdays in a random population, monkeys pounding on keyboards, and games of craps. These tests, however, can be exquisitely sensitive to subtle departures from randomness, and their results can all be expressed as the probability the results obtained would be observed in a genuinely random sequence. Probability values close to zero or one indicate potential problems, while probabilities in the middle of the range are expected for random sequences. Please read the “NOTE” at the top of the results presented below about interpreting the reported probability values: with hundreds of probability values computed, some may be expected, purely by chance, to be close to one or zero.

The Diehard test suite was run on a 11,468,800 byte data set extracted from the beginning of the 16,779,776 HotBits test data set. I limited the data set length to that used by other Diehard examples to avoid possible problems in the code dependent upon the size of the data set.

```
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=1-F(X), where F is the assumed distribution of the sample
random variable X---often normal. But that assumed F is often just
an asymptotic approximation, for which the fit will 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:

FourmilabHotBits.32

1   Birthday Spacings
2   Overlapping Permutations
3   Ranks of 31x31 and 32x32 matrices
4   Ranks of 6x8 Matrices
5   Monkey Tests on 20-bit Words
6   Monkey Tests OPSO,OQSO,DNA
7   Count the 1`s in a Stream of Bytes
8   Count the 1`s in Specific Bytes
9   Parking Lot Test
10  Minimum Distance Test
11  Random Spheres Test
12  The Sqeeze Test
13  Overlapping Sums Test
14  Runs Test
15  The Craps Test
16  All of the above

To choose any particular tests, enter corresponding numbers.
Enter 16 for all tests. If you want to perform all but a few
tests, enter corresponding numbers preceded by "-" sign.
Tests are executed in the order they are entered.

16

|-------------------------------------------------------------|
|           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 integers in the specified file.  Then the|
|file is closed and reopened, then bits 2-25 of the same inte-|
|gers 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 TEST FOR FourmilabHotBits.32
(no_bdays=1024, no_days/yr=2^24, lambda=16.00, sample size=500)

Bits used       mean            chisqr          p-value
1 to 24        15.86           23.2107         0.142517
2 to 25        15.81           13.7843         0.682308
3 to 26        15.76           13.6039         0.694917
4 to 27        15.55           18.6599         0.348401
5 to 28        15.63           25.9381         0.075596
6 to 29        15.83           15.1730         0.583021
7 to 30        16.18           20.5640         0.246401
8 to 31        15.95           13.3762         0.710659
9 to 32        15.92           11.0928         0.851700

degree of freedoms is: 17
---------------------------------------------------------------
p-value for KStest on those 9 p-values: 0.915681

|-------------------------------------------------------------|
|           THE OVERLAPPING 5-PERMUTATION TEST                |
|This is the OPERM5 test.  It looks at a sequence of one 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  |
|equivalent to the likelihood ratio test that the 120 cell    |
|counts came from the specified (asymptotically) normal dis-  |
|tribution with the specified 120x120 covariance matrix (with |
|rank 99).  This version uses 1,000,000 integers, twice.      |
|-------------------------------------------------------------|

OPERM5 test for file
(For samples of 1,000,000 consecutive 5-tuples)

sample 1
chisquare=94.907067 with df=99; p-value= 0.597706
_______________________________________________________________

sample 2
chisquare=60.297852 with df=99; p-value= 0.999246
_______________________________________________________________

|-------------------------------------------------------------|
|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. |
|-------------------------------------------------------------|
Rank test for binary matrices (31x31) from FourmilabHotBits.32

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=28        205             211.4           0.195           0.195
r=29    5155            5134.0          0.086           0.281
r=30    22974           23103.0         0.721           1.001
r=31    11666           11551.5         1.134           2.136

chi-square = 2.136 with df = 3;  p-value = 0.545
--------------------------------------------------------------

|-------------------------------------------------------------|
|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) from FourmilabHotBits.32

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=29        206             211.4           0.139           0.139
r=30    5042            5134.0          1.649           1.788
r=31    23133           23103.0         0.039           1.827
r=32    11619           11551.5         0.394           2.221

chi-square = 2.221 with df = 3;  p-value = 0.528
--------------------------------------------------------------

|-------------------------------------------------------------|
|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 6,5 and (0,...,4) (pooled together).        |
|-------------------------------------------------------------|

Rank test for binary matrices (6x8) from FourmilabHotBits.32

bits  1 to  8

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 924             944.3           0.436           0.436
r=5     21622           21743.9         0.683           1.120
r=6     77454           77311.8         0.262           1.381

chi-square = 1.381 with df = 2;  p-value = 0.501
--------------------------------------------------------------

bits  2 to  9

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 936             944.3           0.073           0.073
r=5     21877           21743.9         0.815           0.888
r=6     77187           77311.8         0.201           1.089

chi-square = 1.089 with df = 2;  p-value = 0.580
--------------------------------------------------------------

bits  3 to 10

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 955             944.3           0.121           0.121
r=5     21949           21743.9         1.935           2.056
r=6     77096           77311.8         0.602           2.658

chi-square = 2.658 with df = 2;  p-value = 0.265
--------------------------------------------------------------

bits  4 to 11

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 964             944.3           0.411           0.411
r=5     21689           21743.9         0.139           0.550
r=6     77347           77311.8         0.016           0.566

chi-square = 0.566 with df = 2;  p-value = 0.754
--------------------------------------------------------------

bits  5 to 12

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 945             944.3           0.001           0.001
r=5     21697           21743.9         0.101           0.102
r=6     77358           77311.8         0.028           0.129

chi-square = 0.129 with df = 2;  p-value = 0.937
--------------------------------------------------------------

bits  6 to 13

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 999             944.3           3.169           3.169
r=5     21929           21743.9         1.576           4.744
r=6     77072           77311.8         0.744           5.488

chi-square = 5.488 with df = 2;  p-value = 0.064
--------------------------------------------------------------

bits  7 to 14

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 956             944.3           0.145           0.145
r=5     21790           21743.9         0.098           0.243
r=6     77254           77311.8         0.043           0.286

chi-square = 0.286 with df = 2;  p-value = 0.867
--------------------------------------------------------------

bits  8 to 15

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 950             944.3           0.034           0.034
r=5     21435           21743.9         4.388           4.423
r=6     77615           77311.8         1.189           5.612

chi-square = 5.612 with df = 2;  p-value = 0.060
--------------------------------------------------------------

bits  9 to 16

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 950             944.3           0.034           0.034
r=5     21739           21743.9         0.001           0.036
r=6     77311           77311.8         0.000           0.036

chi-square = 0.036 with df = 2;  p-value = 0.982
--------------------------------------------------------------

bits 10 to 17

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 936             944.3           0.073           0.073
r=5     21681           21743.9         0.182           0.255
r=6     77383           77311.8         0.066           0.320

chi-square = 0.320 with df = 2;  p-value = 0.852
--------------------------------------------------------------

bits 11 to 18

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 892             944.3           2.897           2.897
r=5     21752           21743.9         0.003           2.900
r=6     77356           77311.8         0.025           2.925

chi-square = 2.925 with df = 2;  p-value = 0.232
--------------------------------------------------------------

bits 12 to 19

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 950             944.3           0.034           0.034
r=5     21614           21743.9         0.776           0.810
r=6     77436           77311.8         0.200           1.010

chi-square = 1.010 with df = 2;  p-value = 0.604
--------------------------------------------------------------

bits 13 to 20

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 945             944.3           0.001           0.001
r=5     21658           21743.9         0.339           0.340
r=6     77397           77311.8         0.094           0.434

chi-square = 0.434 with df = 2;  p-value = 0.805
--------------------------------------------------------------

bits 14 to 21

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 955             944.3           0.121           0.121
r=5     21683           21743.9         0.171           0.292
r=6     77362           77311.8         0.033           0.324

chi-square = 0.324 with df = 2;  p-value = 0.850
--------------------------------------------------------------

bits 15 to 22

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 922             944.3           0.527           0.527
r=5     21719           21743.9         0.029           0.555
r=6     77359           77311.8         0.029           0.584

chi-square = 0.584 with df = 2;  p-value = 0.747
--------------------------------------------------------------

bits 16 to 23

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 904             944.3           1.720           1.720
r=5     21900           21743.9         1.121           2.841
r=6     77196           77311.8         0.173           3.014

chi-square = 3.014 with df = 2;  p-value = 0.222
--------------------------------------------------------------

bits 17 to 24

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 924             944.3           0.436           0.436
r=5     21714           21743.9         0.041           0.478
r=6     77362           77311.8         0.033           0.510

chi-square = 0.510 with df = 2;  p-value = 0.775
--------------------------------------------------------------

bits 18 to 25

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 935             944.3           0.092           0.092
r=5     21747           21743.9         0.000           0.092
r=6     77318           77311.8         0.000           0.093

chi-square = 0.093 with df = 2;  p-value = 0.955
--------------------------------------------------------------

bits 19 to 26

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 950             944.3           0.034           0.034
r=5     21716           21743.9         0.036           0.070
r=6     77334           77311.8         0.006           0.077

chi-square = 0.077 with df = 2;  p-value = 0.962
--------------------------------------------------------------

bits 20 to 27

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 923             944.3           0.480           0.480
r=5     21745           21743.9         0.000           0.481
r=6     77332           77311.8         0.005           0.486

chi-square = 0.486 with df = 2;  p-value = 0.784
--------------------------------------------------------------

bits 21 to 28

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 889             944.3           3.238           3.238
r=5     21903           21743.9         1.164           4.403
r=6     77208           77311.8         0.139           4.542

chi-square = 4.542 with df = 2;  p-value = 0.103
--------------------------------------------------------------

bits 22 to 29

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 952             944.3           0.063           0.063
r=5     21753           21743.9         0.004           0.067
r=6     77295           77311.8         0.004           0.070

chi-square = 0.070 with df = 2;  p-value = 0.965
--------------------------------------------------------------

bits 23 to 30

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 986             944.3           1.841           1.841
r=5     22062           21743.9         4.654           6.495
r=6     76952           77311.8         1.674           8.170

chi-square = 8.170 with df = 2;  p-value = 0.017
--------------------------------------------------------------

bits 24 to 31

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 977             944.3           1.132           1.132
r=5     21823           21743.9         0.288           1.420
r=6     77200           77311.8         0.162           1.582

chi-square = 1.582 with df = 2;  p-value = 0.453
--------------------------------------------------------------

bits 25 to 32

RANK    OBSERVED        EXPECTED        (O-E)^2/E       SUM

r<=4 968             944.3           0.595           0.595
r=5     21885           21743.9         0.916           1.510
r=6     77147           77311.8         0.351           1.862

chi-square = 1.862 with df = 2;  p-value = 0.394
--------------------------------------------------------------
TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
These should be 25 uniform [0,1] random variates:

0.501240        0.580088        0.264713        0.753662        0.937402
0.064310        0.866791        0.060452        0.982397        0.851939
0.231666        0.603516        0.805025        0.850268        0.746786
0.221576        0.774874        0.954789        0.962434        0.784356
0.103210        0.965486        0.016827        0.453440        0.394211
The KS test for those 25 supposed UNI's yields
KS p-value = 0.088531

|-------------------------------------------------------------|
|                  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 FourmilabHotBits.32
(20 bits/word, 2097152 words 20 bitstreams. No. missing words
should average 141909.33 with sigma=428.00.)
----------------------------------------------------------------
BITSTREAM test results for FourmilabHotBits.32.

Bitstream       No. missing words       z-score         p-value
1            142235                   0.76           0.223355
2            141920                   0.02           0.490055
3            142064                   0.36           0.358908
4            142338                   1.00           0.158277
5            141548                  -0.84           0.800729
6            141988                   0.18           0.427082
7            141888                  -0.05           0.519874
8            141930                   0.05           0.480741
9            141610                  -0.70           0.757839
10           142059                   0.35           0.363283
11           142442                   1.24           0.106647
12           142265                   0.83           0.202985
13           142704                   1.86           0.031677
14           141658                  -0.59           0.721472
15           141818                  -0.21           0.584488
16           141810                  -0.23           0.591762
17           142139                   0.54           0.295768
18           142066                   0.37           0.357163
19           141959                   0.12           0.453806
20           141860                  -0.12           0.545879
----------------------------------------------------------------

|-------------------------------------------------------------|
|        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 file FourmilabHotBits.32

Bits used       No. missing words       z-score         p-value
23 to 32                141580          -1.1356         0.871942
22 to 31                141827          -0.2839         0.611755
21 to 30                141592          -1.0942         0.863075
20 to 29                141567          -1.1804         0.881089
19 to 28                141530          -1.3080         0.904569
18 to 27                141664          -0.8460         0.801214
17 to 26                141445          -1.6011         0.945327
16 to 25                141851          -0.2011         0.579705
15 to 24                141327          -2.0080         0.977680
14 to 23                141867          -0.1460         0.558026
13 to 22                141517          -1.3529         0.911950
12 to 21                141589          -1.1046         0.865331
11 to 20                141630          -0.9632         0.832278
10 to 19                141897          -0.0425         0.516957
9 to 18                 141473          -1.5046         0.933785
8 to 17                 141741          -0.5804         0.719194
7 to 16                 141697          -0.7322         0.767968
6 to 15                 142348           1.5127         0.065184
5 to 14                 141311          -2.0632         0.980454
4 to 13                 141515          -1.3598         0.913047
3 to 12                 141648          -0.9011         0.816242
2 to 11                 141756          -0.5287         0.701502
1 to 10                 141559          -1.2080         0.886483
-----------------------------------------------------------------

|------------------------------------------------------------ |
|  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 file FourmilabHotBits.32

Bits used       No. missing words       z-score         p-value
28 to 32                141900          -0.0316         0.512615
27 to 31                141966           0.1921         0.423831
26 to 30                141500          -1.3876         0.917364
25 to 29                141357          -1.8723         0.969418
24 to 28                142099           0.6429         0.260128
23 to 27                141732          -0.6011         0.726119
22 to 26                141330          -1.9638         0.975225
21 to 25                141966           0.1921         0.423831
20 to 24                141817          -0.3130         0.622853
19 to 23                142012           0.3480         0.363907
18 to 22                142448           1.8260         0.033925
17 to 21                142369           1.5582         0.059593
16 to 20                141741          -0.5706         0.715868
15 to 19                141902          -0.0248         0.509912
14 to 18                142162           0.8565         0.195858
13 to 17                141921           0.0396         0.484222
12 to 16                142013           0.3514         0.362635
11 to 15                141955           0.1548         0.438484
10 to 14                141800          -0.3706         0.644536
9 to 13                 141915           0.0192         0.492333
8 to 12                 142330           1.4260         0.076934
7 to 11                 142155           0.8328         0.202484
6 to 10                 142132           0.7548         0.225180
5 to 9                  142252           1.1616         0.122700
4 to 8                  142121           0.7175         0.236525
3 to 7                  142450           1.8328         0.033418
2 to 6                  141697          -0.7198         0.764164
1 to 5                  142027           0.3989         0.344990
-----------------------------------------------------------------

|------------------------------------------------------------ |
|    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 file FourmilabHotBits.32

Bits used       No. missing words       z-score         p-value
31 to 32                142299           1.1495         0.125181
30 to 31                141702          -0.6116         0.729596
29 to 30                141705          -0.6027         0.726660
28 to 29                141910           0.0020         0.499211
27 to 28                141889          -0.0600         0.523910
26 to 27                141104          -2.3756         0.991240
25 to 26                142020           0.3265         0.372038
24 to 25                141725          -0.5437         0.706692
23 to 24                141376          -1.5732         0.942169
22 to 23                141444          -1.3727         0.915070
21 to 22                142121           0.6244         0.266184
20 to 21                141715          -0.5732         0.716760
19 to 20                141282          -1.8505         0.967881
18 to 19                141857          -0.1544         0.561339
17 to 18                141921           0.0344         0.486269
16 to 17                142055           0.4297         0.333705
15 to 16                142639           2.1524         0.015682
14 to 15                141865          -0.1308         0.552020
13 to 14                142382           1.3943         0.081612
12 to 13                141725          -0.5437         0.706692
11 to 12                141746          -0.4818         0.685026
10 to 11                141594          -0.9302         0.823860
9 to 10                 142113           0.6008         0.273988
8 to 9                  142013           0.3058         0.379874
7 to 8                  141896          -0.0393         0.515683
6 to 7                  141759          -0.4435         0.671280
5 to 6                  141432          -1.4081         0.920442
4 to 5                  141645          -0.7797         0.782226
3 to 4                  142413           1.4858         0.068672
2 to 3                  142099           0.5595         0.287911
1 to 2                  141658          -0.7414         0.770770
-----------------------------------------------------------------

|-------------------------------------------------------------|
|    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 for the byte stream from FourmilabHotBits.32
(Degrees of freedom: 5^4-5^3=2500; sample size: 2560000)

chisquare       z-score         p-value
2493.54         -0.091          0.536401

|-------------------------------------------------------------|
|    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|
|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 from FourmilabHotBits.32
(Degrees of freedom: 5^4-5^3=2500; sample size: 256000)

bits used       chisquare       z-score         p-value
1 to 8          2370.17         -1.836          0.966830
2 to 9          2590.32          1.277          0.100746
3 to 10         2513.16          0.186          0.426206
4 to 11         2452.96         -0.665          0.747062
5 to 12         2598.68          1.396          0.081421
6 to 13         2462.08         -0.536          0.704108
7 to 14         2423.88         -1.076          0.859147
8 to 15         2596.54          1.365          0.086087
9 to 16         2501.68          0.024          0.490549
10 to 17        2475.61         -0.345          0.634949
11 to 18        2535.65          0.504          0.307060
12 to 19        2467.15         -0.465          0.678858
13 to 20        2484.22         -0.223          0.588298
14 to 21        2606.47          1.506          0.066070
15 to 22        2458.25         -0.590          0.722563
16 to 23        2546.29          0.655          0.256356
17 to 24        2531.40          0.444          0.328502
18 to 25        2519.09          0.270          0.393567
19 to 26        2526.07          0.369          0.356168
20 to 27        2595.14          1.346          0.089231
21 to 28        2443.93         -0.793          0.786095
22 to 29        2469.71         -0.428          0.665820
23 to 30        2485.01         -0.212          0.583966
24 to 31        2518.65          0.264          0.395996
25 to 32        2530.07          0.425          0.335318
|-------------------------------------------------------------|
|              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 is very close |
|to normally distributed.  Thus (k-3523)/21.9 should be a st- |
|andard normal variable, which, converted to a uniform varia- |
|ble, provides input to a KSTEST based on a sample of 10.     |
|-------------------------------------------------------------|

CDPARK: result of 10 tests on file FourmilabHotBits.32
(Of 12000 tries, the average no. of successes should be
3523.0 with sigma=21.9)

No. succeses         z-score         p-value
3543             0.9132         0.180558
3506            -0.7763         0.781201
3533             0.4566         0.323972
3524             0.0457         0.481790
3526             0.1370         0.445521
3510            -0.5936         0.723613
3543             0.9132         0.180558
3532             0.4110         0.340551
3533             0.4566         0.323972
3533             0.4566         0.323972
Square side=100, avg. no. parked=3528.30 sample std.=11.70
p-value of the KSTEST for those 10 p-values: 0.002300

|-------------------------------------------------------------|
|              THE MINIMUM DISTANCE TEST                      |
|It does this 100 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 be uniform on [0,1) and |
|a KSTEST on the resulting 100 values serves as a test of uni-|
|formity for random points in the square. Test numbers=0 mod 5|
|are printed but the KSTEST is based on the full set of 100   |
|random choices of 8000 points in the 10000x10000 square.     |
|-------------------------------------------------------------|

This is the MINIMUM DISTANCE test for file FourmilabHotBits.32

Sample no.       d^2             mean           equiv uni
5            0.0066          0.7395          0.006640
10           3.0735          1.2046          0.954449
15           0.1273          0.9023          0.120123
20           5.4762          1.1602          0.995928
25           0.9729          1.2054          0.623852
30           0.7707          1.1989          0.539084
35           0.0274          1.2205          0.027205
40           0.0148          1.1269          0.014762
45           1.8536          1.1399          0.844779
50           0.3695          1.1442          0.310181
55           0.4188          1.1592          0.343554
60           0.2675          1.1129          0.235763
65           3.1837          1.1337          0.959224
70           2.3059          1.1826          0.901478
75           1.0426          1.1678          0.649299
80           0.0274          1.1610          0.027178
85           3.4117          1.1430          0.967576
90           3.3879          1.1558          0.966792
95           0.4765          1.1227          0.380508
100          1.1551          1.1143          0.686808
--------------------------------------------------------------
Result of KS test on 100 transformed mindist^2's: p-value=0.297975

|-------------------------------------------------------------|
|             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-|
|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 file FourmilabHotBits.32

sample no       r^3             equiv. uni.
1            95.178          0.958106
2            102.500         0.967178
3            0.896           0.029438
4            47.776          0.796592
5            31.541          0.650536
6            20.136          0.488897
7            20.419          0.493707
8            7.454           0.220000
9            11.429          0.316806
10           32.907          0.666098
11           10.626          0.298265
12           38.004          0.718272
13           6.657           0.199018
14           11.230          0.312259
15           47.004          0.791291
16           1.492           0.048505
17           0.666           0.021962
18           20.268          0.491147
19           0.664           0.021891
20           42.920          0.760853
--------------------------------------------------------------
p-value for KS test on those 20 p-values: 0.539573

|-------------------------------------------------------------|
|                 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 FourmilabHotBits.32

Table of standardized frequency counts
(obs-exp)^2/exp  for j=(1,..,6), 7,...,47,(48,...)
-0.1     0.5    -1.1     0.5    -1.3     0.4
-1.4    -1.2     0.6    -0.9    -1.2    -1.9
0.7    -1.8     1.7    -2.2     0.6     1.3
0.8    -0.8     0.1     0.1     0.1     0.6
0.6     0.4     3.1    -1.7     1.0     0.0
1.4     1.0     0.2    -0.2     0.9    -1.6
1.2    -0.1     0.1     0.4     1.6     0.0
1.8
Chi-square with 42 degrees of freedom:55.429856
z-score=1.465317, p-value=0.080144
_____________________________________________________________

|-------------------------------------------------------------|
|            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 FourmilabHotBits.32

Test no                 p-value
1                     0.222186
2                     0.841877
3                     0.064437
4                     0.936267
5                     0.676322
6                     0.305676
7                     0.679858
8                     0.181487
9                     0.992939
10                    0.653249
_____________________________________________________________

p-value for 10 kstests on 100 kstests:0.642506

|-------------------------------------------------------------|
|    This is the RUNS test.  It counts runs up, and runs down,|
|in a sequence of uniform [0,1) variables, obtained by float- |
|ing the 32-bit integers in the specified file. This example  |
|shows how runs are counted: .123,.357,.789,.425,.224,.416,.95|
|contains an up-run of length 3, a down-run of length 2 and an|
|up-run of (at least) 2, depending on the next values.  The   |
|covariance matrices for the runs-up and runs-down are well   |
|weak inverses of the covariance matrices.  Runs are counted  |
|for sequences of length 10,000.  This is done ten times. Then|
|another three sets of ten.                                   |
|-------------------------------------------------------------|

The RUNS test for file FourmilabHotBits.32
(Up and down runs in a sequence of 10000 numbers)
Set 1
runs up; ks test for 10 p's: 0.005535
runs down; ks test for 10 p's: 0.144368
Set 2
runs up; ks test for 10 p's: 0.157631
runs down; ks test for 10 p's: 0.668033

|-------------------------------------------------------------|
|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 FourmilabHotBits.32
No. of wins:  Observed  Expected
98599        98585.858586
z-score= 0.059, pvalue=0.47657

Analysis of Throws-per-Game:

Throws  Observed        Expected        Chisq    Sum of (O-E)^2/E
1       66719           66666.7         0.041           0.041
2       37257           37654.3         4.192           4.234
3       27201           26954.7         2.250           6.484
4       19333           19313.5         0.020           6.503
5       13682           13851.4         2.072           8.575
6       10159           9943.5          4.669           13.244
7       7134            7145.0          0.017           13.261
8       5036            5139.1          2.067           15.328
9       3669            3699.9          0.257           15.586
10      2741            2666.3          2.093           17.679
11      1948            1923.3          0.316           17.995
12      1423            1388.7          0.845           18.840
13      1048            1003.7          1.954           20.794
14      720             726.1           0.052           20.846
15      518             525.8           0.117           20.963
16      386             381.2           0.062           21.025
17      281             276.5           0.072           21.097
18      201             200.8           0.000           21.097
19      151             146.0           0.172           21.269
20      104             106.2           0.046           21.315
21      289             287.1           0.012           21.328

Chisq=  21.33 for 20 degrees of freedom, p= 0.37808

SUMMARY of craptest on FourmilabHotBits.32
p-value for no. of wins: 0.476565
p-value for throws/game: 0.378076
_____________________________________________________________

```

## Tests with the NIST Statistical Test Suite (SP 800-22)

The following results were produced by testing a sequence of 16,779,776 bytes from the HotBits generator with version 1.8 of the U.S. National Institute of Standards and Technology Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications as described in NIST Special Publication 800-22 [PDF]. This test suite is supplied as source and executable binaries compiled with Microsoft Visual C++. In testing the program, I encountered numerous crashes due to buffer overflows in editing messages. I corrected these in the source code, rebuilt with Visual C++ .NET, and used my patched version for the tests. Since the executable supplied by NIST crashed on my machine, even running the examples, I had no option but to rebuild it. If you're interested in building your own version that doesn't crash, you can download my patches to the source code.

The data set used in the following tests is identical in the first 11,468,800 bytes to that used in the ENT and Diehard tests above. This is the complete data set I generated for testing; due to the sensitivity of the Diehard tests to the data set size, I limited the data I tested to be the same as that used in the Diehard examples and tested that data set with ENT. The NIST tests seem to have no problems with large data sets, so I used the complete data set in these tests.

Version 1.8 of the NIST test suite has a rudimentary graphical user interface which runs only under Microsoft Windows. I performed these tests on my development machine, on which I booted Windows XP SP2. The test suite was compiled on a Windows 2000 machine using Microsoft Visual C++ .NET. The first configuration page of the test suite selects the data to be tested. You can choose from a variety of pseudorandom generators, data from a binary file, or a text file containing a sequence of bits represented as ASCII 0 and 1 characters.

The next page selects the tests to be performed and specifies the parameters to those tests which require them. I enabled all tests and used the parameters from the examples in the paper documenting the tests. Unless you really understand what you're doing, have studied the math underlying the test, and have read the code that implements it, it's a bad idea to get too creative changing these parameters. The LavaRnd people, for example, chose a block length of 14 for the Approximate Entropy test and discovered that with that setting even the “gold standard” Blum-Blum-Shub pseudorandom generator failed this test.

The tests selected and parameters are then displayed on confirmation page, and if accepted the tests begin. On my 3.4 GHz Pentium 4 machine, they run for a couple of hours.

After all the tests have completed, a summary report is written, which is reproduced below. The key values to look at for each test are the “P-value”: the probability the results obtained were due to chance, and the “Proportion” of sequences generated which were deemed to pass the test. Extremal P-values (very close to zero or one), and Proportions which are significantly less than one are indicative of potential problems

```------------------------------------------------------------------------------
RESULTS FOR THE UNIFORMITY OF P-VALUES AND THE PROPORTION OF PASSING SEQUENCES
------------------------------------------------------------------------------

------------------------------------------------------------------------------
C1  C2  C3  C4  C5  C6  C7  C8  C9 C10  P-VALUE  PROPORTION  STATISTICAL TEST
------------------------------------------------------------------------------
15  15  14  16   9  10   8  10   9  22  0.090936   0.9766    frequency
15  12  12  15  19   7   8  14  10  16  0.275709   1.0000    block-frequency
14  16  17  14   8  11  17   6  10  15  0.232760   0.9766    cumulative-sums
15  17  15  17  11   6  20   7   8  12  0.041438   0.9766    cumulative-sums
28  17   8  16   9   8  11  12  10   9  0.000569   0.9609    runs
14  13  12  13   6  14  16  17   8  15  0.407091   1.0000    longest-run
13   8  11  14  14  12  18  15  12  11  0.739918   0.9844    rank
10  17  18  10  13  12  14  10  12  12  0.689019   0.9922    fft
15  24  14  13  15  10  10   8   9  10  0.048716   0.9844    nonperiodic-templates
15  12  17   6  12  13  14  15  12  12  0.637119   0.9766    nonperiodic-templates
15  11  15  12  15  15   9  11  11  14  0.888137   0.9922    nonperiodic-templates
4   7  14  14  10  15  13  12  23  16  0.014216   1.0000    nonperiodic-templates
17  13  13   7  10  11  17  15  11  14  0.534146   0.9922    nonperiodic-templates
14  14  10  17  16   9  13  11   9  15  0.654467   0.9922    nonperiodic-templates
8  11  21   9   7  10  13  15  18  16  0.057146   1.0000    nonperiodic-templates
12  12  16  14  15  14  14  10   8  13  0.848588   0.9922    nonperiodic-templates
10  15  10  10  15  18  16   9  15  10  0.468595   0.9844    nonperiodic-templates
14  16  13  10  18  11   7  16   7  16  0.213309   0.9844    nonperiodic-templates
17  16  15  12   8   9  14  11  11  15  0.585209   0.9922    nonperiodic-templates
9  15   9   8   7  13  19  11  17  20  0.043745   1.0000    nonperiodic-templates
13  11  11  12  13  13  10  17  14  14  0.941144   1.0000    nonperiodic-templates
14  15  11  13  14  11  10  14  13  13  0.980883   0.9922    nonperiodic-templates
10  15  12   6  10  18  12  17  16  12  0.287306   1.0000    nonperiodic-templates
20  14  10   8  15  10  11  11  11  18  0.232760   0.9766    nonperiodic-templates
15  10  17   9  14  15  13  15   9  11  0.671779   0.9844    nonperiodic-templates
12  13  12  11  14  14  13  15  15   9  0.957319   0.9844    nonperiodic-templates
10  15  15  12  14   8  13  15  13  13  0.875539   0.9766    nonperiodic-templates
14  10  17  15   7  16   8  11  14  16  0.350485   0.9844    nonperiodic-templates
20  12  10   9  14  14  18  11  12   8  0.242986   0.9609    nonperiodic-templates
14  11  15   9  14  17   9  14   9  16  0.585209   0.9766    nonperiodic-templates
17  18  11   9  10  12  11   6  13  21  0.063482   0.9922    nonperiodic-templates
13  10   9  15  15  21   8   7  18  12  0.070445   1.0000    nonperiodic-templates
11   8  18  15  10   6  11  18  15  16  0.134686   1.0000    nonperiodic-templates
15  14  17  12   9  12   9  10  14  16  0.671779   0.9766    nonperiodic-templates
17  10  18  11  16  10  10  11   9  16  0.378138   0.9844    nonperiodic-templates
15  12  15  14  17  12   8   9  12  14  0.706149   1.0000    nonperiodic-templates
12  10  14  12  14   8  16  13  12  17  0.756476   0.9922    nonperiodic-templates
13  11  15  13   8  15  10  12  12  19  0.585209   1.0000    nonperiodic-templates
13  14  14  16  11  12  16  10  11  11  0.911413   0.9766    nonperiodic-templates
14  12  14  11   7  13  18  11  16  12  0.602458   0.9922    nonperiodic-templates
13  11  12  15   7  11  11  16  17  15  0.602458   0.9688    nonperiodic-templates
14   6  12  16  16  14  15  13  12  10  0.585209   1.0000    nonperiodic-templates
12   9  10  10  14  14  15  10  15  19  0.534146   1.0000    nonperiodic-templates
6  15  11  17  19  14  11  10  13  12  0.287306   0.9922    nonperiodic-templates
12  16   9  13  13  14  11  11  13  16  0.900104   1.0000    nonperiodic-templates
8  10  14  10  15   8  17  25  13   8  0.008879   0.9922    nonperiodic-templates
9   6  25   8  19   8  14  12  15  12  0.002316   0.9844    nonperiodic-templates
16  11  13  14  14  14   9   7  17  13  0.585209   0.9844    nonperiodic-templates
16  10  13  15   7  14  14  13   8  18  0.378138   0.9844    nonperiodic-templates
10  19  14  13  12  11  12  13   7  17  0.422034   0.9922    nonperiodic-templates
17  10   9  13  11  10  13  14  16  15  0.723129   1.0000    nonperiodic-templates
13  10  12  19  13  20   9   9  12  11  0.242986   0.9922    nonperiodic-templates
6  18  11  11  11  11  17  19  12  12  0.186566   1.0000    nonperiodic-templates
12  12  18  14   9  11  15  15   8  14  0.602458   0.9922    nonperiodic-templates
23  10  12  10  12  12  11  17   8  13  0.110952   0.9766    nonperiodic-templates
13  15  11  15   7  15   9  12  24   7  0.022503   0.9766    nonperiodic-templates
8   8  13  17  16  10  15   9  18  14  0.253551   0.9922    nonperiodic-templates
8  10  10  13  11  18  10  18  17  13  0.299251   1.0000    nonperiodic-templates
13  10  13  15  13  10  14  17   9  14  0.819544   0.9922    nonperiodic-templates
18  11   7  12  12  14  14  17  10  13  0.500934   0.9766    nonperiodic-templates
16  16  14  11  16   8  12  14  11  10  0.689019   0.9844    nonperiodic-templates
11  19  14  14  14  15   9   7  15  10  0.364146   0.9922    nonperiodic-templates
18  15  13  14   6  10  16  16  12   8  0.242986   0.9922    nonperiodic-templates
11  15  13  14  17  17   7  12  12  10  0.551026   1.0000    nonperiodic-templates
12  17  11  14  17  12   7  12  14  12  0.637119   0.9844    nonperiodic-templates
9  13  13   5  10  12  15  20  12  19  0.078086   0.9922    nonperiodic-templates
11  11  11  17  18  16  15   6   9  14  0.242986   1.0000    nonperiodic-templates
14  13   8  15  10  15   9  14  13  17  0.654467   0.9922    nonperiodic-templates
18   7  12  15   7  16  15  13  10  15  0.264458   0.9844    nonperiodic-templates
3  13  15  17  13  16  20   5  17   9  0.005490   1.0000    nonperiodic-templates
10  16  15  13  10  19  13   8  12  12  0.500934   0.9844    nonperiodic-templates
16  16  10  12   7  12   7  16  14  18  0.222869   0.9844    nonperiodic-templates
12   6  15  20  17  12   7  16  11  12  0.100508   0.9844    nonperiodic-templates
12  15  10  14  12  15  14  14  13   9  0.931952   0.9766    nonperiodic-templates
6  10  11  17  14  12  13  11  20  14  0.232760   0.9922    nonperiodic-templates
11  20  13   9  16   8  13  11  10  17  0.242986   1.0000    nonperiodic-templates
18   8   5  12  17  16  18   9  15  10  0.054199   1.0000    nonperiodic-templates
11  13  10  17  14   7  15  14  15  12  0.654467   0.9922    nonperiodic-templates
14  10   9  13   9   8  20  17  14  14  0.232760   0.9766    nonperiodic-templates
12  10  16  10  11  12  14  12  12  19  0.689019   0.9844    nonperiodic-templates
10  13  13  12  16  12  15  10  11  16  0.888137   0.9844    nonperiodic-templates
13  10  14  14  18  17  12  12   6  12  0.422034   0.9922    nonperiodic-templates
15  24  14  13  15  10  10   8   9  10  0.048716   0.9844    nonperiodic-templates
14  10  13  16  11  15  11   6  15  17  0.452799   0.9766    nonperiodic-templates
8  14   9  13   8  14  21  13  13  15  0.222869   1.0000    nonperiodic-templates
17  16   9  14  11  11  14  10  12  14  0.772760   0.9688    nonperiodic-templates
11  12  11  10  14  11  15  19  15  10  0.654467   1.0000    nonperiodic-templates
19   7   8  12  11  14  20  11  13  13  0.141256   0.9844    nonperiodic-templates
12  15  19  15   8   8  13  12  18   8  0.178278   0.9844    nonperiodic-templates
8  16  11  13  13   9  15  20  10  13  0.337162   0.9922    nonperiodic-templates
9   8  15  15   8  17  16  17   9  14  0.242986   0.9922    nonperiodic-templates
18  12   6  12  12   9  13  24  12  10  0.023812   0.9922    nonperiodic-templates
6  11  11  15  10  17  11  21  18   8  0.041438   0.9922    nonperiodic-templates
7  13  14  14  12  14   4  16  18  16  0.116519   1.0000    nonperiodic-templates
11   9  17  11  14  13  13  14  13  13  0.911413   1.0000    nonperiodic-templates
15   9  11  16  15   5  19  13  12  13  0.213309   0.9688    nonperiodic-templates
13  18   8  12  11   8  14  18  14  12  0.392456   0.9766    nonperiodic-templates
17  11  16  10  13  14   9  14  12  12  0.804337   0.9844    nonperiodic-templates
15  16  11   8  12  14  13  18   6  15  0.299251   0.9766    nonperiodic-templates
12  10   9  13  17  10  14  12  18  13  0.637119   0.9844    nonperiodic-templates
11  12  17  12  11  11  14  16   9  15  0.788728   0.9844    nonperiodic-templates
7  14  17  13  10  14  14  13  14  12  0.739918   0.9922    nonperiodic-templates
19  13  14  10  14  12  18  12   9   7  0.275709   0.9766    nonperiodic-templates
16   8   8  13   9  15  18  14  14  13  0.407091   0.9844    nonperiodic-templates
10  14  15  12   9  12  12  12  20  12  0.585209   0.9844    nonperiodic-templates
11  10  11  22   8  15  10  16  14  11  0.162606   0.9922    nonperiodic-templates
13  18  11   9  16  16  15   8  12  10  0.437274   0.9922    nonperiodic-templates
17  10  10  11   8  18  11  13  13  17  0.392456   0.9922    nonperiodic-templates
11  15  14  12  17   8  13  13  17   8  0.517442   1.0000    nonperiodic-templates
15   8  13  10   9  13  12  15  21  12  0.287306   1.0000    nonperiodic-templates
9   8   9  20  17  12   7  15  17  14  0.078086   0.9922    nonperiodic-templates
16  15  13  10  12  15  10   8  17  12  0.637119   0.9766    nonperiodic-templates
6  11  10  16  12  15  17  11  15  15  0.422034   0.9844    nonperiodic-templates
11  16   6  13  15   5  15  17  16  14  0.128379   1.0000    nonperiodic-templates
14  10  16  12   9   9  10  17  18  13  0.437274   0.9766    nonperiodic-templates
12  12  11   9  18  18  12  10  12  14  0.585209   0.9844    nonperiodic-templates
12  12  14  16  12  12  11  21   7  11  0.299251   0.9922    nonperiodic-templates
15  16  12   8  15  14  10  11  18   9  0.468595   1.0000    nonperiodic-templates
11  17  18  12  17   7   9  14  12  11  0.311542   1.0000    nonperiodic-templates
20  16  12  11  17  10  13  12   8   9  0.253551   0.9844    nonperiodic-templates
10  14  17  10  13   7  15  15   8  19  0.204076   1.0000    nonperiodic-templates
14  17  14   8  15  13  11   9  14  13  0.723129   0.9922    nonperiodic-templates
11  15  16  16  11  11  19  10  11   8  0.392456   0.9766    nonperiodic-templates
16  10  16   7   9  17  10  13  13  17  0.311542   0.9844    nonperiodic-templates
12  17  14  10  17   9  12  11  11  15  0.689019   0.9844    nonperiodic-templates
9  11  14  18   9  17  14  17  10   9  0.311542   0.9766    nonperiodic-templates
8  16  15   7  11  10  11  15  15  20  0.170294   1.0000    nonperiodic-templates
16  11  13  14   3  16  15  12  12  16  0.213309   0.9766    nonperiodic-templates
12  22  11  15  12   9  19  11   7  10  0.057146   1.0000    nonperiodic-templates
14  17   7  13   7  13  17  18   9  13  0.178278   0.9922    nonperiodic-templates
14  12  10  15  12  12  14  10  13  16  0.941144   1.0000    nonperiodic-templates
12  14  10  16   9  10  17  12  12  16  0.689019   1.0000    nonperiodic-templates
14  10  11   9  20  10  10  19  12  13  0.232760   1.0000    nonperiodic-templates
12  15  17   9  11  11  12  12  17  12  0.756476   0.9844    nonperiodic-templates
12  16  19  13  11  14   9  15  12   7  0.392456   0.9922    nonperiodic-templates
16  10  11  13  11  11  16  20  10  10  0.407091   1.0000    nonperiodic-templates
17   8  19  12  13  16   8  13  12  10  0.299251   0.9922    nonperiodic-templates
15  15  12  12  16   9  19  14   4  12  0.148094   0.9922    nonperiodic-templates
15  14  11  10  10  21   7  13  13  14  0.264458   0.9922    nonperiodic-templates
16  16  13  14  13  10  10   9  18   9  0.500934   0.9844    nonperiodic-templates
19   5  16  13  14   6  13  14  16  12  0.100508   0.9922    nonperiodic-templates
13  13  17  11  13  12  11  14  12  12  0.970538   1.0000    nonperiodic-templates
16  13   8  12   9  15  18  13  15   9  0.452799   1.0000    nonperiodic-templates
14  13  11   9  19  14  10  12  10  16  0.568055   1.0000    nonperiodic-templates
10  13  18  12  20  13  13  10  10   9  0.324180   1.0000    nonperiodic-templates
9  18   9  12  13  12  13  14  12  16  0.706149   1.0000    nonperiodic-templates
11  15  12  13  14  11  18  11  14   9  0.788728   0.9922    nonperiodic-templates
11  19  12  14  10  10  17  14  10  11  0.534146   1.0000    nonperiodic-templates
9  14  12   9  18   8  13  13  15  17  0.422034   1.0000    nonperiodic-templates
11  13   8  13  17   8  12  18  18  10  0.253551   0.9922    nonperiodic-templates
10  15  17  16   8  18  12   5  11  16  0.110952   0.9922    nonperiodic-templates
7  14  11  10  10  25  16  11  14  10  0.022503   0.9922    nonperiodic-templates
7  18  12  14   8  10  15  18  11  15  0.232760   1.0000    nonperiodic-templates
16  17  11  16  11  14  15  12   8   8  0.468595   0.9844    nonperiodic-templates
12  11  14  13  12  13   5  20  14  14  0.299251   0.9688    nonperiodic-templates
13  10  15  13  18  17  12  12   6  12  0.407091   0.9922    nonperiodic-templates
14  16  17  10  14  10  14   8  14  11  0.654467   0.9766    overlapping-templates
17  10  16  11  16   9  13  11   9  16  0.517442   0.9609    universal
17  12  10  18  11  12  13  11  11  13  0.756476   1.0000    apen
7   5  10   9  11   9   7   8   7   8  0.947557   1.0000    random-excursions
7  13   5   5   6  12   4  10  11   8  0.235285   0.9877    random-excursions
3   4   7  14   8   7   9   9   7  13  0.146359   1.0000    random-excursions
11   6   6   9   8   9   7  10   6   9  0.934318   1.0000    random-excursions
6   6   9   9   5  10   8   8   8  12  0.845066   1.0000    random-excursions
7  16   4   6   5   8   8  11  12   4  0.050710   0.9877    random-excursions
10   6   6   6   8  12  11   7   6   9  0.752361   1.0000    random-excursions
6  10   7   6  10   7   9  11  10   5  0.823278   1.0000    random-excursions
7  10   7   7  12   7   7   9   6   9  0.919445   0.9877    random-excursions-variant
8   5   8  10   7  11   4  13   7   8  0.521600   0.9877    random-excursions-variant
8   7   7   8   8  11   4   6   9  13  0.624107   0.9877    random-excursions-variant
8   7   8   9   8  10   4  10  10   7  0.919445   1.0000    random-excursions-variant
7  10   7   6   7   5  11  13   7   8  0.650132   1.0000    random-excursions-variant
7   5  12   9  11   5   3   7   7  15  0.087559   1.0000    random-excursions-variant
7   8   5  13   8   7  10  11   5   7  0.598138   1.0000    random-excursions-variant
5   8   7   8   6  14  11   5   8   9  0.472584   0.9753    random-excursions-variant
5   6   9  10   6   9  11   9   8   8  0.902994   0.9753    random-excursions-variant
6   7  11   9   9   8   7   6  10   8  0.959132   0.9877    random-excursions-variant
5   8  14  10   9   7   5  12   4   7  0.235285   1.0000    random-excursions-variant
9  10   7  10   6   6   9   8   8   8  0.984058   0.9877    random-excursions-variant
12   6   3  17   7   3   6   9   7  11  0.013217   0.9877    random-excursions-variant
11   5   6   7  14   7  11   7   5   8  0.360699   0.9753    random-excursions-variant
5   9   8  10   9   8   3   8  13   8  0.521600   0.9630    random-excursions-variant
5   8   8  10  13   3   7  11   8   8  0.425817   0.9753    random-excursions-variant
5   7  13   9  11   6  10   8   9   3  0.360699   0.9877    random-excursions-variant
5  10  10   7  13   9   8   2  10   7  0.302291   0.9877    random-excursions-variant
11  19  13  16  12  12  10  14  10  11  0.671779   0.9922    serial
17  10   9  14  12  21  17  11   7  10  0.095617   0.9922    serial
15  10  13  12  12  14  11  18   8  15  0.671779   0.9922    linear-complexity

- - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - -
The minimum pass rate for each statistical test with the exception of the random
excursion (variant) test is approximately = 0.963616 for a sample size = 128
binary sequences.

The minimum pass rate for the random excursion (variant) test is approximately
0.956834 for a sample size = 81 binary sequences.

For further guidelines construct a probability table using the MAPLE program
provided in the addendum section of the documentation.
- - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - -
```

For each test a chart is generated which shows the expectation for a random sequence as a solid line with the results from the data being tested plotted as a series of red dots for each sub-sequence tested. The following are the charts for the HotBits data.

Frequency

Block Frequency

Frequency Test within a Block

Runs

Longest Run of Ones

Binary Matrix Rank

Discrete Fourier Transform

Nonperiodic Template Matching

Overlapping Template Matching

Maurer's “Universal Statistical” Test

Approximate Entropy

Random Excursions

Random Excursions Variant

Serial

Linear Complexity

In addition to the test summary and graphs, detailed results from the various tests are written to a number of data files. You can download a ZIPped archive containing all of these results, and the HotBits data stream which was tested to produce them from the links below.

## Generator Quality Monitoring

Each HotBits generator machine performs a continuous quality control check on the random data it produces. For each block of random data generated, a specified percentage (10% for the Fourmilab generators) of additional data is generated and subjected to the ENT tests described above. The results of these tests, both for all bytes produced since the HotBits generator was started and those generated since the last status report was requested, are reported in the server status report as shown below. This status report is available only on the Fourmilab local network—it is not accessible over the Internet.

Quality control testing is performed on bytes generated expressly for that purpose and discarded immediately after being tested. The random bytes produced for inventory and returned to requesters have not been examined by any program or human before being delivered.

# HotBits Server Status Report

Server hotbits0.dmz.fourmilab.ch up since Monday, 25 September 2006 15:12:14 CEST
Requests processed: 10006, 10001 OK, 5 rejected
Unauthorised access attempts: 0
Bytes returned: 10239757
Inventory length: 7103488
Inventory file (/server/var/hbproxy/inventory.dat) length: 7103488, valid.
Bytes in current buffer: 243
Bytes built for inventory: 9696256
Seconds spent building inventory: 92926, 104 bytes/second
Quota queue length: 0

Quality measurements on a sample of 965838 bytes generated
since the server was started:

Entropy: 7.999825 bits per byte.
Chi-square distribution: 233.74 (random exceeds this 75.00% of the time).
Mean value: 127.5374 (expectation: 127.5).
Monte Carlo π value: 3.141470930 (error 0.00 percent).
Serial correlation coefficient: 0.000885 (expectation 0).

Quality measurements on a sample of 159018 bytes generated
since Friday, 29 September 2006 13:39:55 CEST:

Entropy: 7.998742 bits per byte.
Chi-square distribution: 277.54 (random exceeds this 25.00% of the time).
Mean value: 127.8514 (expectation: 127.5).
Monte Carlo π value: 3.138663548 (error 0.09 percent).
Serial correlation coefficient: 0.002713 (expectation 0).

Release 3.0, September 2006
Built on Sep 25 2006 at 15:12:09

by John Walker
September, 2006