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The following archive was created by hippie-mail 7.98617-22 on Thu May 27 1999 - 23:44:22
mgraffam@idsi.net
Fri, 9 Apr 1999 17:01:37 -0400 (EDT)
Statistical Test of the Linux (2.2.3) /dev/random
Michael Graffam (mgraffam@idsi.net) April 9th 1999
The machine used in this test is an Intel P5 based machine, rated at
150 MHz but overclocked to 200 MHz, using a Triton VX chipset. The IDE
drives are a Seagate Medalist ST31277A (hda) and a Western Digital Caviar
AC31600 (hdb). This box is running the Linux 2.2.3 kernel on an otherwise
essentially (for the purposes here) stock Redhat 5.2 system.
To generate 11M of random bits from /dev/random the machine was issued
the following commands:
bash# dd if=/dev/random of=random.output &
bash# cat < /dev/hda1 > /dev/null &
bash# cat < /dev/hdb2 > /dev/null &
bash# cat < /dev/hda2 > /dev/null &
bash# cat < /dev/hdb1 > /dev/null &
The hard drives were ran as such in order to generate enough IRQ firings
to keep /dev/random generating bits, the machine was also under general
use at this time generating further entropy via keystrokes and
mouse movements. The cat commands were repeated as needed in order to
get the required 11M to run Diehard.
This can be see as a less than perfect scenario, by running the drives in
this manner, the IRQ timings may become predictable to an attacker. An
attack like this could be conducted by forcing the machine to eat swap
space. Such an attack could be done remotely, perhaps by exploiting a
memory leak in a daemon. As the machine starts swapping, the attacker
may be able to piece together what the entropy pool will looks like, and
can then stands a chance of predicting the output of /dev/random;
allowing an attack on, say, IPSec. This is especially true of a server
where no human input will be available.
The average bit rate from /dev/random under these circumstances was
around 1024 to 1523 bytes/sec.
The author does not believe that statistical tests on the output of
/dev/random is cryptographically very meaningful. Knowing that the
/dev/random entropy pool is hashed with SHA prior to being exported to the
character device, it is clear that the output will be uniformly
distributed. To judge the usefulness of /dev/random one would really
need to have a detailed analysis of the estimated entropy gained by the
IRQ timings and such to ensure that the random.c code is not generating
more bits than it reasonably can from the given source material.
But, for the sake of the statistics, the Diehard output is appended below.
The /dev/random output passed all of the Diehard tests.
************************************************************************
BIRTHDAY SPACINGS TEST, M= 512 N=2**24 LAMBDA= 2.0000
Results for random.output
For a sample of size 500: mean
random.output using bits 1 to 24 1.948
duplicate number number
spacings observed expected
0 76. 67.668
1 131. 135.335
2 140. 135.335
3 79. 90.224
4 52. 45.112
5 15. 18.045
6 to INF 7. 8.282
Chisquare with 6 d.o.f. = 4.49 p-value= .388760
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 2 to 25 2.028
duplicate number number
spacings observed expected
0 59. 67.668
1 140. 135.335
2 140. 135.335
3 87. 90.224
4 47. 45.112
5 19. 18.045
6 to INF 8. 8.282
Chisquare with 6 d.o.f. = 1.69 p-value= .053817
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 3 to 26 1.938
duplicate number number
spacings observed expected
0 70. 67.668
1 136. 135.335
2 151. 135.335
3 73. 90.224
4 46. 45.112
5 18. 18.045
6 to INF 6. 8.282
Chisquare with 6 d.o.f. = 5.83 p-value= .557618
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 4 to 27 1.918
duplicate number number
spacings observed expected
0 71. 67.668
1 142. 135.335
2 134. 135.335
3 94. 90.224
4 35. 45.112
5 19. 18.045
6 to INF 5. 8.282
Chisquare with 6 d.o.f. = 4.28 p-value= .361313
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 5 to 28 2.090
duplicate number number
spacings observed expected
0 61. 67.668
1 133. 135.335
2 132. 135.335
3 94. 90.224
4 52. 45.112
5 17. 18.045
6 to INF 11. 8.282
Chisquare with 6 d.o.f. = 2.94 p-value= .183906
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 6 to 29 2.002
duplicate number number
spacings observed expected
0 61. 67.668
1 143. 135.335
2 131. 135.335
3 98. 90.224
4 41. 45.112
5 20. 18.045
6 to INF 6. 8.282
Chisquare with 6 d.o.f. = 3.12 p-value= .205790
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 7 to 30 2.026
duplicate number number
spacings observed expected
0 63. 67.668
1 136. 135.335
2 143. 135.335
3 80. 90.224
4 47. 45.112
5 25. 18.045
6 to INF 6. 8.282
Chisquare with 6 d.o.f. = 5.31 p-value= .494844
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 8 to 31 1.982
duplicate number number
spacings observed expected
0 70. 67.668
1 125. 135.335
2 147. 135.335
3 92. 90.224
4 44. 45.112
5 14. 18.045
6 to INF 8. 8.282
Chisquare with 6 d.o.f. = 2.85 p-value= .173022
:::::::::::::::::::::::::::::::::::::::::
For a sample of size 500: mean
random.output using bits 9 to 32 1.994
duplicate number number
spacings observed expected
0 75. 67.668
1 127. 135.335
2 134. 135.335
3 93. 90.224
4 46. 45.112
5 14. 18.045
6 to INF 11. 8.282
Chisquare with 6 d.o.f. = 3.22 p-value= .219586
:::::::::::::::::::::::::::::::::::::::::
The 9 p-values were
.388760 .053817 .557618 .361313 .183906
.205790 .494844 .173022 .219586
A KSTEST for the 9 p-values yields .959368
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
--------------------------------------------------------------------------------
OPERM5 test for file random.output
For a sample of 1,000,000 consecutive 5-tuples,
chisquare for 99 degrees of freedom= 85.480; p-value= .168319
OPERM5 test for file random.output
For a sample of 1,000,000 consecutive 5-tuples,
chisquare for 99 degrees of freedom= 81.640; p-value= .102675
--------------------------------------------------------------------------------
Binary rank test for random.output
Rank test for 31x31 binary matrices:
rows from leftmost 31 bits of each 32-bit integer
rank observed expected (o-e)^2/e sum
28 214 211.4 .031533 .032
29 5208 5134.0 1.066317 1.098
30 23080 23103.0 .022991 1.121
31 11498 11551.5 .248007 1.369
chisquare= 1.369 for 3 d. of f.; p-value= .408153
--------------------------------------------------------------
Binary rank test for random.output
Rank test for 32x32 binary matrices:
rows from leftmost 32 bits of each 32-bit integer
rank observed expected (o-e)^2/e sum
29 224 211.4 .748784 .749
30 5112 5134.0 .094361 .843
31 23162 23103.0 .150433 .994
32 11502 11551.5 .212324 1.206
chisquare= 1.206 for 3 d. of f.; p-value= .384484
--------------------------------------------------------------
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
--------------------------------------------------------------------------------
Binary Rank Test for random.output
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 1 to 8
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 973 944.3 .872 .872
r =5 22043 21743.9 4.114 4.986
r =6 76984 77311.8 1.390 6.376
p=1-exp(-SUM/2)= .95875
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 2 to 9
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 985 944.3 1.754 1.754
r =5 21621 21743.9 .695 2.449
r =6 77394 77311.8 .087 2.536
p=1-exp(-SUM/2)= .71862
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 3 to 10
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 919 944.3 .678 .678
r =5 21767 21743.9 .025 .702
r =6 77314 77311.8 .000 .703
p=1-exp(-SUM/2)= .29620
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 4 to 11
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 997 944.3 2.941 2.941
r =5 21687 21743.9 .149 3.090
r =6 77316 77311.8 .000 3.090
p=1-exp(-SUM/2)= .78670
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 5 to 12
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 929 944.3 .248 .248
r =5 21887 21743.9 .942 1.190
r =6 77184 77311.8 .211 1.401
p=1-exp(-SUM/2)= .50366
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 6 to 13
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 966 944.3 .499 .499
r =5 21791 21743.9 .102 .601
r =6 77243 77311.8 .061 .662
p=1-exp(-SUM/2)= .28175
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 7 to 14
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 968 944.3 .595 .595
r =5 21852 21743.9 .537 1.132
r =6 77180 77311.8 .225 1.357
p=1-exp(-SUM/2)= .49259
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 8 to 15
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 979 944.3 1.275 1.275
r =5 21708 21743.9 .059 1.334
r =6 77313 77311.8 .000 1.334
p=1-exp(-SUM/2)= .48683
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 9 to 16
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 971 944.3 .755 .755
r =5 21710 21743.9 .053 .808
r =6 77319 77311.8 .001 .808
p=1-exp(-SUM/2)= .33249
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 10 to 17
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 915 944.3 .909 .909
r =5 21937 21743.9 1.715 2.624
r =6 77148 77311.8 .347 2.971
p=1-exp(-SUM/2)= .77362
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 11 to 18
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 961 944.3 .295 .295
r =5 21488 21743.9 3.012 3.307
r =6 77551 77311.8 .740 4.047
p=1-exp(-SUM/2)= .86781
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 12 to 19
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 954 944.3 .100 .100
r =5 21629 21743.9 .607 .707
r =6 77417 77311.8 .143 .850
p=1-exp(-SUM/2)= .34620
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 13 to 20
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 910 944.3 1.246 1.246
r =5 21825 21743.9 .302 1.548
r =6 77265 77311.8 .028 1.577
p=1-exp(-SUM/2)= .54543
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 14 to 21
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 957 944.3 .171 .171
r =5 21648 21743.9 .423 .594
r =6 77395 77311.8 .090 .683
p=1-exp(-SUM/2)= .28939
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 15 to 22
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 983 944.3 1.586 1.586
r =5 21718 21743.9 .031 1.617
r =6 77299 77311.8 .002 1.619
p=1-exp(-SUM/2)= .55490
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 16 to 23
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 951 944.3 .048 .048
r =5 21781 21743.9 .063 .111
r =6 77268 77311.8 .025 .136
p=1-exp(-SUM/2)= .06557
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 17 to 24
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 959 944.3 .229 .229
r =5 21546 21743.9 1.801 2.030
r =6 77495 77311.8 .434 2.464
p=1-exp(-SUM/2)= .70830
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 18 to 25
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 916 944.3 .848 .848
r =5 21599 21743.9 .966 1.814
r =6 77485 77311.8 .388 2.202
p=1-exp(-SUM/2)= .66743
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 19 to 26
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 944 944.3 .000 .000
r =5 21748 21743.9 .001 .001
r =6 77308 77311.8 .000 .001
p=1-exp(-SUM/2)= .00053
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 20 to 27
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 950 944.3 .034 .034
r =5 21652 21743.9 .388 .423
r =6 77398 77311.8 .096 .519
p=1-exp(-SUM/2)= .22853
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 21 to 28
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 909 944.3 1.320 1.320
r =5 21679 21743.9 .194 1.513
r =6 77412 77311.8 .130 1.643
p=1-exp(-SUM/2)= .56028
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 22 to 29
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 921 944.3 .575 .575
r =5 21706 21743.9 .066 .641
r =6 77373 77311.8 .048 .689
p=1-exp(-SUM/2)= .29159
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 23 to 30
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 942 944.3 .006 .006
r =5 21706 21743.9 .066 .072
r =6 77352 77311.8 .021 .093
p=1-exp(-SUM/2)= .04523
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 24 to 31
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 957 944.3 .171 .171
r =5 21627 21743.9 .628 .799
r =6 77416 77311.8 .140 .940
p=1-exp(-SUM/2)= .37490
Rank of a 6x8 binary matrix,
rows formed from eight bits of the RNG random.output
b-rank test for bits 25 to 32
OBSERVED EXPECTED (O-E)^2/E SUM
r<=4 862 944.3 7.173 7.173
r =5 21557 21743.9 1.607 8.780
r =6 77581 77311.8 .937 9.717
p=1-exp(-SUM/2)= .99224
TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices
These should be 25 uniform [0,1] random variables:
.958754 .718624 .296198 .786698 .503656
.281746 .492592 .486834 .332486 .773624
.867808 .346202 .545427 .289389 .554897
.065571 .708301 .667430 .000528 .228527
.560283 .291594 .045229 .374898 .992237
brank test summary for random.output
The KS test for those 25 supposed UNI's yields
KS p-value= .247997
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
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THE OVERLAPPING 20-tuples BITSTREAM TEST,
20 BITS PER WORD, 2^21 words.
This test samples the bitstream 20 times.
No. missing words should average 141909. with sigma=428.
----------------------------------- ---------------
tst no 1: 142546 missing words, 1.49 sigmas from mean, p-value= .93157
tst no 2: 141851 missing words, -.14 sigmas from mean, p-value= .44580
tst no 3: 142209 missing words, .70 sigmas from mean, p-value= .75809
tst no 4: 141864 missing words, -.11 sigmas from mean, p-value= .45783
tst no 5: 142122 missing words, .50 sigmas from mean, p-value= .69037
tst no 6: 142860 missing words, 2.22 sigmas from mean, p-value= .98683
tst no 7: 141913 missing words, .01 sigmas from mean, p-value= .50342
tst no 8: 142106 missing words, .46 sigmas from mean, p-value= .67707
tst no 9: 141059 missing words, -1.99 sigmas from mean, p-value= .02348
tst no 10: 141669 missing words, -.56 sigmas from mean, p-value= .28722
tst no 11: 141780 missing words, -.30 sigmas from mean, p-value= .38126
tst no 12: 141524 missing words, -.90 sigmas from mean, p-value= .18398
tst no 13: 141732 missing words, -.41 sigmas from mean, p-value= .33932
tst no 14: 141096 missing words, -1.90 sigmas from mean, p-value= .02870
tst no 15: 142276 missing words, .86 sigmas from mean, p-value= .80420
tst no 16: 142650 missing words, 1.73 sigmas from mean, p-value= .95823
tst no 17: 142341 missing words, 1.01 sigmas from mean, p-value= .84341
tst no 18: 141632 missing words, -.65 sigmas from mean, p-value= .25850
tst no 19: 141930 missing words, .05 sigmas from mean, p-value= .51926
tst no 20: 142146 missing words, .55 sigmas from mean, p-value= .70986
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:: This is the COUNT-THE-1's TEST on a stream of bytes. ::
:: Consider the file under test as a stream of bytes (four per ::
:: 32 bit integer). Each byte can contain from 0 to 8 1's, ::
:: with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let ::
:: the stream of bytes provide a string of overlapping 5-letter ::
:: words, each "letter" taking values A,B,C,D,E. The letters are ::
:: determined by the number of 1's in a byte:: 0,1,or 2 yield A,::
:: 3 yields B, 4 yields C, 5 yields D and 6,7 or 8 yield E. Thus ::
:: we have a monkey at a typewriter hitting five keys with vari- ::
:: ous probabilities (37,56,70,56,37 over 256). There are 5^5 ::
:: possible 5-letter words, and from a string of 256,000 (over- ::
:: lapping) 5-letter words, counts are made on the frequencies ::
:: for each word. The quadratic form in the weak inverse of ::
:: the covariance matrix of the cell counts provides a chisquare ::
:: test:: Q5-Q4, the difference of the naive Pearson sums of ::
:: (OBS-EXP)^2/EXP on counts for 5- and 4-letter cell counts. ::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Test results for random.output
Chi-square with 5^5-5^4=2500 d.of f. for sample size:2560000
chisquare equiv normal p-value
Results fo COUNT-THE-1's in successive bytes:
byte stream for random.output 2382.04 -1.668 .047633
byte stream for random.output 2478.50 -.304 .380516
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
--------------------------------------------------------------------------------
Chi-square with 5^5-5^4=2500 d.of f. for sample size: 256000
chisquare equiv normal p value
Results for COUNT-THE-1's in specified bytes:
bits 1 to 8 2460.93 -.553 .290295
bits 2 to 9 2610.08 1.557 .940242
bits 3 to 10 2510.73 .152 .560281
bits 4 to 11 2559.15 .836 .798555
bits 5 to 12 2614.67 1.622 .947556
bits 6 to 13 2411.15 -1.256 .104473
bits 7 to 14 2419.33 -1.141 .126953
bits 8 to 15 2555.44 .784 .783502
bits 9 to 16 2466.57 -.473 .318209
bits 10 to 17 2445.70 -.768 .221263
bits 11 to 18 2455.42 -.630 .264199
bits 12 to 19 2494.95 -.071 .471512
bits 13 to 20 2591.01 1.287 .900974
bits 14 to 21 2436.46 -.899 .184426
bits 15 to 22 2590.58 1.281 .899912
bits 16 to 23 2485.96 -.199 .421304
bits 17 to 24 2422.45 -1.097 .136392
bits 18 to 25 2461.10 -.550 .291102
bits 19 to 26 2510.45 .148 .558756
bits 20 to 27 2479.97 -.283 .388466
bits 21 to 28 2440.30 -.844 .199267
bits 22 to 29 2469.72 -.428 .334234
bits 23 to 30 2695.85 2.770 .997195
bits 24 to 31 2539.18 .554 .710231
bits 25 to 32 2506.48 .092 .536484
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
--------------------------------------------------------------------------------
CDPARK: result of ten tests on file random.output
Of 12,000 tries, the average no. of successes
should be 3523 with sigma=21.9
Successes: 3509 z-score: -.639 p-value: .261324
Successes: 3537 z-score: .639 p-value: .738676
Successes: 3522 z-score: -.046 p-value: .481790
Successes: 3504 z-score: -.868 p-value: .192812
Successes: 3520 z-score: -.137 p-value: .445521
Successes: 3529 z-score: .274 p-value: .607947
Successes: 3530 z-score: .320 p-value: .625377
Successes: 3497 z-score: -1.187 p-value: .117571
Successes: 3494 z-score: -1.324 p-value: .092718
Successes: 3520 z-score: -.137 p-value: .445521
square size avg. no. parked sample sigma
100. 3516.200 13.810
KSTEST for the above 10: p= .630992
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
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This is the MINIMUM DISTANCE test
for random integers in the file random.output
Sample no. d^2 avg equiv uni
5 .8575 1.1329 .577597
10 1.9224 .9637 .855147
15 .1965 .9798 .179207
20 2.5432 .9529 .922381
25 2.6101 .9813 .927433
30 1.7595 1.1496 .829377
35 .0016 1.2551 .001589
40 .6109 1.2359 .458815
45 .2836 1.2844 .247974
50 .3028 1.2476 .262375
55 .1153 1.2972 .109418
60 .1577 1.2494 .146582
65 .0104 1.2058 .010380
70 1.0942 1.2675 .667024
75 .5521 1.2626 .425883
80 2.5079 1.2635 .919578
85 .2438 1.2072 .217306
90 .9556 1.1759 .617254
95 .7719 1.1443 .539672
100 1.6521 1.1496 .809943
MINIMUM DISTANCE TEST for random.output
Result of KS test on 20 transformed mindist^2's:
p-value= .540837
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
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The 3DSPHERES test for file random.output
sample no: 1 r^3= 12.086 p-value= .33161
sample no: 2 r^3= 12.527 p-value= .34136
sample no: 3 r^3= 35.984 p-value= .69864
sample no: 4 r^3= 134.105 p-value= .98855
sample no: 5 r^3= 7.042 p-value= .20923
sample no: 6 r^3= 34.232 p-value= .68052
sample no: 7 r^3= 86.645 p-value= .94432
sample no: 8 r^3= .918 p-value= .03014
sample no: 9 r^3= 12.210 p-value= .33436
sample no: 10 r^3= 1.014 p-value= .03324
sample no: 11 r^3= 6.093 p-value= .18379
sample no: 12 r^3= 15.354 p-value= .40059
sample no: 13 r^3= 34.789 p-value= .68640
sample no: 14 r^3= 8.875 p-value= .25608
sample no: 15 r^3= 7.544 p-value= .22233
sample no: 16 r^3= 81.033 p-value= .93287
sample no: 17 r^3= 6.423 p-value= .19273
sample no: 18 r^3= 21.696 p-value= .51481
sample no: 19 r^3= .204 p-value= .00676
sample no: 20 r^3= 14.489 p-value= .38305
A KS test is applied to those 20 p-values.
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3DSPHERES test for file random.output p-value= .783965
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RESULTS OF SQUEEZE TEST FOR random.output
Table of standardized frequency counts
( (obs-exp)/sqrt(exp) )^2
for j taking values <=6,7,8,...,47,>=48:
1.3 -.3 -1.1 .9 -.8 -1.4
-.7 -.1 1.2 1.2 .3 -.3
.2 .2 .9 .9 -.6 -1.9
.3 -.6 1.8 -.8 -.3 1.7
-.3 -.1 -1.4 -.4 -1.1 -1.1
-1.3 2.7 -2.8 .3 .9 -1.7
-1.2 2.4 .1 .4 -1.3 2.0
-1.1
Chi-square with 42 degrees of freedom: 62.598
z-score= 2.247 p-value= .978755
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Test no. 1 p-value .426868
Test no. 2 p-value .584803
Test no. 3 p-value .197034
Test no. 4 p-value .037916
Test no. 5 p-value .219429
Test no. 6 p-value .556347
Test no. 7 p-value .839419
Test no. 8 p-value .024590
Test no. 9 p-value .516325
Test no. 10 p-value .530974
Results of the OSUM test for random.output
KSTEST on the above 10 p-values: .682059
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The RUNS test for file random.output
Up and down runs in a sample of 10000
_________________________________________________
Run test for random.output :
runs up; ks test for 10 p's: .183344
runs down; ks test for 10 p's: .670685
Run test for random.output :
runs up; ks test for 10 p's: .858145
runs down; ks test for 10 p's: .094168
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Results of craps test for random.output
No. of wins: Observed Expected
98687 98585.86
98687= No. of wins, z-score= .452 pvalue= .67450
Analysis of Throws-per-Game:
Chisq= 14.25 for 20 degrees of freedom, p= .18257
Throws Observed Expected Chisq Sum
1 66774 66666.7 .173 .173
2 37705 37654.3 .068 .241
3 27020 26954.7 .158 .399
4 19410 19313.5 .483 .882
5 13776 13851.4 .411 1.292
6 9998 9943.5 .298 1.590
7 6980 7145.0 3.811 5.402
8 5142 5139.1 .002 5.404
9 3691 3699.9 .021 5.425
10 2637 2666.3 .322 5.747
11 1849 1923.3 2.873 8.619
12 1406 1388.7 .215 8.834
13 980 1003.7 .560 9.394
14 745 726.1 .490 9.884
15 533 525.8 .098 9.982
16 366 381.2 .602 10.584
17 262 276.5 .764 11.348
18 208 200.8 .256 11.604
19 154 146.0 .440 12.044
20 97 106.2 .800 12.844
21 267 287.1 1.409 14.253
SUMMARY FOR random.output
p-value for no. of wins: .674495
p-value for throws/game: .182568
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Test completed. File random.output
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The following archive was created by hippie-mail 7.98617-22 on Thu May 27 1999 - 23:44:22