Producing True Random Numbers at 44Kbits/sec on a common PC I found that FM radio hiss can be digitized and processed to yield true, quality random numbers which pass the Diehard randomness tests. A cheap FM radio was tuned to hiss at the high end of the FM band. The radio's earphone-out signal was fed into ordinary computer soundcard and digitized at 16bits/sample at 22Ksamples/sec. Various spectral-analysis programs showed that the FM hiss was fairly white analog noise, however the raw data did not pass Diehard tests. I then processed the raw data: The parity of each raw byte was shifted into a register until a byte accumulated, which was then output. The resulting data, 1/8 the size of the raw data, passes Diehard randomness tests with no additional processing. Diehard output: BIRTHDAY SPACINGS TEST, M= 512 N=2**24 LAMBDA= 2.0000 bigsimple.bin using bits 1 to 24 p-value= .215128 bigsimple.bin using bits 2 to 25 p-value= .749499 bigsimple.bin using bits 3 to 26 p-value= .096196 bigsimple.bin using bits 4 to 27 p-value= .658912 bigsimple.bin using bits 5 to 28 p-value= .017458 bigsimple.bin using bits 6 to 29 p-value= .574795 bigsimple.bin using bits 7 to 30 p-value= .057843 bigsimple.bin using bits 8 to 31 p-value= .755589 bigsimple.bin using bits 9 to 32 p-value= .088504 The 9 p-values were .215128 .749499 .096196 .658912 .017458 .574795 .057843 .755589 .088504 A KSTEST for the 9 p-values yields .896831 ---------------------------------------------------------------------------- ---- OPERM5 test for file bigsimple.bin chisquare for 99 degrees of freedom= 87.229; p-value= .204786 OPERM5 test for file bigsimple.bin chisquare for 99 degrees of freedom=159.365; p-value= .999882 ---------------------------------------------------------------------------- ---- Binary rank test for bigsimple.bin Rank test for 31x31 binary matrices: rows from leftmost 31 bits of each 32-bit integer rank observed expected (o-e)^2/e sum 28 199 211.4 .729394 .729 29 5138 5134.0 .003101 .732 30 23023 23103.0 .277344 1.010 31 11640 11551.5 .677653 1.687 chisquare= 1.687 for 3 d. of f.; p-value= .457248 Binary rank test for bigsimple.bin Rank test for 32x32 binary matrices: rows from leftmost 32 bits of each 32-bit integer rank observed expected (o-e)^2/e sum 29 206 211.4 .138848 .139 30 5156 5134.0 .094185 .233 31 23196 23103.0 .373989 .607 32 11442 11551.5 1.038443 1.645 chisquare= 1.645 for 3 d. of f.; p-value= .450664 ---------------------------------------------------------------------------- ---- b-rank test for bits 1 to 8 p=1-exp(-SUM/2)= .14013 b-rank test for bits 2 to 9 p=1-exp(-SUM/2)= .95786 b-rank test for bits 3 to 10 p=1-exp(-SUM/2)= .23685 b-rank test for bits 4 to 11 p=1-exp(-SUM/2)= .54937 b-rank test for bits 5 to 12 p=1-exp(-SUM/2)= .60969 b-rank test for bits 6 to 13 p=1-exp(-SUM/2)= .62964 b-rank test for bits 7 to 14 p=1-exp(-SUM/2)= .89250 b-rank test for bits 8 to 15 p=1-exp(-SUM/2)= .78444 b-rank test for bits 9 to 16 p=1-exp(-SUM/2)= .55013 b-rank test for bits 10 to 17 p=1-exp(-SUM/2)= .23741 b-rank test for bits 11 to 18 p=1-exp(-SUM/2)= .03669 b-rank test for bits 12 to 19 p=1-exp(-SUM/2)= .04415 b-rank test for bits 13 to 20 p=1-exp(-SUM/2)= .62760 b-rank test for bits 14 to 21 p=1-exp(-SUM/2)= .61393 b-rank test for bits 15 to 22 p=1-exp(-SUM/2)= .96515 b-rank test for bits 16 to 23 p=1-exp(-SUM/2)= .08958 b-rank test for bits 17 to 24 p=1-exp(-SUM/2)= .41657 b-rank test for bits 18 to 25 p=1-exp(-SUM/2)= .57191 b-rank test for bits 19 to 26 p=1-exp(-SUM/2)= .88550 b-rank test for bits 20 to 27 p=1-exp(-SUM/2)= .08824 b-rank test for bits 21 to 28 p=1-exp(-SUM/2)= .71721 b-rank test for bits 22 to 29 p=1-exp(-SUM/2)= .54291 b-rank test for bits 23 to 30 p=1-exp(-SUM/2)= .67989 b-rank test for bits 24 to 31 p=1-exp(-SUM/2)= .53742 b-rank test for bits 25 to 32 p=1-exp(-SUM/2)= .17990 TEST SUMMARY, 25 tests on 100,000 random 6x8 matrices These should be 25 uniform [0,1] random variables: .140129 .957865 .236846 .549372 .609694 .629638 .892499 .784442 .550129 .237412 .036688 .044150 .627598 .613930 .965150 .089578 .416568 .571912 .885501 .088241 .717215 .542912 .679889 .537418 .179904 brank test summary for bigsimple.bin The KS test for those 25 supposed UNI's yields KS p-value= .206759 ---------------------------------------------------------------------------- ---- No. missing words should average 141909. with sigma=428. tst no 1: 142384 missing words, 1.11 sigmas from mean, p-value= .86629 tst no 2: 142777 missing words, 2.03 sigmas from mean, p-value= .97868 tst no 3: 143026 missing words, 2.61 sigmas from mean, p-value= .99546 tst no 4: 142308 missing words, .93 sigmas from mean, p-value= .82420 tst no 5: 142173 missing words, .62 sigmas from mean, p-value= .73107 tst no 6: 142022 missing words, .26 sigmas from mean, p-value= .60382 tst no 7: 141931 missing words, .05 sigmas from mean, p-value= .52019 tst no 8: 142314 missing words, .95 sigmas from mean, p-value= .82780 tst no 9: 141482 missing words, -1.00 sigmas from mean, p-value= .15904 tst no 10: 141443 missing words, -1.09 sigmas from mean, p-value= .13796 tst no 11: 141796 missing words, -.26 sigmas from mean, p-value= .39559 tst no 12: 142268 missing words, .84 sigmas from mean, p-value= .79899 tst no 13: 141365 missing words, -1.27 sigmas from mean, p-value= .10172 tst no 14: 142147 missing words, .56 sigmas from mean, p-value= .71066 tst no 15: 141423 missing words, -1.14 sigmas from mean, p-value= .12792 tst no 16: 142599 missing words, 1.61 sigmas from mean, p-value= .94645 tst no 17: 142520 missing words, 1.43 sigmas from mean, p-value= .92318 tst no 18: 141225 missing words, -1.60 sigmas from mean, p-value= .05492 tst no 19: 141413 missing words, -1.16 sigmas from mean, p-value= .12310 tst no 20: 141194 missing words, -1.67 sigmas from mean, p-value= .04733 ---------------------------------------------------------------------------- ---- OPSO for bigsimple.bin using bits 23 to 32 141802 -.370 .3557 OPSO for bigsimple.bin using bits 22 to 31 141789 -.415 .3391 OPSO for bigsimple.bin using bits 21 to 30 141377 -1.836 .0332 OPSO for bigsimple.bin using bits 20 to 29 142196 .989 .8386 OPSO for bigsimple.bin using bits 19 to 28 142155 .847 .8015 OPSO for bigsimple.bin using bits 18 to 27 142171 .902 .8166 OPSO for bigsimple.bin using bits 17 to 26 141331 -1.994 .0231 OPSO for bigsimple.bin using bits 16 to 25 141565 -1.187 .1175 OPSO for bigsimple.bin using bits 15 to 24 141550 -1.239 .1077 OPSO for bigsimple.bin using bits 14 to 23 142197 .992 .8394 OPSO for bigsimple.bin using bits 13 to 22 141930 .071 .5284 OPSO for bigsimple.bin using bits 12 to 21 141593 -1.091 .1377 OPSO for bigsimple.bin using bits 11 to 20 141585 -1.118 .1317 OPSO for bigsimple.bin using bits 10 to 19 141418 -1.694 .0451 OPSO for bigsimple.bin using bits 9 to 18 141044 -2.984 .0014 OPSO for bigsimple.bin using bits 8 to 17 141908 -.005 .4982 OPSO for bigsimple.bin using bits 7 to 16 141648 -.901 .1838 OPSO for bigsimple.bin using bits 6 to 15 141884 -.087 .4652 OPSO for bigsimple.bin using bits 5 to 14 142362 1.561 .9407 OPSO for bigsimple.bin using bits 4 to 13 141924 .051 .5202 OPSO for bigsimple.bin using bits 3 to 12 141289 -2.139 .0162 OPSO for bigsimple.bin using bits 2 to 11 142145 .813 .7918 OPSO for bigsimple.bin using bits 1 to 10 141728 -.625 .2659 OQSO for bigsimple.bin using bits 28 to 32 142157 .840 .7994 OQSO for bigsimple.bin using bits 27 to 31 141539 -1.255 .1047 OQSO for bigsimple.bin using bits 26 to 30 141511 -1.350 .0885 OQSO for bigsimple.bin using bits 25 to 29 141902 -.025 .4901 OQSO for bigsimple.bin using bits 24 to 28 141674 -.798 .2125 OQSO for bigsimple.bin using bits 23 to 27 141837 -.245 .4032 OQSO for bigsimple.bin using bits 22 to 26 142276 1.243 .8931 OQSO for bigsimple.bin using bits 21 to 25 142511 2.040 .9793 OQSO for bigsimple.bin using bits 20 to 24 142223 1.063 .8562 OQSO for bigsimple.bin using bits 19 to 23 141683 -.767 .2215 OQSO for bigsimple.bin using bits 18 to 22 141936 .090 .5360 OQSO for bigsimple.bin using bits 17 to 21 141337 -1.940 .0262 OQSO for bigsimple.bin using bits 16 to 20 141959 .168 .5669 OQSO for bigsimple.bin using bits 15 to 19 141975 .223 .5881 OQSO for bigsimple.bin using bits 14 to 18 142166 .870 .8079 OQSO for bigsimple.bin using bits 13 to 17 141738 -.581 .2807 OQSO for bigsimple.bin using bits 12 to 16 141628 -.954 .1701 OQSO for bigsimple.bin using bits 11 to 15 141781 -.435 .3318 OQSO for bigsimple.bin using bits 10 to 14 142112 .687 .7540 OQSO for bigsimple.bin using bits 9 to 13 141920 .036 .5144 OQSO for bigsimple.bin using bits 8 to 12 141668 -.818 .2067 OQSO for bigsimple.bin using bits 7 to 11 141655 -.862 .1943 OQSO for bigsimple.bin using bits 6 to 10 141984 .253 .5999 OQSO for bigsimple.bin using bits 5 to 9 141585 -1.099 .1358 OQSO for bigsimple.bin using bits 4 to 8 142013 .351 .6374 OQSO for bigsimple.bin using bits 3 to 7 141833 -.259 .3979 OQSO for bigsimple.bin using bits 2 to 6 141671 -.808 .2096 OQSO for bigsimple.bin using bits 1 to 5 142007 .331 .6297 DNA for bigsimple.bin using bits 31 to 32 142254 1.017 .8454 DNA for bigsimple.bin using bits 30 to 31 142326 1.229 .8905 DNA for bigsimple.bin using bits 29 to 30 142408 1.471 .9294 DNA for bigsimple.bin using bits 28 to 29 142186 .816 .7928 DNA for bigsimple.bin using bits 27 to 28 141882 -.081 .4679 DNA for bigsimple.bin using bits 26 to 27 142132 .657 .7444 DNA for bigsimple.bin using bits 25 to 26 141958 .144 .5571 DNA for bigsimple.bin using bits 24 to 25 141857 -.154 .4387 DNA for bigsimple.bin using bits 23 to 24 141521 -1.146 .1260 DNA for bigsimple.bin using bits 22 to 23 142664 2.226 .9870 DNA for bigsimple.bin using bits 21 to 22 141448 -1.361 .0868 DNA for bigsimple.bin using bits 20 to 21 142090 .533 .7030 DNA for bigsimple.bin using bits 19 to 20 141867 -.125 .4503 DNA for bigsimple.bin using bits 18 to 19 141894 -.045 .4820 DNA for bigsimple.bin using bits 17 to 18 141463 -1.317 .0940 DNA for bigsimple.bin using bits 16 to 17 141587 -.951 .1708 DNA for bigsimple.bin using bits 15 to 16 142088 .527 .7009 DNA for bigsimple.bin using bits 14 to 15 141372 -1.585 .0565 DNA for bigsimple.bin using bits 13 to 14 141896 -.039 .4843 DNA for bigsimple.bin using bits 12 to 13 142256 1.023 .8468 DNA for bigsimple.bin using bits 11 to 12 141172 -2.175 .0148 DNA for bigsimple.bin using bits 10 to 11 141524 -1.137 .1278 DNA for bigsimple.bin using bits 9 to 10 141700 -.617 .2685 DNA for bigsimple.bin using bits 8 to 9 142180 .798 .7877 DNA for bigsimple.bin using bits 7 to 8 141782 -.376 .3536 DNA for bigsimple.bin using bits 6 to 7 142054 .427 .6652 DNA for bigsimple.bin using bits 5 to 6 142370 1.359 .9129 DNA for bigsimple.bin using bits 4 to 5 142032 .362 .6413 DNA for bigsimple.bin using bits 3 to 4 141471 -1.293 .0980 DNA for bigsimple.bin using bits 2 to 3 142013 .306 .6201 DNA for bigsimple.bin using bits 1 to 2 141754 -.458 .3234 ---------------------------------------------------------------------------- ---- Test results for bigsimple.bin 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 bigsimple.bin 2480.79 -.272 .392963 byte stream for bigsimple.bin 2544.61 .631 .735960 ---------------------------------------------------------------------------- ---- 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 2539.84 .563 .713407 bits 2 to 9 2462.74 -.527 .299128 bits 3 to 10 2475.45 -.347 .364216 bits 4 to 11 2352.58 -2.085 .018545 bits 5 to 12 2486.65 -.189 .425128 bits 6 to 13 2473.02 -.382 .351384 bits 7 to 14 2655.21 2.195 .985918 bits 8 to 15 2536.20 .512 .695677 bits 9 to 16 2532.48 .459 .677002 bits 10 to 17 2314.27 -2.627 .004311 bits 11 to 18 2605.76 1.496 .932629 bits 12 to 19 2588.76 1.255 .895302 bits 13 to 20 2486.16 -.196 .422396 bits 14 to 21 2496.38 -.051 .479582 bits 15 to 22 2425.24 -1.057 .145196 bits 16 to 23 2490.05 -.141 .444052 bits 17 to 24 2373.30 -1.792 .036578 bits 18 to 25 2356.86 -2.024 .021470 bits 19 to 26 2557.84 .818 .793310 bits 20 to 27 2590.32 1.277 .899259 bits 21 to 28 2467.57 -.459 .323273 bits 22 to 29 2496.07 -.056 .477834 bits 23 to 30 2530.74 .435 .668114 bits 24 to 31 2474.58 -.360 .359599 bits 25 to 32 2580.29 1.135 .871906 ---------------------------------------------------------------------------- ---- CDPARK: result of ten tests on file bigsimple.bin Of 12,000 tries, the average no. of successes should be 3523 with sigma=21.9 Successes: 3529 z-score: .274 p-value: .607947 Successes: 3521 z-score: -.091 p-value: .463618 Successes: 3526 z-score: .137 p-value: .554479 Successes: 3536 z-score: .594 p-value: .723613 Successes: 3517 z-score: -.274 p-value: .392053 Successes: 3539 z-score: .731 p-value: .767486 Successes: 3524 z-score: .046 p-value: .518210 Successes: 3513 z-score: -.457 p-value: .323972 Successes: 3541 z-score: .822 p-value: .794438 Successes: 3527 z-score: .183 p-value: .572463 square size avg. no. parked sample sigma 100. 3527.300 8.736 KSTEST for the above 10: p= .804330 ---------------------------------------------------------------------------- ---- This is the MINIMUM DISTANCE test for random integers in the file bigsimple.bin Sample no. d^2 avg equiv uni 5 .1930 1.0055 .176285 10 .5596 .9379 .430160 15 1.1844 1.0280 .695890 20 .5422 .8555 .420105 25 .9861 .9044 .628811 30 .2861 .8935 .249863 35 .1485 .9897 .138682 40 .2950 .9306 .256592 45 .6921 .8989 .501193 50 .1139 .9307 .108141 55 1.0898 .9325 .665562 60 .0072 .9084 .007231 65 .6066 .8989 .456462 70 .7895 .9008 .547730 75 .3097 .8698 .267450 80 .4391 .8918 .356821 85 .8823 .8699 .588018 90 .2776 .9121 .243421 95 1.3020 .9192 .729777 100 2.3327 .9325 .904094 MINIMUM DISTANCE TEST for bigsimple.bin Result of KS test on 20 transformed mindist^2's: p-value= .157745 ---------------------------------------------------------------------------- ---- The 3DSPHERES test for file bigsimple.bin sample no: 1 r^3= 61.647 p-value= .87189 sample no: 2 r^3= 61.307 p-value= .87043 sample no: 3 r^3= 21.447 p-value= .51076 sample no: 4 r^3= 2.684 p-value= .08559 sample no: 5 r^3= 1.207 p-value= .03942 sample no: 6 r^3= 12.684 p-value= .34479 sample no: 7 r^3= 24.098 p-value= .55214 sample no: 8 r^3= .287 p-value= .00951 sample no: 9 r^3= 68.533 p-value= .89817 sample no: 10 r^3= 15.201 p-value= .39751 sample no: 11 r^3= 37.780 p-value= .71616 sample no: 12 r^3= 48.397 p-value= .80076 sample no: 13 r^3= 26.584 p-value= .58775 sample no: 14 r^3= 29.700 p-value= .62842 sample no: 15 r^3= 8.969 p-value= .25841 sample no: 16 r^3= 43.698 p-value= .76697 sample no: 17 r^3= 61.285 p-value= .87034 sample no: 18 r^3= 30.428 p-value= .63733 sample no: 19 r^3= 54.608 p-value= .83802 sample no: 20 r^3= 19.998 p-value= .48655 3DSPHERES test for file bigsimple.bin p-value= .595668 ---------------------------------------------------------------------------- ---- RESULTS OF SQUEEZE TEST FOR bigsimple.bin Table of standardized frequency counts ( (obs-exp)/sqrt(exp) )^2 for j taking values <=6,7,8,...,47,>=48: 2.0 .1 1.3 1.6 .8 -1.0 .0 -.1 -2.2 -1.0 -1.9 .7 .5 -.3 .9 -.4 .6 -1.3 .5 -.8 .4 -.2 1.6 -.5 .5 1.6 -.1 .0 -1.1 1.2 .8 .6 .2 -1.0 -.1 2.2 .3 1.1 .5 -1.8 -.6 .0 -1.1 Chi-square with 42 degrees of freedom: 45.808 z-score= .415 p-value= .683048 ______________________________________________________________ ---------------------------------------------------------------------------- ---- Test no. 1 p-value .130689 Test no. 2 p-value .697905 Test no. 3 p-value .312460 Test no. 4 p-value .031315 Test no. 5 p-value .961343 Test no. 6 p-value .483192 Test no. 7 p-value .554386 Test no. 8 p-value .960960 Test no. 9 p-value .767135 Test no. 10 p-value .146497 Results of the OSUM test for bigsimple.bin KSTEST on the above 10 p-values: .135933 ---------------------------------------------------------------------------- ---- The RUNS test for file bigsimple.bin Up and down runs in a sample of 10000 _________________________________________________ Run test for bigsimple.bin : runs up; ks test for 10 p's: .368663 runs down; ks test for 10 p's: .177188 Run test for bigsimple.bin : runs up; ks test for 10 p's: .199189 runs down; ks test for 10 p's: .860525 ---------------------------------------------------------------------------- ---- Results of craps test for bigsimple.bin No. of wins: Observed Expected 98315 98585.86 Chisq= 26.08 for 20 degrees of freedom, p= .83666 Throws Observed Expected Chisq Sum SUMMARY FOR bigsimple.bin p-value for no. of wins: .112863 p-value for throws/game: .836658 Test completed. File bigsimple.bin :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::: honig@alum.mit.edu "Speech is not protected simply because it is written in a language" Federal Misjudge Gwin on the Bernstein Case