Investigation of the statistical probability of a bit in non-bitcoin



    The sha256 (sha256 (BlockHeader)) function, which is used in the bitcoin hashing algorithm, is considered to be very reliable. It is so reliable that there is only one way to find a golden non for the next block in the blockchain - by going through the mining process.

    Currently, the Bitcoin blockchain already has more than 530 thousand blocks. This is quite enough to conduct a statistical analysis and answer the question: "Are the values ​​of bits in previously found nones equally likely?".

    The simplest statistical analysis can be done not on the whole blockchain, but only on the last few thousand blocks. To do this, you need to take this information, for example, from the site btc.com. This blockchain explorer has a well-developed api, which we will use. Let's write a simple script that, using the requests library, downloads information about the last 2000 blocks. I am interested in only one value - nonce, which I will write to a text file along with the block number.

    import requests
    target_blk_num=538800
    num_block_for_study=2000
    txt=open("bitcoin/nonces.txt","w")
    blk_num=target_blk_num-num_block_for_study
    print("Fetching blocks from "+str(blk_num) +" to "+str(target_blk_num) )
    while blk_num<target_blk_num:
        blk_num_=format(blk_num,'d')
        url_hash="https://chain.api.btc.com/v3/block/"+blk_num_
        block=requests.get(url_hash)
        j=block.json()
        data=j['data']
        nonce=data['nonce']
        print(blk_num,format(nonce,'x'))
        txt.write("%d " % blk_num)
        txt.write("0x%08X " % nonce)
        txt.write("%d" % nonce)
        txt.write("\n")
        blk_num=blk_num+1
    txt.close()
    

    You need to run the script and wait for it to finish. Let's get this file with 2 thousand lines:

    Values ​​of nonse last blocks:
    536800 0x0A75F108 175501576
    536801 0x64046EE3 1678012131
    536802 0x43842181 1132732801
    536803 0x69D1FE65 1775369829
    536804 0x5F3A5706 1597658886
    536805 0x77E7FC5A 2011692122
    536806 0x29FC3572 704394610
    536807 0x6FA83F41 1873297217
    536808 0xB498D106 3029913862
    536809 0xCFCADF12 3486179090
    536810 0x3217C81D 840419357
    536811 0x0A4DD1A5 172872101
    536812 0xC422E8B4 3290622132
    536813 0x9512DA88 2501040776
    536814 0x5606B5C9 1443280329
    536815 0x4EAD3CC6 1319976134
    536816 0x421482FF 1108640511
    536817 0x15311B34 355539764
    536818 0xE2879E05 3800538629
    536819 0x41EB3E73 1105935987
    536820 0x29ED08CD 703400141
    536821 0x8B4FB3F7 2337256439
    536822 0x0FF2B634 267564596
    536823 0x32E51D55 853876053
    536824 0xE09DBAE1 3768433377
    536825 0xD97F44E5 3648996581
    536826 0x90A4EC5E 2426727518
    536827 0xCBF74735 3421980469
    536828 0x6D2F10F0 1831801072
    536829 0xE595A073 3851788403
    536830 0xC8765E94 3363200660
    536831 0x09DC9851 165451857
    536832 0xC2EFEA16 3270502934
    536833 0x7897622F 2023186991
    536834 0x706949F8 1885948408
    536835 0x1B619051 459378769
    536836 0xE707A346 3876037446
    536837 0xB3B0D216 3014709782
    536838 0x51B8F9E8 1371077096
    536839 0xE2207F9D 3793780637
    536840 0xD0D15245 3503379013
    536841 0x904FE5C5 2421155269
    536842 0xFF326B4B 4281494347
    536843 0x312BAA45 824945221
    536844 0x88702768 2289051496
    536845 0xD8924803 3633465347
    536846 0xEFAB5E6B 4020985451
    536847 0x76779ACB 1987549899
    536848 0x81436738 2168678200
    536849 0x02896E6A 42561130
    536850 0x76D3ED09 1993600265
    536851 0x8F2052C7 2401260231
    536852 0x6A9984CA 1788445898
    536853 0xD541C515 3577857301
    536854 0x6AC2384F 1791113295
    536855 0x214031F1 557855217
    536856 0xF028BB76 4029201270
    536857 0xF821479C 4162930588
    536858 0xEF9C2E42 4019990082
    536859 0x9E02734E 2650960718
    536860 0x20C786E9 549947113
    536861 0xEB84DD18 3951353112
    536862 0x422720B2 1109860530
    536863 0xA59EC71A 2778646298
    536864 0x2FFF3B42 805256002
    536865 0x5D606431 1566598193
    536866 0xF4FB4B01 4110109441
    536867 0x68AC0CD5 1756105941
    536868 0x6C35C2A6 1815462566
    536869 0xF38F0409 4086236169
    536870 0xABE37AC2 2883812034
    536871 0xF60096CA 4127233738
    536872 0x8E3B0BA3 2386234275
    536873 0x2904FF1A 688193306
    536874 0xCE6A1E32 3463061042
    536875 0xB110BD24 2970664228
    536876 0xF28289CC 4068641228
    536877 0x2719E083 656007299
    536878 0xD14BF4E7 3511416039
    536879 0x046C418A 74203530
    536880 0xF239BEC6 4063870662
    536881 0x619CDB26 1637669670
    536882 0x0FAB9F09 262905609
    536883 0xC8A14FC1 3366014913
    536884 0x2BDC9BF0 735878128
    536885 0x551D0A18 1427966488
    536886 0x43BDC8E5 1136511205
    536887 0x93236DB3 2468572595
    536888 0xA6DF3DD6 2799648214
    536889 0xD956E81A 3646351386
    536890 0xA292405D 2727493725
    536891 0x4ECF97EA 1322227690
    536892 0x6C289307 1814598407
    536893 0x0FE7B408 266843144
    536894 0xD7A54779 3617933177
    536895 0x49A575B6 1235580342
    536896 0x660B7018 1712025624
    536897 0x5A90DE66 1519443558
    536898 0x24602343 610280259
    536899 0xEA91EE92 3935432338
    536900 0x2A7DEBC9 712895433
    536901 0x7F8919BF 2139691455
    536902 0x85DD793B 2245884219
    536903 0x508B343F 1351300159
    536904 0xA7541C57 2807307351
    536905 0x902EAF9E 2418978718
    536906 0x2932BE12 691191314
    536907 0x304C8808 810321928
    536908 0xB0DC682A 2967234602
    536909 0x4F53EFD7 1330900951
    536910 0xF042D46A 4030911594
    536911 0x84726E0D 2222091789
    536912 0xC8667200 3362157056
    536913 0x4CCEAD50 1288613200
    536914 0x60D8E12E 1624826158
    536915 0x40BC2101 1086071041
    536916 0x5758F201 1465446913
    536917 0x45BD335F 1170027359
    536918 0x56E151A9 1457607081
    536919 0x11DE7F25 299794213
    536920 0x361D50CE 907890894
    536921 0xFBC73CE0 4224138464
    536922 0x57232881 1461921921
    536923 0x83AB7ACF 2209053391
    536924 0x8F7E47AF 2407417775
    536925 0x6425150A 1680151818
    536926 0x596C4C78 1500269688
    536927 0xA2B5D934 2729826612
    536928 0xBC673E6A 3160882794
    536929 0xE8116AB6 3893455542
    536930 0x92661734 2456164148
    536931 0x65904416 1703953430
    536932 0xD2DC8C64 3537669220
    536933 0xAC14A926 2887035174
    536934 0x98B9F2D6 2562323158
    536935 0xB85796F5 3092748021
    536936 0x08D3AC55 148089941
    536937 0x08532D08 139668744
    536938 0x58022FF2 1476538354
    536939 0x2C1D0FA9 740102057
    536940 0x83668A2B 2204535339
    536941 0x938CBC67 2475474023
    536942 0x49A14B37 1235307319
    536943 0x8FA2492A 2409777450
    536944 0x822227E8 2183276520
    536945 0xB6AE2123 3064865059
    536946 0xE83D8756 3896346454
    536947 0xD6437316 3594744598
    536948 0x731E7A99 1931377305
    536949 0x9F63EDE8 2674126312
    536950 0xA9B67DCD 2847309261
    536951 0x2B77E7F6 729278454
    536952 0x6F3E3B5F 1866349407
    536953 0xF1FF973E 4060059454
    536954 0x28AC9397 682398615
    536955 0x5920CE0A 1495322122
    536956 0x916C0AA2 2439776930
    536957 0x5CDF71F2 1558147570
    536958 0x9A41B702 2587997954
    536959 0xB16D4B56 2976729942
    536960 0x0C88EE3E 210300478
    536961 0x6A6D7474 1785558132
    536962 0x7544CFAE 1967443886
    536963 0xD14824C2 3511166146
    536964 0xA1FF9FFA 2717884410
    536965 0xA1CABEE0 2714418912
    536966 0xED5BD1E0 3982217696
    536967 0x3095185B 815077467
    536968 0x87A71339 2275873593
    536969 0x877DB48E 2273162382
    536970 0x4ABD04F2 1253901554
    536971 0xC29C3B16 3265018646
    536972 0xF0C21FF7 4039254007
    536973 0x4B60F69E 1264645790
    536974 0x1297C086 311935110
    536975 0xB2DF972E 3000997678
    536976 0xA3746A7F 2742315647
    536977 0x5690FA49 1452341833
    536978 0xDD69D9F1 3714701809
    536979 0x16C9F432 382334002
    536980 0x776F7186 2003792262
    536981 0xE5DC34D6 3856413910
    536982 0x07B9F683 129627779
    536983 0x37ADFF70 934150000
    536984 0x020655F6 33969654
    536985 0x828F3054 2190422100
    536986 0x6830DB63 1748032355
    536987 0xC8BDAD1D 3367873821
    536988 0x456172DD 1164014301
    536989 0x4EEF786C 1324316780
    536990 0x047313AB 74650539
    536991 0x4A9C6F2E 1251766062
    536992 0x16AE3E21 380517921
    536993 0x75E6BBE9 1978055657
    536994 0x4DB86F32 1303932722
    536995 0x537A470F 1400522511
    536996 0x09D57784 164984708
    536997 0x225542F8 576013048
    536998 0x9285CB22 2458241826
    536999 0x8A5D251C 2321360156
    537000 0x4EF17C51 1324448849
    537001 0xF2EE1FD0 4075691984
    537002 0x21F69ED9 569810649
    537003 0xA3D38623 2748548643
    537004 0x437747C6 1131890630
    537005 0x2DAE22B4 766386868
    537006 0xA564238C 2774803340
    537007 0x9B0DA8F5 2601363701
    537008 0x9BF14D85 2616282501
    537009 0xEF7EFDCD 4018077133
    537010 0x1D3654BE 490099902
    537011 0x284DA6AE 676177582
    537012 0xCCACDD6D 3433880941
    537013 0x85DC2096 2245795990
    537014 0x23720223 594674211
    537015 0xBD5A6697 3176818327
    537016 0x02ABD9F2 44816882
    537017 0xCD810703 3447785219
    537018 0x2594C210 630506000
    537019 0x04D87CA6 81296550
    537020 0x21BF189D 566171805
    537021 0x0B3E23E2 188621794
    537022 0x7852C9A1 2018691489
    537023 0xE55F03D2 3848209362
    537024 0xE0DE42DC 3772662492
    537025 0xDDE9DF00 3723091712
    537026 0x9A857C48 2592439368
    537027 0x6506143F 1694897215
    537028 0x8A1E2D28 2317233448
    537029 0x9D56D3A8 2639713192
    537030 0xE65D60AE 3864879278
    537031 0xCC66C1EE 3429286382
    537032 0x21A20619 564266521
    537033 0x007C5EA1 8150689
    537034 0x726CE93C 1919740220
    537035 0x707D5E32 1887264306
    537036 0x7AE9C992 2062141842
    537037 0x161EA8C2 371108034
    537038 0x5DD540CB 1574256843
    537039 0x42DA383D 1121597501
    537040 0x6A852D47 1787112775
    537041 0xE003B830 3758340144
    537042 0xA9CE3085 2848862341
    537043 0xEFBD3268 4022153832
    537044 0xF54E55A9 4115551657
    537045 0x7149A940 1900652864
    537046 0x701E3E47 1881030215
    537047 0x758F5F80 1972330368
    537048 0xFA3F88E0 4198467808
    537049 0xA0C77A3A 2697427514
    537050 0x97D20967 2547124583
    537051 0x44CDB6A6 1154332326
    537052 0x53281BD2 1395137490
    537053 0x49305D48 1227906376
    537054 0x843D2581 2218599809
    537055 0x9440B117 2487267607
    537056 0xC38E9622 3280901666
    537057 0x82869D41 2189860161
    537058 0x2DD924D9 769205465
    537059 0xCE8C480A 3465299978
    537060 0xECD53611 3973395985
    537061 0xAD78CFBC 2910375868
    537062 0xFFE73547 4293342535
    537063 0x833FDB8A 2202000266
    537064 0x60BB4400 1622885376
    537065 0xDE2D780E 3727521806
    537066 0x7373797F 1936947583
    537067 0xAE525433 2924631091
    537068 0x8A91E5AA 2324817322
    537069 0x8CC62047 2361794631
    537070 0x6092E498 1620239512
    537071 0x83370EB6 2201423542
    537072 0x19C2F477 432206967
    537073 0x3536E714 892790548
    537074 0x68FC8978 1761380728
    537075 0xB146C1B2 2974204338
    537076 0x86A5FF39 2259025721
    537077 0xC7006B6A 3338693482
    537078 0xA9B14363 2846966627
    537079 0x43DA311B 1138372891
    537080 0xEEC44C12 4005841938
    537081 0x3CB1A0AC 1018273964
    537082 0x9E875D82 2659671426
    537083 0x5734A4B3 1463067827
    537084 0x50449436 1346671670
    537085 0x74A2CF44 1956826948
    537086 0x161DDBE8 371055592
    537087 0x85D01F7A 2245009274
    537088 0x6DA2C81C 1839384604
    537089 0x6A4E9C84 1783536772
    537090 0x898D87F9 2307753977
    537091 0xBE49168C 3192460940
    537092 0x77CCF6DF 2009921247
    537093 0x8BAB207C 2343247996
    537094 0xD1C8CDCB 3519598027
    537095 0xF85BEAEE 4166773486
    537096 0xD3B6A308 3551961864
    537097 0x1F830312 528679698
    537098 0x6230DB94 1647369108
    537099 0x5F9BC800 1604044800
    537100 0x77AF542B 2007979051
    537101 0x599B39D0 1503345104
    537102 0x97121813 2534545427
    537103 0x2FE9C1B5 803848629
    537104 0xA58C9DC2 2777456066
    537105 0xA650952E 2790298926
    537106 0xA6DE4F97 2799587223
    537107 0xDFF1F9B9 3757177273
    537108 0x5980E9F1 1501620721
    537109 0x94B2003D 2494693437
    537110 0xB6FA8716 3069871894
    537111 0x7532A699 1966253721
    537112 0x717D6C2E 1904045102
    537113 0x10B523B8 280306616
    537114 0x25E93A02 636041730
    537115 0x78600DE3 2019560931
    537116 0xD0A4DFB9 3500466105
    537117 0x1965022E 426050094
    537118 0x8C2B3EFA 2351644410
    537119 0xAF7FD1E9 2944389609
    537120 0x420A3B31 1107966769
    537121 0xABE37683 2883810947
    537122 0x7832BBEB 2016590827
    537123 0xC83E2CEB 3359517931
    537124 0x8EA76ED7 2393337559
    537125 0xE4655C3C 3831847996
    537126 0x91DA6CBB 2447011003
    537127 0x69A5ABC4 1772465092
    537128 0x9879C300 2558116608
    537129 0x02A7C845 44550213
    537130 0x4622E285 1176691333
    537131 0x352397C3 891525059
    537132 0x9E2D2803 2653759491
    537133 0xACB24867 2897365095
    537134 0xCC4D68A9 3427625129
    537135 0x005CD031 6082609
    537136 0x228665C3 579233219
    537137 0xB8C5D596 3099973014
    537138 0x1B1599A4 454400420
    537139 0xC33197EC 3274807276
    537140 0x028C60A4 42754212
    537141 0x0ADF69A2 182413730
    537142 0xEE31513C 3996209468
    537143 0x711A3D10 1897544976
    537144 0x190F447D 420430973
    537145 0x4A34F6CE 1244985038
    537146 0xB8CF8EC1 3100610241
    537147 0x096E8D7B 158240123
    537148 0x4A41FA12 1245837842
    537149 0x6F94338A 1871983498
    537150 0x49582F6E 1230516078
    537151 0x181CCB1C 404540188
    537152 0x69991C6B 1771641963
    537153 0x44C5E9AA 1153821098
    537154 0x264DB929 642627881
    537155 0xDF328C43 3744631875
    537156 0xE62FF280 3861901952
    537157 0x90FFD450 2432685136
    537158 0xDF8FAC57 3750734935
    537159 0xA1D156DF 2714851039
    537160 0x71894209 1904820745
    537161 0x9E8C4A0E 2659994126
    537162 0x4EEAE41E 1324016670
    537163 0xB39E40D2 3013492946
    537164 0x83A1CB40 2208418624
    537165 0xC5990BB2 3315141554
    537166 0x62C16D29 1656843561
    537167 0xDFBA4DD4 3753528788
    537168 0xF27FF0AC 4068470956
    537169 0x4D297BEF 1294564335
    537170 0xD3CDC811 3553478673
    537171 0xEB5BB554 3948655956
    537172 0x1F035838 520312888
    537173 0x0F8008F0 260049136
    537174 0x1D0D54EB 487412971
    537175 0x0AD7DAA8 181918376
    537176 0xDB4243B5 3678553013
    537177 0xEB87623B 3951518267
    537178 0xA6E29A9D 2799868573
    537179 0x657DEC41 1702751297
    537180 0xDEBF78A9 3737090217
    537181 0x9A0E3E42 2584624706
    537182 0x4D81B479 1300345977
    537183 0xEE237F29 3995303721
    537184 0x64483418 1682453528
    537185 0x70CE3A54 1892563540
    537186 0x0407E621 67626529
    537187 0xE87F58A8 3900659880
    537188 0x8A572922 2320967970
    537189 0xA1DA472C 2715436844
    537190 0x1DB85112 498618642
    537191 0xA5B332E0 2779984608
    537192 0x8C6EB381 2356065153
    537193 0x011D856C 18711916
    537194 0x4FFB15C5 1341855173
    537195 0x8BB55113 2343915795
    537196 0x1F88FDBC 529071548
    537197 0x71E8295D 1911040349
    537198 0xDD326D1C 3711069468
    537199 0x4997E7A1 1234692001
    537200 0x0C25B238 203797048
    537201 0xDC02F821 3691182113
    537202 0x7E0272FA 2114089722
    537203 0x7428DA05 1948834309
    537204 0x5375694E 1400203598
    537205 0xE4895B26 3834207014
    537206 0xF2E27798 4074928024
    537207 0x2750C3E9 659604457
    537208 0xC491CE90 3297889936
    537209 0xB27D5F42 2994560834
    537210 0x48DEEAE7 1222568679
    537211 0xD52B16F8 3576370936
    537212 0x109162BA 277963450
    537213 0x06F78C0D 116886541
    537214 0xC3364C98 3275115672
    537215 0x6BC8BF8E 1808318350
    537216 0xDFB8B924 3753425188
    537217 0x74FCDF85 1962729349
    537218 0x89B3E526 2310268198
    537219 0xB2B2756B 2998039915
    537220 0x0BF45620 200562208
    537221 0xF73D1ACF 4147976911
    537222 0xC0C860CC 3234357452
    537223 0xDE29DC19 3727285273
    537224 0xB20C3EDC 2987146972
    537225 0xE676C194 3866542484
    537226 0x845B30F8 2220568824
    537227 0x8B71E0B3 2339496115
    537228 0xE8DB2A11 3906677265
    537229 0x229AB7E1 580564961
    537230 0x51D93418 1373189144
    537231 0x74E81286 1961366150
    537232 0x683E8338 1748927288
    537233 0x94CAB9B2 2496313778
    537234 0xA8A33C31 2829270065
    537235 0x43B4A14E 1135911246
    537236 0xA93CD929 2839337257
    537237 0x5350FBDC 1397816284
    537238 0xD261E675 3529631349
    537239 0xC9E11D00 3386973440
    537240 0x62D26A99 1657957017
    537241 0x92A069C1 2459986369
    537242 0x2517A475 622306421
    537243 0xCE77FA34 3463969332
    537244 0x0CCCB54F 214742351
    537245 0x4CDF0B6B 1289685867
    537246 0x23481E45 591928901
    537247 0x9F02003A 2667708474
    537248 0xD3BDA440 3552420928
    537249 0x2DED0928 770509096
    537250 0x2EA05107 782258439
    537251 0x9A2B9830 2586548272
    537252 0x098813F0 159912944
    537253 0x245C53F8 610030584
    537254 0xD29B8F24 3533410084
    537255 0x890D4C28 2299350056
    537256 0x685F61BA 1751081402
    537257 0x12B43DA1 313802145
    537258 0xD6882535 3599246645
    537259 0x251D0329 622658345
    537260 0x0E98327A 244855418
    537261 0x5A2DA66F 1512941167
    537262 0x00E45870 14964848
    537263 0x91E53ADF 2447719135
    537264 0xE501A114 3842089236
    537265 0xB3119D1A 3004275994
    537266 0x1FF020F0 535830768
    537267 0xC748BF35 3343433525
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    538767 0x4399D940 1134156096
    538768 0x97E2F94F 2548234575
    538769 0x1F48A827 524855335
    538770 0x88C31D10 2294488336
    538771 0xF44D4700 4098705152
    538772 0xE464DA81 3831814785
    538773 0x6E7A6187 1853514119
    538774 0x448F6E9A 1150250650
    538775 0xED003AA2 3976215202
    538776 0x1AF65C81 452353153
    538777 0x36DA69E0 920283616
    538778 0x417C436A 1098662762
    538779 0x8B5DEA00 2338187776
    538780 0x97ECE049 2548883529
    538781 0x52F85F9B 1392009115
    538782 0x80370F89 2151092105
    538783 0x2796E165 664199525
    538784 0x8D019EE9 2365693673
    538785 0xE1E18CCD 3789655245
    538786 0x86B198AB 2259785899
    538787 0x6BEBF3F0 1810625520
    538788 0x3D4B8620 1028359712
    538789 0x6F8F030C 1871643404
    538790 0xC6798969 3329853801
    538791 0xEB9FFD16 3953130774
    538792 0xDD760436 3715499062
    538793 0x2030335F 540029791
    538794 0xC7E80160 3353870688
    538795 0x0894B200 143962624
    538796 0x326EE03C 846127164
    538797 0x43C4DD73 1136975219
    538798 0xE58D6273 3851248243
    538799 0x1B71803F 460423231


    Now we divide this sample into groups of one hundred values ​​and produce a statistical analysis for each group. Nonse is a 32 bit number. According to common sense, every bit nonce is equally likely to take the value one or zero. That's what I'm going to check. I will write here such a script that counts the number of units in each bit position for a group of 100 nonsense.

    import sys
    import os
    folder=""
    stat_N=0
    stat=[]
    definit_stat():global stat
        global stat_N
        stat_N=0
        i=0while i<64:
           stat.append(0.0)
           i=i+1defclear_stat():global stat
        global stat_N
        stat_N=0
        i=0while i<64:
           stat[i]=0.0
           i=i+1defadd_stat(nonce):global stat
        global stat_N
        stat_N=stat_N+1
        i=0while i<64:
            if nonce & (1<<i):
                stat[i]=stat[i]+1
            i=i+1defnonce_num_bits():
        i=32while i<64:
            if stat[i]:
                return64
            i=i+1return32defprint_stat():global stat
        global stat_N
        i=0
        sz = nonce_num_bits()
        while i<sz:
            print(i," ",stat[i]/stat_N)
            i=i+1definit_stat_file():global folder
        txt=open(folder+"/stat.txt","w")
        txt.close()
    defwrite_stat():global folder
        global stat
        global stat_N
        txt=open(folder+"/stat.txt","a")
        i=0
        sz = nonce_num_bits()
        while i<sz:
            s=stat[i]/stat_N
            txt.write("%.2f " % s )
            i=i+1
        txt.write("\n")
        txt.close()
    #--------------------------------
    args = sys.argv
    if len(args)<2:
        print("Error: need argument name of folder with list of nonces")
        exit()
    folder=args[1]
    print( "Working on \"" + folder +"\"" )
    if os.path.isdir(folder)==False:
        print("Error: folder \""+folder+"\" does not exist")
        exit()
    init_stat()
    init_stat_file()
    f = open(folder+"/nonces.txt","r")
    content = f.read().split("\n")
    for line in content:
        word = line.split(" ")
        if len(word)<2:
            break
        nonce=int(word[1],16)
        add_stat(nonce)
        if stat_N==100:
            write_stat()
            clear_stat()
    f.close
    

    This script reads the bitcoin / nonces.txt file and creates a new bitcoin / stst.txt file of approximately the following form: Here, each column is the probability that a unit will appear in the corresponding bit position. For example, look at the column representing bit1, bit6, and bit7 of nonse: Expectancy 0.459, 0.446, and 0.458. Now it remains to somehow display the resulting statistics in the form of graphs. For this I have another plot.py script

    0.49 0.50 0.45 0.39 0.44 0.40 0.54 0.43 0.47 0.55 0.50 0.54 0.53 0.50 0.54 0.51 0.56 0.47 0.48 0.44 0.44 0.51 0.41 0.52 0.50 0.54 0.42 0.47 0.39 0.52 0.61 0.48
    0.40 0.61 0.52 0.48 0.49 0.55 0.45 0.43 0.55 0.56 0.54 0.55 0.56 0.60 0.52 0.42 0.52 0.46 0.61 0.54 0.52 0.52 0.55 0.49 0.46 0.48 0.41 0.42 0.45 0.42 0.47 0.47
    0.39 0.50 0.43 0.45 0.46 0.47 0.40 0.51 0.48 0.44 0.52 0.51 0.46 0.45 0.51 0.45 0.45 0.53 0.59 0.51 0.53 0.44 0.52 0.55 0.53 0.47 0.49 0.47 0.41 0.51 0.52 0.52
    0.51 0.43 0.39 0.49 0.46 0.53 0.40 0.50 0.49 0.42 0.47 0.56 0.49 0.51 0.57 0.44 0.58 0.52 0.50 0.54 0.48 0.50 0.43 0.53 0.52 0.49 0.48 0.56 0.43 0.42 0.49 0.49
    0.44 0.33 0.43 0.57 0.52 0.52 0.44 0.45 0.53 0.45 0.43 0.50 0.52 0.59 0.44 0.47 0.49 0.40 0.50 0.58 0.55 0.55 0.47 0.51 0.45 0.48 0.44 0.42 0.41 0.47 0.50 0.48
    0.49 0.46 0.51 0.42 0.57 0.45 0.45 0.35 0.53 0.42 0.51 0.52 0.47 0.51 0.47 0.45 0.48 0.49 0.54 0.48 0.50 0.49 0.46 0.56 0.44 0.48 0.47 0.55 0.46 0.45 0.51 0.50
    0.51 0.45 0.52 0.50 0.59 0.41 0.50 0.34 0.47 0.54 0.56 0.48 0.54 0.49 0.54 0.44 0.49 0.48 0.50 0.58 0.44 0.48 0.51 0.47 0.37 0.45 0.47 0.47 0.50 0.52 0.46 0.43
    0.51 0.43 0.52 0.57 0.58 0.51 0.47 0.36 0.51 0.44 0.47 0.47 0.49 0.47 0.43 0.39 0.47 0.50 0.56 0.46 0.47 0.47 0.46 0.63 0.48 0.43 0.44 0.46 0.43 0.51 0.45 0.49
    0.49 0.49 0.40 0.49 0.53 0.53 0.49 0.50 0.47 0.50 0.41 0.55 0.48 0.48 0.49 0.46 0.54 0.51 0.49 0.55 0.53 0.48 0.50 0.53 0.50 0.41 0.45 0.42 0.41 0.47 0.44 0.51
    0.47 0.47 0.47 0.58 0.52 0.42 0.53 0.53 0.59 0.55 0.43 0.48 0.50 0.42 0.46 0.50 0.52 0.53 0.43 0.48 0.58 0.44 0.43 0.42 0.54 0.42 0.58 0.42 0.39 0.45 0.46 0.56
    0.51 0.47 0.45 0.38 0.48 0.51 0.39 0.48 0.50 0.57 0.52 0.54 0.54 0.46 0.67 0.47 0.53 0.54 0.48 0.53 0.57 0.55 0.46 0.54 0.43 0.53 0.49 0.37 0.60 0.46 0.50 0.49
    0.45 0.46 0.47 0.46 0.47 0.51 0.41 0.53 0.59 0.45 0.51 0.43 0.49 0.46 0.46 0.55 0.52 0.51 0.56 0.51 0.49 0.46 0.45 0.49 0.50 0.55 0.48 0.54 0.44 0.49 0.59 0.49
    0.39 0.46 0.53 0.43 0.42 0.54 0.49 0.46 0.52 0.56 0.38 0.46 0.47 0.56 0.60 0.46 0.50 0.59 0.39 0.57 0.50 0.55 0.44 0.54 0.45 0.47 0.52 0.53 0.45 0.41 0.47 0.42
    0.48 0.46 0.49 0.45 0.52 0.43 0.43 0.46 0.43 0.52 0.53 0.50 0.52 0.61 0.43 0.51 0.56 0.51 0.57 0.48 0.51 0.48 0.61 0.52 0.52 0.54 0.42 0.45 0.48 0.49 0.46 0.48
    0.55 0.46 0.50 0.47 0.42 0.40 0.34 0.58 0.52 0.50 0.45 0.45 0.55 0.48 0.41 0.47 0.47 0.46 0.49 0.53 0.50 0.57 0.41 0.58 0.38 0.55 0.47 0.43 0.43 0.43 0.51 0.51
    0.39 0.49 0.56 0.45 0.45 0.51 0.42 0.49 0.45 0.51 0.48 0.44 0.48 0.45 0.53 0.47 0.53 0.45 0.48 0.42 0.54 0.51 0.47 0.59 0.49 0.37 0.48 0.47 0.43 0.44 0.47 0.40
    0.42 0.38 0.56 0.45 0.61 0.42 0.49 0.45 0.52 0.49 0.55 0.44 0.46 0.43 0.50 0.47 0.51 0.47 0.58 0.51 0.55 0.45 0.56 0.47 0.52 0.53 0.42 0.47 0.49 0.47 0.50 0.51
    0.49 0.49 0.49 0.57 0.52 0.48 0.44 0.38 0.50 0.48 0.50 0.52 0.44 0.48 0.46 0.43 0.51 0.52 0.53 0.50 0.44 0.46 0.46 0.46 0.39 0.49 0.52 0.38 0.42 0.53 0.52 0.49
    0.48 0.42 0.44 0.51 0.54 0.49 0.48 0.43 0.50 0.44 0.41 0.45 0.54 0.54 0.49 0.42 0.52 0.45 0.44 0.55 0.51 0.54 0.46 0.59 0.51 0.49 0.51 0.41 0.44 0.42 0.49 0.52
    0.42 0.41 0.45 0.48 0.45 0.50 0.36 0.50 0.51 0.48 0.45 0.61 0.45 0.52 0.49 0.52 0.45 0.47 0.47 0.57 0.48 0.52 0.54 0.46 0.54 0.54 0.56 0.51 0.39 0.41 0.59 0.55










    import sys
    import os
    import numpy as np
    import matplotlib.mlab as mlab
    import matplotlib.pyplot as plt
    folder=""#--------------------------------
    args = sys.argv
    if len(args)<2:
        print("Error: need argument name of folder with list of nonces")
        exit()
    folder=args[1]
    print( "Working on \"" + folder +"\"" )
    if os.path.isdir(folder)==False:
        print("Error: folder \""+folder+"\" does not exist")
        exit()
    stat_N=0
    stat=[]
    definit_stat():global stat
        global stat_N
        stat_N=0
        i=0while i<64:
           stat.append([])
           i=i+1defdraw_stat(start_bit,ax=None):
        print("draw_stat ",start_bit)
        save_single=0if ax isNone:
            fig = plt.figure( figsize=(16,9))
            ax = fig.gca()
            save_single=1
        ax.set_yticks( np.arange( 0.0, 1.0, 0.1 ) )
        plt.title("Probability of specific bit in nonce")
        plt.ylabel("Probability")
        plt.xlabel("Every X-point presents group nonces averaged")
        plt.ylim(0.0,1.0)
        plt.grid()
        i=0while i<8:
            color="#"if i&1:
                color=color+"FF"else:
                color=color+"00"if i&2:
                color=color+"FF"else:
                color=color+"00"if i&4:
                color=color+"FF"else:
                color=color+"00"if i==7:
                color="#A0A0A0"
            ax.plot(stat[start_bit+i],color,label="bit"+str(start_bit+i) )
            i=i+1
        ax.legend(loc="upper left",ncol=2)
        #plt.show()if save_single :
            plt.savefig(folder+"/bit"+str(start_bit)+"-"+str(start_bit+7)+".png",dpi=300)
    defdraw_hist(start_bit,ax=None):
        print("draw_hist ",start_bit)
        save_single=0if ax isNone:
            fig = plt.figure( figsize=(16,9))
            ax = fig.gca()
            save_single=1
        ax.set_xticks( np.arange( 0.0, 1.0, 0.1 ) )
        x=np.arange( 0.0, 1.0, 0.05 )
        plt.title("Probability Density of specific bit in nonce")
        plt.ylabel("Density")
        plt.xlabel("Probability")
        plt.grid()
        i=0
        max_val=0while i<8:
            histogram=[0]*20
            k=0
            stat_column=stat[start_bit+i]
            n=len(stat_column)
            while k<n:
                val=stat_column[k]
                idx=int(round(val*20))
                histogram[idx]=histogram[idx]+1if max_val<histogram[idx]:
                    max_val=histogram[idx];
                k=k+1
            color="#"if i&1:
                color=color+"FF"else:
                color=color+"00"if i&2:
                color=color+"FF"else:
                color=color+"00"if i&4:
                color=color+"FF"else:
                color=color+"00"if i==7:
                color="#A0A0A0"
            ax.plot(x,histogram,color,label="bit"+str(start_bit+i) )
            i=i+1
        plt.ylim(0.0,max_val*1.2)
        ax.set_yticks( np.arange( 0.0, max_val*1.2, 1.0 ) )
        ax.legend(loc="upper right")
        #plt.show()if save_single :
            plt.savefig(folder+"/density"+str(start_bit)+"-"+str(start_bit+7)+".png",dpi=300)
    num_bits=32
    init_stat()
    f = open(folder+"/stat.txt","r")
    content = f.read().split("\n")
    for line in content:
        word = line.split(" ")
        nb=len(word)-1if nb==32or nb==64:
            num_bits=nb
        else:
            break
        i=0while i<num_bits:
            stat[i].append( float(word[i]) )
            i=i+1
    f.close
    print("Num Bits in Nonce:",num_bits)
    draw_hist(0)
    draw_hist(8)
    draw_hist(16)
    draw_hist(24)
    add_subplots = 0
    fig = plt.figure( figsize=(16,9*num_bits/32))
    ax1 = plt.subplot(221+add_subplots)
    draw_hist(0,ax1)
    ax2 = plt.subplot(222+add_subplots)
    draw_hist(8,ax2)
    ax3 = plt.subplot(223+add_subplots)
    draw_hist(16,ax3)
    ax4 = plt.subplot(224+add_subplots)
    draw_hist(24,ax4)
    plt.savefig(folder+"/density_all.png",dpi=300)
    draw_stat(0)
    draw_stat(8)
    draw_stat(16)
    draw_stat(24)
    fig = plt.figure( figsize=(16,9*num_bits/32))
    ax1 = plt.subplot(221+add_subplots)
    draw_stat(0,ax1)
    ax2 = plt.subplot(222+add_subplots)
    draw_stat(8,ax2)
    ax3 = plt.subplot(223+add_subplots)
    draw_stat(16,ax3)
    ax4 = plt.subplot(224+add_subplots)
    draw_stat(24,ax4)
    plt.savefig(folder+"/bits_all.png",dpi=300)
    

    Now it's time to review these charts. If you display them all 32 on one page, you will get incomprehensible porridge. Therefore, the script builds 4 graphs and each displays 8-bit statistics. So it will be more convenient to consider.

    Here is everything:






    It can be seen that some bits in non-Bitcoin are not equally probable in their meaning and the graphics are shifted down. So the value of the bit is still slightly more likely zero than one. Below are the probability distribution graphs.






    Here you can see everything the same - on average, the probability of the value of some bits in Bitcoin's found golden nonce is not equal to 0.5, as I expected, but rather shifted towards zero. Bits 1, 15, 22 stand out in particular. What is the reason for this, unfortunately I cannot say. Nevertheless, this fact allows us to think about the redistribution of power miners across the ranges of values ​​more likely. For example, like this: there are a hundred hardware miners (video cards or asikov), 65 miners are sent to search nonce with values ​​of 22 bit equal to zero and 35 miners of minenits in ranges with value of bit 22 equal to one. Further, each selected range can be divided into smaller subranges according to the probability of bit 15 ... and so on. The idea is to look for golden nonses in more likely ranges according to the latest accumulated statistics.

    Then the idea arose to look at the statistics more deeply, not only for the last 2000 blocks, but for the entire history of the blockchain bitcoin. The whole blockchain is already almost 200 GB, so all the sharp-witted terms “Bitcoin, blockchain, bigdata” are already present here.

    I installed a bitcoin wallet and deflated the entire blockchain.

    Further I used such a project .
    This project allows the python script to go around the blockchain and extract the data of interest to us.

    Slightly changed the script from the example so that it writes statistics to a text file. Now the entire range of blocks of about 530 thousand is divided into groups of 1000 blocks. In each block, I look at the average probability of a bit nonsense value.

    import sys
    sys.path.append('..')
    from blockchain_parser.blockchain import Blockchain
    stat_N=0
    stat=[]
    definit_stat():global stat
        global stat_N
        stat_N=0
        i=0while i<32:
           stat.append(0)
           i=i+1defclear_stat():global stat
        global stat_N
        stat_N=0
        i=0while i<32:
           stat[i]=0
           i=i+1defadd_stat(nonce):global stat
        global stat_N
        stat_N=stat_N+1
        i=0while i<32:
            if nonce & (1<<i):
                stat[i]=stat[i]+1
            i=i+1defprint_stat():global stat
        global stat_N
        i=0while i<32:
            print(i," ",stat[i]/stat_N)
            i=i+1definit_stat_file():
        txt=open("stat.txt","w")
        txt.close()
    defwrite_stat():global stat
        global stat_N
        txt=open("stat.txt","a")
        i=0while i<32:
            s=stat[i]/stat_N
            txt.write("%.4f " % s )
            i=i+1
        txt.write("\n")
        txt.close()
    init_stat()
    init_stat_file()
    # Instantiate the Blockchain by giving the path to the directory # containing the .blk files created by bitcoind
    blockchain = Blockchain(sys.argv[1])
    height=0# To get the blocks ordered by height, you need to provide the path of the# `index` directory (LevelDB index) being maintained by bitcoind. It contains# .ldb files and is present inside the `blocks` directoryfor block in blockchain.get_ordered_blocks(sys.argv[1] + '/index', end=538000):
        print("")
        height=block.height
        print("height=%d" % (block.height) )
        print("header=",end='')
        arr=block.hex[:80]
        for a in arr:
            print(format(a,'02X'),end='')
        print("")
        print("hash  =%s" %  (block.hash) )
        hdr=block.header
        nonce=hdr.nonce
        print("nonce =", format(nonce,'08X'))
        add_stat(nonce)
        if stat_N==1000:
            write_stat()
            clear_stat()
        print("")
    

    This script also generates a statistics file of probability non-bit throughout the entire history of the bitcoin blockchain. It turns out quite an interesting picture.

    Here's how the probability of bits nonce from zero to fifteenth changed over time:



    But from the sixteenth to thirty-first bits of nonse:



    Here, each point horizontally is one thousand blocks in the blockchain, and for this thousand, the average probability of occurrence of one in the bit position nonce is considered.

    Honestly, I don’t even know how to comment on this. At times, bits in non-oh how not equally probable. And if so, then why the miners are still just going over the nonsense? Is it possible to distribute the power of miners according to the pattern found?

    If you are interested in such studies, you can still read my other articles:

    1. Bitcoin Probabilistic Mining Method
    2. Can bitcoins be calculated faster, easier or easier?
    3. Miner Bitcoin on the Rover Mars3


    UPDATE:
    I did additional research on Ethereum. The picture is even more interesting.
    If you look at the previously found nons, then with the naked eye you can see the uneven distribution: The hash algorithm of the ether is more complicated than that of Bitcoin. Non-air 64x bit. The middle of nonse (bits 28-36) is almost empty. At the same time, in the miner's source code that for Nvidia Cuda, that for AMD OpenCL, there seems to be no indication that the nonsense should be matched in the middle - there is a uniform build-up from zero to 0xFFFFFFFFFFFFFFFFFF. Here is the probability distribution of bits of non-ether:

    5844149 0xb431dfa8061eedc5
    5844150 0xac0badf80082e518
    5844151 0x64404b000042af4f
    5844152 0x96d84c7bd66ae434
    5844153 0xa6dd55a000364353
    5844154 0x516a29f005a56197
    5844155 0x4c0a1000006d50aa
    5844156 0x9e61733806e46e10
    5844157 0x8191b4d80925a334
    5844158 0x53a42df800b24a72
    5844159 0x9eb85a3c0031e03b
    5844160 0xb707255405787b53
    5844161 0xbf6bd50011069dc9
    5844162 0x83171430153262b1
    5844163 0x2e1c8a302ad94fc2
    5844164 0x84c971180db7252e
    5844165 0x47d606ф00110d3bb
    5844166 0x7b1f78bc127bb7cb







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