Prefix match length (bits)

Fit to analytical expectation

Expected distribution assumes MD5 acts like a random function. The leading run of matching bits has a truncated geometric distribution on 128 Bernoulli(1/2) trials — each additional matching bit halves the probability.

Expected (now) scales that analytical distribution to the hashes computed so far; Expected (end) scales it to the full 2¹²⁸ keyspace. Residual is the Pearson residual (Observed − Expected) / √Expected in units of σ. A p-value near 1 means the observed distribution is consistent with random; near 0 means it is statistically surprising.

Distribution moments

	Observed	Expected
Mean	1.000	1.000
Std. deviation	1.414	1.414

Pearson χ² test

Samples	13,403,914,325,983,232
Reduced χ²	1.013
Degrees of freedom	51
p-value	0.447

Per-bucket detail

Bucket	Observed	Expected (now)	Expected (end)	Residual
0	6,701,957,085,610,166	6,701,957,162,991,616	170,141,183,460,469,231,731,687,303,715,884,105,728	−0.95σ
1	3,350,978,617,283,210	3,350,978,581,495,808	85,070,591,730,234,615,865,843,651,857,942,052,864	+0.62σ
2	1,675,489,352,913,561	1,675,489,290,747,904	42,535,295,865,117,307,932,921,825,928,971,026,432	+1.52σ
3	837,744,663,339,306	837,744,645,373,952	21,267,647,932,558,653,966,460,912,964,485,513,216	+0.62σ
4	418,872,312,976,803	418,872,322,686,976	10,633,823,966,279,326,983,230,456,482,242,756,608	−0.47σ
5	209,436,165,884,023	209,436,161,343,488	5,316,911,983,139,663,491,615,228,241,121,378,304	+0.31σ
6	104,718,071,145,834	104,718,080,671,744	2,658,455,991,569,831,745,807,614,120,560,689,152	−0.93σ
7	52,359,035,864,527	52,359,040,335,872	1,329,227,995,784,915,872,903,807,060,280,344,576	−0.62σ
8	26,179,507,707,554	26,179,520,167,936	664,613,997,892,457,936,451,903,530,140,172,288	−2.44σ
9	13,089,753,726,467	13,089,760,083,968	332,306,998,946,228,968,225,951,765,070,086,144	−1.76σ
10	6,544,877,338,220	6,544,880,041,984	166,153,499,473,114,484,112,975,882,535,043,072	−1.06σ
11	3,272,443,308,044	3,272,440,020,992	83,076,749,736,557,242,056,487,941,267,521,536	+1.82σ
12	1,636,220,062,828	1,636,220,010,496	41,538,374,868,278,621,028,243,970,633,760,768	+0.04σ
13	818,109,489,496	818,110,005,248	20,769,187,434,139,310,514,121,985,316,880,384	−0.57σ
14	409,054,707,659	409,055,002,624	10,384,593,717,069,655,257,060,992,658,440,192	−0.46σ
15	204,527,737,580	204,527,501,312	5,192,296,858,534,827,628,530,496,329,220,096	+0.52σ
16	102,263,615,687	102,263,750,656	2,596,148,429,267,413,814,265,248,164,610,048	−0.42σ
17	51,131,753,022	51,131,875,328	1,298,074,214,633,706,907,132,624,082,305,024	−0.54σ
18	25,565,721,833	25,565,937,664	649,037,107,316,853,453,566,312,041,152,512	−1.35σ
19	12,782,835,404	12,782,968,832	324,518,553,658,426,726,783,156,020,576,256	−1.18σ
20	6,391,443,727	6,391,484,416	162,259,276,829,213,363,391,578,010,288,128	−0.51σ
21	3,195,767,110	3,195,742,208	81,129,638,414,606,681,695,789,005,144,064	+0.44σ
22	1,597,875,090	1,597,871,104	40,564,819,207,303,340,847,894,502,572,032	+0.10σ
23	798,951,314	798,935,552	20,282,409,603,651,670,423,947,251,286,016	+0.56σ
24	399,470,407	399,467,776	10,141,204,801,825,835,211,973,625,643,008	+0.13σ
25	199,706,896	199,733,888	5,070,602,400,912,917,605,986,812,821,504	−1.91σ
26	99,864,807	99,866,944	2,535,301,200,456,458,802,993,406,410,752	−0.21σ
27	49,948,302	49,933,472	1,267,650,600,228,229,401,496,703,205,376	+2.10σ
28	24,972,945	24,966,736	633,825,300,114,114,700,748,351,602,688	+1.24σ
29	12,480,106	12,483,368	316,912,650,057,057,350,374,175,801,344	−0.92σ
30	6,239,764	6,241,684	158,456,325,028,528,675,187,087,900,672	−0.77σ
31	3,118,627	3,120,842	79,228,162,514,264,337,593,543,950,336	−1.25σ
32	1,562,031	1,560,421	39,614,081,257,132,168,796,771,975,168	+1.29σ
33	780,506	780,210	19,807,040,628,566,084,398,385,987,584	+0.33σ
34	390,576	390,105	9,903,520,314,283,042,199,192,993,792	+0.75σ
35	194,974	195,053	4,951,760,157,141,521,099,596,496,896	−0.18σ
36	97,420	97,526	2,475,880,078,570,760,549,798,248,448	−0.34σ
37	48,780	48,763	1,237,940,039,285,380,274,899,124,224	+0.08σ
38	24,239	24,382	618,970,019,642,690,137,449,562,112	−0.91σ
39	12,195	12,191	309,485,009,821,345,068,724,781,056	+0.04σ
40	6,042	6,095	154,742,504,910,672,534,362,390,528	−0.68σ
41	3,100	3,048	77,371,252,455,336,267,181,195,264	+0.95σ
42	1,525	1,524	38,685,626,227,668,133,590,597,632	+0.03σ
43	736	762	19,342,813,113,834,066,795,298,816	−0.94σ
44	382	381	9,671,406,556,917,033,397,649,408	+0.05σ
45	204	190	4,835,703,278,458,516,698,824,704	+0.98σ
46	94	95	2,417,851,639,229,258,349,412,352	−0.13σ
47	58	48	1,208,925,819,614,629,174,706,176	+1.50σ
48	32	24	604,462,909,807,314,587,353,088	+1.68σ
49	6	12	302,231,454,903,657,293,676,544	−1.71σ
50	7	6	151,115,727,451,828,646,838,272	+0.43σ
51	3	3	75,557,863,725,914,323,419,136	+0.01σ
52	2	1	37,778,931,862,957,161,709,568	+0.42σ
53	0	1	18,889,465,931,478,580,854,784	−0.86σ
54	0	0	9,444,732,965,739,290,427,392	−0.61σ
55	0	0	4,722,366,482,869,645,213,696	−0.43σ
56	0	0	2,361,183,241,434,822,606,848	−0.30σ
57	0	0	1,180,591,620,717,411,303,424	−0.22σ
58	1	0	590,295,810,358,705,651,712	+6.41σ
59	0	0	295,147,905,179,352,825,856	−0.11σ
60	0	0	147,573,952,589,676,412,928	−0.08σ
61	0	0	73,786,976,294,838,206,464	−0.05σ
62	0	0	36,893,488,147,419,103,232	−0.04σ
63	0	0	18,446,744,073,709,551,616	−0.03σ
64	0	0	9,223,372,036,854,775,808	−0.02σ
65	0	0	4,611,686,018,427,387,904	−0.01σ
66	0	0	2,305,843,009,213,693,952	−0.01σ
67	0	0	1,152,921,504,606,846,976	−0.01σ
68	0	0	576,460,752,303,423,488	−0.00σ
69	0	0	288,230,376,151,711,744	−0.00σ
70	0	0	144,115,188,075,855,872	−0.00σ
71	0	0	72,057,594,037,927,936	−0.00σ
72	0	0	36,028,797,018,963,968	−0.00σ
73	0	0	18,014,398,509,481,984	−0.00σ
74	0	0	9,007,199,254,740,992	−0.00σ
75	0	0	4,503,599,627,370,496	−0.00σ
76	0	0	2,251,799,813,685,248	−0.00σ
77	0	0	1,125,899,906,842,624	−0.00σ
78	0	0	562,949,953,421,312	−0.00σ
79	0	0	281,474,976,710,656	−0.00σ
80	0	0	140,737,488,355,328	−0.00σ
81	0	0	70,368,744,177,664	−0.00σ
82	0	0	35,184,372,088,832	−0.00σ
83	0	0	17,592,186,044,416	−0.00σ
84	0	0	8,796,093,022,208	−0.00σ
85	0	0	4,398,046,511,104	−0.00σ
86	0	0	2,199,023,255,552	−0.00σ
87	0	0	1,099,511,627,776	−0.00σ
88	0	0	549,755,813,888	−0.00σ
89	0	0	274,877,906,944	−0.00σ
90	0	0	137,438,953,472	−0.00σ
91	0	0	68,719,476,736	−0.00σ
92	0	0	34,359,738,368	−0.00σ
93	0	0	17,179,869,184	−0.00σ
94	0	0	8,589,934,592	−0.00σ
95	0	0	4,294,967,296	−0.00σ
96	0	0	2,147,483,648	−0.00σ
97	0	0	1,073,741,824	−0.00σ
98	0	0	536,870,912	−0.00σ
99	0	0	268,435,456	−0.00σ
100	0	0	134,217,728	−0.00σ
101	0	0	67,108,864	−0.00σ
102	0	0	33,554,432	−0.00σ
103	0	0	16,777,216	−0.00σ
104	0	0	8,388,608	−0.00σ
105	0	0	4,194,304	−0.00σ
106	0	0	2,097,152	−0.00σ
107	0	0	1,048,576	−0.00σ
108	0	0	524,288	−0.00σ
109	0	0	262,144	−0.00σ
110	0	0	131,072	−0.00σ
111	0	0	65,536	−0.00σ
112	0	0	32,768	−0.00σ
113	0	0	16,384	−0.00σ
114	0	0	8,192	−0.00σ
115	0	0	4,096	−0.00σ
116	0	0	2,048	−0.00σ
117	0	0	1,024	−0.00σ
118	0	0	512	−0.00σ
119	0	0	256	−0.00σ
120	0	0	128	−0.00σ
121	0	0	64	−0.00σ
122	0	0	32	−0.00σ
123	0	0	16	−0.00σ
124	0	0	8	−0.00σ
125	0	0	4	−0.00σ
126	0	0	2	−0.00σ
127	0	0	1	−0.00σ
128	0	0	1	−0.00σ