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	11,625,647,541,583,872
Reduced χ²	0.842
Degrees of freedom	51
p-value	0.781

Per-bucket detail

Bucket	Observed	Expected (now)	Expected (end)	Residual
0	5,812,823,730,152,067	5,812,823,770,791,936	170,141,183,460,469,231,731,687,303,715,884,105,728	−0.53σ
1	2,906,411,916,105,910	2,906,411,885,395,968	85,070,591,730,234,615,865,843,651,857,942,052,864	+0.57σ
2	1,453,205,968,414,689	1,453,205,942,697,984	42,535,295,865,117,307,932,921,825,928,971,026,432	+0.67σ
3	726,602,984,930,770	726,602,971,348,992	21,267,647,932,558,653,966,460,912,964,485,513,216	+0.50σ
4	363,301,483,773,227	363,301,485,674,496	10,633,823,966,279,326,983,230,456,482,242,756,608	−0.10σ
5	181,650,745,152,483	181,650,742,837,248	5,316,911,983,139,663,491,615,228,241,121,378,304	+0.17σ
6	90,825,365,425,101	90,825,371,418,624	2,658,455,991,569,831,745,807,614,120,560,689,152	−0.63σ
7	45,412,681,010,009	45,412,685,709,312	1,329,227,995,784,915,872,903,807,060,280,344,576	−0.70σ
8	22,706,330,956,124	22,706,342,854,656	664,613,997,892,457,936,451,903,530,140,172,288	−2.50σ
9	11,353,167,369,950	11,353,171,427,328	332,306,998,946,228,968,225,951,765,070,086,144	−1.20σ
10	5,676,581,722,373	5,676,585,713,664	166,153,499,473,114,484,112,975,882,535,043,072	−1.68σ
11	2,838,295,622,702	2,838,292,856,832	83,076,749,736,557,242,056,487,941,267,521,536	+1.64σ
12	1,419,146,009,284	1,419,146,428,416	41,538,374,868,278,621,028,243,970,633,760,768	−0.35σ
13	709,572,668,321	709,573,214,208	20,769,187,434,139,310,514,121,985,316,880,384	−0.65σ
14	354,785,981,218	354,786,607,104	10,384,593,717,069,655,257,060,992,658,440,192	−1.05σ
15	177,393,535,734	177,393,303,552	5,192,296,858,534,827,628,530,496,329,220,096	+0.55σ
16	88,696,625,149	88,696,651,776	2,596,148,429,267,413,814,265,248,164,610,048	−0.09σ
17	44,348,114,542	44,348,325,888	1,298,074,214,633,706,907,132,624,082,305,024	−1.00σ
18	22,174,015,289	22,174,162,944	649,037,107,316,853,453,566,312,041,152,512	−0.99σ
19	11,086,957,172	11,087,081,472	324,518,553,658,426,726,783,156,020,576,256	−1.18σ
20	5,543,492,774	5,543,540,736	162,259,276,829,213,363,391,578,010,288,128	−0.64σ
21	2,771,781,599	2,771,770,368	81,129,638,414,606,681,695,789,005,144,064	+0.21σ
22	1,385,872,125	1,385,885,184	40,564,819,207,303,340,847,894,502,572,032	−0.35σ
23	692,966,739	692,942,592	20,282,409,603,651,670,423,947,251,286,016	+0.92σ
24	346,465,635	346,471,296	10,141,204,801,825,835,211,973,625,643,008	−0.30σ
25	173,214,860	173,235,648	5,070,602,400,912,917,605,986,812,821,504	−1.58σ
26	86,617,779	86,617,824	2,535,301,200,456,458,802,993,406,410,752	−0.00σ
27	43,321,450	43,308,912	1,267,650,600,228,229,401,496,703,205,376	+1.91σ
28	21,654,923	21,654,456	633,825,300,114,114,700,748,351,602,688	+0.10σ
29	10,824,503	10,827,228	316,912,650,057,057,350,374,175,801,344	−0.83σ
30	5,414,792	5,413,614	158,456,325,028,528,675,187,087,900,672	+0.51σ
31	2,705,812	2,706,807	79,228,162,514,264,337,593,543,950,336	−0.60σ
32	1,354,676	1,353,404	39,614,081,257,132,168,796,771,975,168	+1.09σ
33	677,063	676,702	19,807,040,628,566,084,398,385,987,584	+0.44σ
34	338,887	338,351	9,903,520,314,283,042,199,192,993,792	+0.92σ
35	169,221	169,175	4,951,760,157,141,521,099,596,496,896	+0.11σ
36	84,391	84,588	2,475,880,078,570,760,549,798,248,448	−0.68σ
37	42,295	42,294	1,237,940,039,285,380,274,899,124,224	+0.01σ
38	21,017	21,147	618,970,019,642,690,137,449,562,112	−0.89σ
39	10,594	10,573	309,485,009,821,345,068,724,781,056	+0.20σ
40	5,280	5,287	154,742,504,910,672,534,362,390,528	−0.09σ
41	2,691	2,643	77,371,252,455,336,267,181,195,264	+0.93σ
42	1,317	1,322	38,685,626,227,668,133,590,597,632	−0.13σ
43	653	661	19,342,813,113,834,066,795,298,816	−0.31σ
44	330	330	9,671,406,556,917,033,397,649,408	−0.02σ
45	181	165	4,835,703,278,458,516,698,824,704	+1.23σ
46	78	83	2,417,851,639,229,258,349,412,352	−0.51σ
47	49	41	1,208,925,819,614,629,174,706,176	+1.20σ
48	28	21	604,462,909,807,314,587,353,088	+1.62σ
49	5	10	302,231,454,903,657,293,676,544	−1.66σ
50	5	5	151,115,727,451,828,646,838,272	−0.07σ
51	3	3	75,557,863,725,914,323,419,136	+0.26σ
52	2	1	37,778,931,862,957,161,709,568	+0.62σ
53	0	1	18,889,465,931,478,580,854,784	−0.80σ
54	0	0	9,444,732,965,739,290,427,392	−0.57σ
55	0	0	4,722,366,482,869,645,213,696	−0.40σ
56	0	0	2,361,183,241,434,822,606,848	−0.28σ
57	0	0	1,180,591,620,717,411,303,424	−0.20σ
58	1	0	590,295,810,358,705,651,712	+6.90σ
59	0	0	295,147,905,179,352,825,856	−0.10σ
60	0	0	147,573,952,589,676,412,928	−0.07σ
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