Editing Birthday problem (section)

===Probability table===
{{Main|Birthday attack}}

:{| class="wikitable" style="white-space:nowrap;"
|-
! rowspan="2" | length of <br />hex string
! rowspan="2" | no. of<br />bits<br />({{mvar|b}})
! rowspan="2" | hash space<br />size<br />({{math|2<sup>''b''</sup>}})
! colspan="10" | Number of hashed elements such that probability of at least one hash collision&nbsp;≥&nbsp;{{mvar|p}}
|-
! {{mvar|p}} = {{val||e=-18}}
! {{mvar|p}} = {{val||e=-15}}
! {{mvar|p}} = {{val||e=-12}}
! {{mvar|p}} = {{val||e=-9}}
! {{mvar|p}} = {{val||e=-6}}
! {{mvar|p}} = 0.001
! {{mvar|p}} = 0.01
! {{mvar|p}} = 0.25
! {{mvar|p}} = 0.50
! {{mvar|p}} = 0.75
|- align="center"
| bgcolor="#F2F2F2" | 8
| bgcolor="#F2F2F2" | 32
| bgcolor="#F2F2F2" | {{val|4.3|e=9}}
| 2
| 2
| 2
| 2.9
| 93
| {{val|2.9|e=3}}
| {{val|9.3|e=3}}
| {{val|5.0|e=4}}
| {{val|7.7|e=4}}
| {{val|1.1|e=5}}
|- align="center"
| bgcolor="#F2F2F2" | (10)
| bgcolor="#F2F2F2" | (40)
| bgcolor="#F2F2F2" | ({{val|1.1|e=12}})
| 2
| 2
| 2
| 47
| {{val|1.5|e=3}}
| {{val|4.7|e=4}}
| {{val|1.5|e=5}}
| {{val|8.0|e=5}}
| {{val|1.2|e=6}}
| {{val|1.7|e=6}}
|- align="center"
| bgcolor="#F2F2F2" | (12)
| bgcolor="#F2F2F2" | (48)
| bgcolor="#F2F2F2" | ({{val|2.8|e=14}})
| 2
| 2
| 24
| {{val|7.5|e=2}}
| {{val|2.4|e=4}}
| {{val|7.5|e=5}}
| {{val|2.4|e=6}}
| {{val|1.3|e=7}}
| {{val|2.0|e=7}}
| {{val|2.8|e=7}}
|- align="center"
| bgcolor="#F2F2F2" | 16
| bgcolor="#F2F2F2" | 64
| bgcolor="#F2F2F2" | {{val|1.8|e=19}}
| 6.1
| {{val|1.9|e=2}}
| {{val|6.1|e=3}}
| {{val|1.9|e=5}}
| {{val|6.1|e=6}}
| {{val|1.9|e=8}}
| {{val|6.1|e=8}}
| {{val|3.3|e=9}}
| {{val|5.1|e=9}}
| {{val|7.2|e=9}}
|- align="center"
| bgcolor="#F2F2F2" | (24)
| bgcolor="#F2F2F2" | (96)
| bgcolor="#F2F2F2" | ({{val|7.9|e=28}})
| {{val|4.0|e=5}}
| {{val|1.3|e=7}}
| {{val|4.0|e=8}}
| {{val|1.3|e=10}}
| {{val|4.0|e=11}}
| {{val|1.3|e=13}}
| {{val|4.0|e=13}}
| {{val|2.1|e=14}}
| {{val|3.3|e=14}}
| {{val|4.7|e=14}}
|- align="center"
| bgcolor="#F2F2F2" | 32
| bgcolor="#F2F2F2" | 128
| bgcolor="#F2F2F2" | {{val|3.4|e=38}}
| {{val|2.6|e=10}}
| {{val|8.2|e=11}}
| {{val|2.6|e=13}}
| {{val|8.2|e=14}}
| {{val|2.6|e=16}}
| {{val|8.3|e=17}}
| {{val|2.6|e=18}}
| {{val|1.4|e=19}}
| {{val|2.2|e=19}}
| {{val|3.1|e=19}}
|- align="center"
| bgcolor="#F2F2F2" | (48)
| bgcolor="#F2F2F2" | (192)
| bgcolor="#F2F2F2" | ({{val|6.3|e=57}})
| {{val|1.1|e=20}}
| {{val|3.5|e=21}}
| {{val|1.1|e=23}}
| {{val|3.5|e=24}}
| {{val|1.1|e=26}}
| {{val|3.5|e=27}}
| {{val|1.1|e=28}}
| {{val|6.0|e=28}}
| {{val|9.3|e=28}}
| {{val|1.3|e=29}}
|- align="center"
| bgcolor="#F2F2F2" | 64
| bgcolor="#F2F2F2" | 256
| bgcolor="#F2F2F2" | {{val|1.2|e=77}}
| {{val|4.8|e=29}}
| {{val|1.5|e=31}}
| {{val|4.8|e=32}}
| {{val|1.5|e=34}}
| {{val|4.8|e=35}}
| {{val|1.5|e=37}}
| {{val|4.8|e=37}}
| {{val|2.6|e=38}}
| {{val|4.0|e=38}}
| {{val|5.7|e=38}}
|- align="center"
| bgcolor="#F2F2F2" | (96)
| bgcolor="#F2F2F2" | (384)
| bgcolor="#F2F2F2" | ({{val|3.9|e=115}})
| {{val|8.9|e=48}}
| {{val|2.8|e=50}}
| {{val|8.9|e=51}}
| {{val|2.8|e=53}}
| {{val|8.9|e=54}}
| {{val|2.8|e=56}}
| {{val|8.9|e=56}}
| {{val|4.8|e=57}}
| {{val|7.4|e=57}}
| {{val|1.0|e=58}}
|- align="center"
| bgcolor="#F2F2F2" | 128
| bgcolor="#F2F2F2" | 512
| bgcolor="#F2F2F2" | {{val|1.3|e=154}}
| {{val|1.6|e=68}}
| {{val|5.2|e=69}}
| {{val|1.6|e=71}}
| {{val|5.2|e=72}}
| {{val|1.6|e=74}}
| {{val|5.2|e=75}}
| {{val|1.6|e=76}}
| {{val|8.8|e=76}}
| {{val|1.4|e=77}}
| {{val|1.9|e=77}}
|}
[[File:birthday_attack_vs_paradox.svg|thumb|Comparison of the birthday problem (1) and birthday attack (2):{{parabreak}}
In (1), collisions are found within one set, in this case, 3 out of 276 pairings of the 24 lunar astronauts.{{parabreak}}
In (2), collisions are found between two sets, in this case, 1 out of 256 pairings of only the first bytes of SHA-256 hashes of 16 variants each of benign and harmful contracts.]]
The lighter fields in this table show the number of hashes needed to achieve the given probability of collision (column) given a hash space of a certain size in bits (row). Using the birthday analogy: the "hash space size" resembles the "available days", the "probability of collision" resembles the "probability of shared birthday", and the "required number of hashed elements" resembles the "required number of people in a group". One could also use this chart to determine the minimum hash size required (given upper bounds on the hashes and probability of error), or the probability of collision (for fixed number of hashes and probability of error).

For comparison, {{val|e=-18}} to {{val|e=-15}} is the uncorrectable [[bit error rate]] of a typical hard disk.<ref>Jim Gray, Catharine van Ingen. [https://arxiv.org/abs/cs/0701166 Empirical Measurements of Disk Failure Rates and Error Rates]</ref> In theory, 128-bit hash functions, such as [[MD5]], should stay within that range until about {{val|8.2|e=11}} documents, even if its possible outputs are many more.