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Encyclopedia of Cryptography and Security
pp 3637
Birthday Paradox
 Arjen K. Lenstra
The birthday paradox refers to the fact that there is a probability of more than 50% that among a group of at least 23 randomly selected people at least 2 have the same birthday. It follows from
$$
\frac{365}{365}\cdot\frac{3651}{365}\cdots\frac{36522}{365}\approx0.49<0.5;
$$
it is called a paradox because the 23 is felt to be unreasonably small compared to 365. Further, in general, it follows from
$$
\prod_{0\leq i\leq 1.18\sqrt{p}}\frac{pi}{p}\approx 0.5
$$
that it is not unreasonable to expect a duplicate after about \(\sqrt{p}\) elements have been picked at random (and with replacement) from a set of cardinality p. A good exposition of the probability analysis underlying the birthday paradox can be found in Corman et al. [1], Section 5.4.
Under reasonable assumptions about their inputs, common cryptographic kbit hash functions may be assumed to produce random, uniformly distributed kbit outputs. Thus one may expect that a set of the order of ...
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 Title
 Birthday Paradox
 Reference Work Title
 Encyclopedia of Cryptography and Security
 Reference Work Part
 B
 Pages
 pp 3637
 Copyright
 2005
 DOI
 10.1007/0387234837_30
 Print ISBN
 9780387234731
 Online ISBN
 9780387234830
 Publisher
 Springer US
 Copyright Holder
 International Federation for Information Processing
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 Editors

 Henk C. A. van Tilborg ^{(1)}
 Editor Affiliations

 1.
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