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
it is called a paradox because the 23 is felt to be unreasonably small compared to 365. Further, in general, it follows from
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 k-bit hash functions may be assumed to produce random, uniformly distributed k-bit outputs. Thus one may expect that a set of the order of \(2^{k/2}\)inputs contains two...
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References
Cormen, Thomas L., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). Introduction to Algorithms (2nd ed.). MIT Press, Cambridge, MA.
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Lenstra, A.K. (2005). Birthday Paradox. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_30
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