Birthday Paradox

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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

$$ \frac{365}{365}\cdot\frac{365-1}{365}\cdots\frac{365-22}{365}\approx0.49<0.5; $$

it is called a paradox because the 23 is felt to be unreasonably small compared to 365. Further, in general, it follows from

$$ \prod_{0\leq i\leq 1.18\sqrt{p}}\frac{p-i}{p}\approx 0.5 $$

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 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 ...