Birthday Paradox

Related Concepts

Collision Resistance; Generic Attacks Against DLP; Hash Functions; Integer Factoring

Definition

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

$$\frac{365} {365} \cdot \frac{365 - 1} {365} \cdots \frac{365 - 22} {365} \approx 0.49 < 0.5$$

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

$${\prod \limits_{0\le i\le 1.18\sqrt{p}}}\frac{p - i} {p} < 0.5$$

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].

Applications

Under reasonable assumptions about their inputs, common cryptographic k-bit hash functions may be assumed to...