Abstract
In this paper, we study multi-collision probability. For a hash function H:D→R with |R|=n, it has been believed that we can find an s-collision by hashing Q=n(s-1)/s times. We first show that this probability is at most 1/s! for any s, which is very small for large s. (for example, s=n(s-1)/s) Thus the above folklore is wrong for large s. We next show that if s is small, so that we can assume Q-s≈Q, then this probability is at least 1/s!-1/2(s!)2, which is very high for small s (for example, s is a constant). Thus the above folklore is true for small s. Moreover, we show that by hashing (s!)1/s×Q+s-1(≤n) times, an s-collision is found with probability approximately 0.5 for any n and s such that (s!/n)1/s≈0. Note that if s=2, it coincides with the usual birthday paradox. Hence it is a generalization of the birthday paradox to multi-collisions.
