IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Online ISSN : 1745-1337
Print ISSN : 0916-8508
Special Section on Cryptography and Information Security
Birthday Paradox for Multi-Collisions
Kazuhiro SUZUKIDongvu TONIENKaoru KUROSAWAKoji TOYOTA
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2008 Volume E91.A Issue 1 Pages 39-45

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Abstract

In this paper, we study multi-collision probability. For a hash function H:DR 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-sQ, 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.

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© 2008 The Institute of Electronics, Information and Communication Engineers
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