Birthday paradox for multi-collisions

Authors:
Kazuhiro Suzuki;Dongvu Tonien;Kaoru Kurosawa;Koji Toyota
Affiliations:
Venture Business Laboratory, Ibaraki University, Ibaraki, Japan;School of Information Technology and Computer Science, University of Wollongong, Wollongong, Australia;Department of Computer and Information Sciences, Ibaraki University, Ibaraki, Japan;Department of Computer and Information Sciences, Ibaraki University, Ibaraki, Japan
Venue:
ICISC'06 Proceedings of the 9th international conference on Information Security and Cryptology
Year:
2006

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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! which is very small for large s. We next show that by hashing (s!)1/ s ×Q times, an s-collision is found with probability approximately 0.5 for sufficiently large n. Note that if s=2, it coincides with the usual birthday paradox. Hence it is a generalization of the birthday paradox to multi-collisions.