Birthday Paradox for Multi-collisions

  • Kazuhiro Suzuki
  • Dongvu Tonien
  • Kaoru Kurosawa
  • Koji Toyota
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4296)


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! 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.


hash function birthday paradox multi-collision collision resistant 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kazuhiro Suzuki
    • 1
  • Dongvu Tonien
    • 2
  • Kaoru Kurosawa
    • 3
  • Koji Toyota
    • 3
  1. 1.Venture Business LaboratoryIbaraki UniversityIbarakiJapan
  2. 2.School of Information Technology and Computer ScienceUniversity of WollongongWollongongAustralia
  3. 3.Department of Computer and Information SciencesIbaraki UniversityIbarakiJapan

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