Stam’s Conjecture and Threshold Phenomena in Collision Resistance

  • John Steinberger
  • Xiaoming Sun
  • Zhe Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7417)


At CRYPTO 2008 Stam [8] conjectured that if an \((m\!+\!s)\)-bit to s-bit compression function F makes r calls to a primitive f of n-bit input, then a collision for F can be obtained (with high probability) using \(r2^{(nr-m)/(r+1)}\) queries to f, which is sometimes less than the birthday bound. Steinberger [9] proved Stam’s conjecture up to a constant multiplicative factor for most cases in which \(r = 1\) and for certain other cases that reduce to the case \(r = 1\). In this paper we prove the general case of Stam’s conjecture (also up to a constant multiplicative factor). Our result is qualitatively different from Steinberger’s, moreover, as we show the following novel threshold phenomenon: that exponentially many (more exactly, \(2^{s-2(m-n)/(r+1)}\)) collisions are obtained with high probability after \(O(1)r2^{(nr-m)/(r+1)}\) queries. This in particular shows that threshold phenomena observed in practical compression functions such as JH are, in fact, unavoidable for compression functions with those parameters.


Hash Function Compression Function Input Length Provable Security Threshold Phenomenon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© International Association for Cryptologic Research 2012 2012

Authors and Affiliations

  1. 1.Institute of Theoretical Computer ScienceTsinghua UniversityBeijingChina
  2. 2.Institute of Computing TechnologyChina Academy of SciencesBeijingChina
  3. 3.Hulu SoftwareBeijingChina

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