Stam’s Conjecture and Threshold Phenomena in Collision Resistance

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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.