Stam's Conjecture and Threshold Phenomena in Collision Resistance

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 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) collisions are obtained with high probability after O(1)r2 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.