.1 Formalization

Intuitively, a collision resistant function (CRF) f is a function for which it is hard to find two inputs s.t. their images collide: If x 6= x′, then w.h.p. f(x) 6= f(x′). It is easy to see that any one-to-one function satisfies this requirement. However, we are interested in functions that are many-to-one, specifically, functions that map long inputs to short inputs, i.e., |f(x)| < |x|. One immediate use for CRFs would be to verify the integrity of a large database D. Instead of storing the entire data in a safe place, we could use a CRF function f : {0, 1}∗ → {0, 1} and save only d← f(D), which is relatively short. For any future suspicious version D′ of our database, we can now simply test d = f(D′) and see if the version was compromised.