  This paper uses a variant of the notion of \emph{inaccessible entropy}
(Haitner, Reingold, Vadhan and Wee, STOC 2009), to give an alternative
construction and proof for the fundamental result, first proved by Rompel (STOC
1990), that \emph{Universal One-Way Hash Functions (UOWHFs)} can be based on
any one-way functions. We observe that a small tweak of any one-way function
$f$ is already a weak form of a UOWHF: consider the function $F(x,i)$ that
returns the $i$-bit-long prefix of $f(x)$. If $F$ were a UOWHF then given a
random $x$ and $i$ it would be hard to come up with $x'\neq x$ such that
$F(x,i)=F(x',i)$. While this may not be the case, we show (rather easily) that
it is hard to sample $x'$ with almost full entropy among all the possible such
values of $x'$. The rest of our construction simply amplifies and exploits this
basic property.Combined with other recent work, the construction of three
fundamental cryptographic primitives (Pseudorandom Generators, Statistically
Hiding Commitments and UOWHFs) out of one-way functions is now to a large
extent unified. In particular, all three constructions rely on and manipulate
computational notions of entropy in similar ways. Pseudorandom Generators rely
on the well-established notion of pseudoentropy, whereas Statistically Hiding
Commitments and UOWHFs rely on the newer notion of inaccessible entropy.
