Revised: September 6, 2020

Published: October 13, 2020

**Keywords:**cryptography, complexity theory, one-way functions, computational entropy

**ACM Classification:**F.0

**AMS Classification:**03D15

**Abstract:**
[Plain Text Version]

This paper uses a variant of the notion of *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
*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.

In an additional result we reprove the seminal result of Impagliazzo and Levin (FOCS 1989): a reduction from “uniform distribution” average-case complexity problems to ones with arbitrary (polynomial-time samplable) distributions. We do that using techniques similar to those we use to construct UOWHFs from one-way functions, where the source of this similarity is the use of a notion similar to inaccessible entropy. This draws an interesting connection between two seemingly separate lines of research: average-case complexity and universal one-way hash-functions.

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A preliminary version of this paper appeared as “Universal One-Way Hash Functions via Inaccessible Entropy” in EUROCRYPT 2010.