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### Paper:

TR21-092 | 28th June 2021 21:27

#### A Note on One-way Functions and Sparse Languages

TR21-092
Authors: Yanyi Liu, Rafael Pass
Publication: 2nd July 2021 04:24
Keywords:

Abstract:

We show equivalence between the existence of one-way
functions and the existence of a \emph{sparse} language that is
hard-on-average w.r.t. some efficiently samplable high-entropy''
distribution.
In more detail, the following are equivalent:
- The existentence of a $S(\cdot)$-sparse language $L$ that is
hard-on-average with respect to some samplable distribution with
Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq 4\log n$;
- The existentence of a $S(\cdot)$-sparse language $L \in \NP$, that is
hard-on-average with respect to some samplable distribution with
Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq n/3$;
- The existence of one-way functions.

Our results are insipired by, and generalize, the recent elegant paper by Ilango,
Ren and Santhanam (ECCC'21), which presents similar characterizations for
concrete sparse languages.

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