## Cryptology ePrint Archive: Report 2021/890

A Note on One-way Functions and Sparse Languages

Yanyi Liu and Rafael Pass

Abstract: We show equivalence between the existence of one-way functions and the existence of a 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.

Category / Keywords: foundations / one-way functions, average-case complexity