Paper 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.
Metadata
- Available format(s)
- Category
- Foundations
- Publication info
- Preprint. MINOR revision.
- Keywords
- one-way functionsaverage-case complexity
- Contact author(s)
- yl2866 @ cornell edu,rafael @ cs cornell edu
- History
- 2023-02-10: revised
- 2021-06-29: received
- See all versions
- Short URL
- https://ia.cr/2021/890
- License
-
CC BY