On One-Way Functions and Sparse Languages

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 existence of a S(center dot)-sparse language L that is hard-on-average with respect to some samplable distribution with Shannon entropy h(center dot) such that h(n) - log(S(n)) >= 4 log n; - The existence of a S(center dot)-sparse language L is an element of NP, that is hard-on-average with respect to some samplable distribution with Shannon entropy h(center dot) such that h(n) - log(S(n)) >= n/3; - The existence of one-way functions. where a language L is said to be S(center dot)-sparse if | L boolean AND {0, 1}(n)| <= S(n) for all n is an element of N. Our results are inspired by, and generalize, results from the elegant recent paper by Ilango, Ren and Santhanam (IRS, STOC'22), which presents similar connections for specific sparse languages.