We provide a characterization of pseudoentropy in terms of hardness of sampling: Let (XB) be jointly distributed random variables such that B takes values in a polynomial-sized set. We show that B is computationally indistinguishable from a random variable of higher Shannon entropy given X if and only if there is no probabilistic polynomial-time S such that (XS(X)) has small KL divergence from (XB) . This can be viewed as an analogue of the Impagliazzo Hardcore Theorem (FOCS `95) for Shannon entropy (rather than min-entropy).

Using this characterization, we show that if f is a one-way function, then (f(Un)Un) has ``next-bit pseudoentropy'' at least n+logn, establishing a conjecture of Haitner, Reingold, and Vadhan (STOC `10). Plugging this into the construction of Haitner et al., this yields a simpler construction of pseudorandom generators from one-way functions. In particular, the construction only performs hashing once, and only needs the hash functions that are randomness extractors (e.g.\ universal hash functions) rather than needing them to support ``local list-decoding'' (as in the Goldreich--Levin hardcore predicate, STOC `89).

With an additional idea, we also show how to improve the seed length of the pseudorandom generator to O(n3), compared to O(n4) in the construction of Haitner et al