Abstract: 
Pseudoentropy has found a lot of important applications to cryptography and complexity theory. In this paper we focus on the foundational problem that has not been investigated so far, namely by how much pseudoentropy (the amount seen by computationally bounded attackers) diﬀers from its informationtheoretic counterpart (seen by unbounded observers), given certain limits on attacker’s computational power? We provide the following answer for HILL pseudoentropy, which exhibits a threshold behavior around the size exponential in the entropy amount:– If the attacker size (s) and advantage () satisfy s (formula presented) where k is the claimed amount of pseudoentropy, then the pseudoentropy boils down to the informationtheoretic smooth entropy. – If s (formula presented) then pseudoentropy could be arbitrarily bigger than the informationtheoretic smooth entropy. Besides answering the posted question, we show an elegant application of our result to the complexity theory, namely that it implies the classical result on the existence of functions hard to approximate (due to Pippenger). In our approach we utilize nonconstructive techniques: the duality of linear programming and the probabilistic method.
