A Tight Transformation between HILL and Metric Conditional Pseudoentropy

HILL Entropy and Metric Entropy are generalizations of the information-theoretic notion of min-entropy to the realistic setting where adversaries are computationally bounded. The notion of HILL Entropy appeared in the breakthrough construction of a PRG from any one-way function (H̊astad et al.), and has become the most important and most widely used variant of computational entropy. In turn, Metric Entropy defined as a relaxation of HILL Entropy, has been proven to be much easier to handle, in particular in the context of computational generalizations of the Green-Tao-Ziegler Dense Model Theorem which find applications in leakage-resilient cryptography, memory delegation or deterministic encryption. Fortunately, Metric Entropy can be converted, with some loss in quality, to HILL Entropy as shown by Barak, Shaltiel and Wigderson. In this paper we improve their result, reducing the loss in quality of entropy. Our bound is tight and, interestingly, independent of size of the probability space. As an interesting example of application we derive the computational dense model theorem with best possible parameters. Our approach is based on the theory of convex approximation in Lspaces.