On pseudoentropy versus compressibility

A source is comprehensible if we can efficiently compute short descriptions of strings in the support and efficiently recover the strings from the descriptions. A source has high pseudo-entropy if it is computationally distinguishable from a source of high entropy. In this paper, we present a technique for proving lower bounds on compressibility in an oracle setting, which yields the following results: 1. We exhibit oracles relative to which there exists samplable sources over {0, 1}ⁿ of low pseudoentropy (say n/2) that cannot be compressed to length less than n - ω(log n) by polynomial size circuits. This matches the upper bounds in (Goldberg and Sipser, 1991, Trevisan et al., 2004), and provides an oracle separation between compressibility and pseudoentropy, thereby partially addressing an open problem posed in (Impagliazzo, 1999). 2. We also provide a separation between 1/s-metric-type pseudoentropy and 1/s-Yao-type pseudoentropy - which are two computational analogues of entropy introduced in (Barak et al., 2003) - for the class of oracle circuits of sizes (s polynomially bounded). This is the first known separation result for metric-type and Yao-type pseudoentropy. 3. In the random oracle model, we show that there exists in compressible functions as defined in (Dwork et al., 1996) where any substantial compression of the output of the function must reveal something about the seed. This yields the first known practical realization of incompressible functions, under the assumption that random oracles may be realized using cryptographic hash functions. Finally, we show that computational assumptions are needed to separate compressibility and pseudoentropy for samplable sources. In particular, if one-way functions do not exist, then any samplable flat source of entropy k can be compressed by circuits to length k + 0(log n); furthermore, any such source has 1/2-Yao-type pseudoentropy k + 0(log n).