A distribution is k-incompressible, Yao [FOCS '82], if no efficient compression scheme compresses it to less than k bits. While being a natural measure, its relation to other computational analogs of entropy such as pseudoentropy, Hastad, Impagliazzo, Levin, and Luby [SICOMP '99], and to other cryptographic hardness assumptions, was unclear. We advance towards a better understating of this notion, showing that a k-incompressible distribution has (k-2) bits of next-block pseudoentropy, a refinement of pseudoentropy introduced by Haitner, Reingold, and Vadhan [SICOMP '13]. We deduce that a samplable distribution X that is (H(X)+2)-incompressible, implies the existence of one-way functions.
@InProceedings{haitner_et_al:LIPIcs.ITCS.2023.66, author = {Haitner, Iftach and Mazor, Noam and Silbak, Jad}, title = {{Incompressiblity and Next-Block Pseudoentropy}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {66:1--66:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.66}, URN = {urn:nbn:de:0030-drops-175697}, doi = {10.4230/LIPIcs.ITCS.2023.66}, annote = {Keywords: incompressibility, next-block pseudoentropy, sparse languages} }
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