{"doi":"10.1109/isit.2019.8849240","_id":"7136","department":[{"_id":"KrPi"}],"conference":{"start_date":"2019-07-07","location":"Paris, France","name":"ISIT: International Symposium on Information Theory","end_date":"2019-07-12"},"publisher":"IEEE","author":[{"last_name":"Skórski","id":"EC09FA6A-02D0-11E9-8223-86B7C91467DD","full_name":"Skórski, Maciej","first_name":"Maciej"}],"scopus_import":"1","date_updated":"2023-09-06T11:15:41Z","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"It is well established that the notion of min-entropy fails to satisfy the \\emph{chain rule} of the form H(X,Y)=H(X|Y)+H(Y), known for Shannon Entropy. Such a property would help to analyze how min-entropy is split among smaller blocks. Problems of this kind arise for example when constructing extractors and dispersers.\r\nWe show that any sequence of variables exhibits a very strong strong block-source structure (conditional distributions of blocks are nearly flat) when we \\emph{spoil few correlated bits}. This implies, conditioned on the spoiled bits, that \\emph{splitting-recombination properties} hold. In particular, we have many nice properties that min-entropy doesn't obey in general, for example strong chain rules, \"information can't hurt\" inequalities, equivalences of average and worst-case conditional entropy definitions and others. Quantitatively, for any sequence X1,…,Xt of random variables over an alphabet X we prove that, when conditioned on m=t⋅O(loglog|X|+loglog(1/ϵ)+logt) bits of auxiliary information, all conditional distributions of the form Xi|X2019 IEEE International Symposium on Information Theory, Paris, France, 2019.","ista":"Skórski M. 2019. Strong chain rules for min-entropy under few bits spoiled. 2019 IEEE International Symposium on Information Theory. ISIT: International Symposium on Information Theory, 8849240.","mla":"Skórski, Maciej. “Strong Chain Rules for Min-Entropy under Few Bits Spoiled.” 2019 IEEE International Symposium on Information Theory, 8849240, IEEE, 2019, doi:10.1109/isit.2019.8849240.","chicago":"Skórski, Maciej. “Strong Chain Rules for Min-Entropy under Few Bits Spoiled.” In 2019 IEEE International Symposium on Information Theory. IEEE, 2019. https://doi.org/10.1109/isit.2019.8849240.","short":"M. Skórski, in:, 2019 IEEE International Symposium on Information Theory, IEEE, 2019.","ama":"Skórski M. Strong chain rules for min-entropy under few bits spoiled. In: 2019 IEEE International Symposium on Information Theory. IEEE; 2019. doi:10.1109/isit.2019.8849240","apa":"Skórski, M. (2019). Strong chain rules for min-entropy under few bits spoiled. In 2019 IEEE International Symposium on Information Theory. Paris, France: IEEE. https://doi.org/10.1109/isit.2019.8849240"},"publication":"2019 IEEE International Symposium on Information Theory","external_id":{"isi":["000489100301043"],"arxiv":["1702.08476"]},"date_created":"2019-11-28T10:19:21Z","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","month":"07","oa_version":"Preprint","article_processing_charge":"No","publication_identifier":{"isbn":["9781538692912"]},"title":"Strong chain rules for min-entropy under few bits spoiled","main_file_link":[{"url":"https://arxiv.org/abs/1702.08476","open_access":"1"}],"date_published":"2019-07-01T00:00:00Z","isi":1,"type":"conference"}