{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:10:52Z","timestamp":1750306252413,"version":"3.41.0"},"reference-count":46,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2017,4,27]],"date-time":"2017-04-27T00:00:00Z","timestamp":1493251200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"BSF","award":["2010120"],"award-info":[{"award-number":["2010120"]}]},{"name":"ISF","award":["864\/11"],"award-info":[{"award-number":["864\/11"]}]},{"name":"ERC","award":["279559"],"award-info":[{"award-number":["279559"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2017,6,30]]},"abstract":"\n A sampling procedure for a distribution\n P<\/jats:italic>\n over {0, 1}\n \u2113<\/jats:sup>\n is a function\n C<\/jats:italic>\n : {0, 1}\n \n n<\/jats:italic>\n <\/jats:sup>\n \u2192 {0, 1}\n \u2113<\/jats:sup>\n such that the distribution\n C<\/jats:italic>\n (\n \n U\n n<\/jats:sub>\n <\/jats:italic>\n ) (obtained by applying\n C<\/jats:italic>\n on the uniform distribution\n \n U\n n<\/jats:sub>\n <\/jats:italic>\n ) is the \u201cdesired distribution\u201d\n P<\/jats:italic>\n . Let\n n<\/jats:italic>\n >\n r<\/jats:italic>\n \u2265 \u2113 =\n n<\/jats:italic>\n \u03a9(1)<\/jats:sup>\n . An \u03f5-\n nb-PRG<\/jats:italic>\n (defined by Dubrov and Ishai [2006]) is a function\n G<\/jats:italic>\n : {0, 1}\n \n r<\/jats:italic>\n <\/jats:sup>\n \u2192 {0, 1}\n \n n<\/jats:italic>\n <\/jats:sup>\n such that for every\n C<\/jats:italic>\n : {0, 1}\n \n n<\/jats:italic>\n <\/jats:sup>\n \u2192 {0, 1}\n \u2113<\/jats:sup>\n in some class of \u201cinteresting sampling procedures,\u201d\n C<\/jats:italic>\n \u2032(\n \n U\n r<\/jats:sub>\n <\/jats:italic>\n ) =\n C<\/jats:italic>\n (\n G<\/jats:italic>\n (\n \n U\n r<\/jats:sub>\n <\/jats:italic>\n )) is \u03f5-close to\n C<\/jats:italic>\n (\n \n U\n n<\/jats:sub>\n <\/jats:italic>\n ) in\n statistical distance<\/jats:italic>\n .\n <\/jats:p>\n \n We construct poly-time computable nb-PRGs with\n r<\/jats:italic>\n =\n O<\/jats:italic>\n (\u2113) for poly-size circuits relying on the assumption that there exists \u03b2 > 0 and a problem\n L<\/jats:italic>\n in E = DTIME(2\n \n O<\/jats:italic>\n (\n n<\/jats:italic>\n )\n <\/jats:sup>\n ) such that for every large enough\n n<\/jats:italic>\n , nondeterministic circuits of size 2\n \n \u03b2\n n<\/jats:italic>\n <\/jats:sup>\n that have NP-gates cannot solve\n L<\/jats:italic>\n on inputs of length\n n<\/jats:italic>\n . This assumption is a scaled nonuniform analog of (the widely believed) EXP \u2260 \u03a3\n 2<\/jats:sub>\n P<\/jats:sup>\n , and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and Ishai have\n r<\/jats:italic>\n = \u03a9(\u2113\n 2<\/jats:sup>\n ) and are based on very strong cryptographic assumptions or, alternatively, on nonstandard assumptions regarding incompressibility of functions on random inputs. When restricting to poly-size circuits\n C<\/jats:italic>\n : {0, 1}\n \n n<\/jats:italic>\n <\/jats:sup>\n \u2192 {0, 1}\n \u2113<\/jats:sup>\n with Shannon entropy\n H<\/jats:italic>\n (\n C<\/jats:italic>\n (\n \n U\n n<\/jats:sub>\n <\/jats:italic>\n )) \u2a7d\n k<\/jats:italic>\n , for \u2113 >\n k<\/jats:italic>\n =\n n<\/jats:italic>\n \u03a9(1)<\/jats:sup>\n , our nb-PRGs have\n r<\/jats:italic>\n =\n O<\/jats:italic>\n (\n k<\/jats:italic>\n ). The nb-PRGs of Dubrov and Ishai use seed length\n r<\/jats:italic>\n = \u03a9(\n k<\/jats:italic>\n 2<\/jats:sup>\n ) and require that the probability distribution of\n C<\/jats:italic>\n (\n \n U\n n<\/jats:sub>\n <\/jats:italic>\n ) is efficiently computable.\n <\/jats:p>\n \n Our nb-PRGs follow from a notion of \u201cconditional PRGs,\u201d which may be of independent interest. These are PRGs where\n G<\/jats:italic>\n (\n \n U\n r<\/jats:sub>\n <\/jats:italic>\n ) remains pseudorandom even when conditioned on a \u201clarge\u201d event {\n A<\/jats:italic>\n (\n G<\/jats:italic>\n (\n \n U\n r<\/jats:sub>\n <\/jats:italic>\n )) = 1}, for an arbitrary poly-size circuit\n A<\/jats:italic>\n . A related notion was considered by Shaltiel and Umans [2005] in a different setting, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai.\n <\/jats:p>\n \n We also give an unconditional construction of poly-time computable nb-PRGs for poly(\n n<\/jats:italic>\n )-size, depth\n d<\/jats:italic>\n circuits\n C<\/jats:italic>\n : {0, 1}\n \n n<\/jats:italic>\n <\/jats:sup>\n \u2192 {0, 1}\n \u2113<\/jats:sup>\n with\n r<\/jats:italic>\n =\n O<\/jats:italic>\n (\u2113 \u00b7 log\n \n d<\/jats:italic>\n +\n O<\/jats:italic>\n (1)\n <\/jats:sup>\n n<\/jats:italic>\n ). This improves upon the previous work of Dubrov and Ishai that has\n r<\/jats:italic>\n \u2265 \u2113\n 2<\/jats:sup>\n . This result follows by adapting a recent PRG construction of Trevisan and Xue [2013] to the case of nb-PRGs. We also show that this PRG can be implemented by a uniform family of constant-depth circuits with slightly increased seed length.\n <\/jats:p>","DOI":"10.1145\/3018057","type":"journal-article","created":{"date-parts":[[2017,4,28]],"date-time":"2017-04-28T12:38:23Z","timestamp":1493383103000},"page":"1-26","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Pseudorandom Generators with Optimal Seed Length for Non-Boolean Poly-Size Circuits"],"prefix":"10.1145","volume":"9","author":[{"given":"Sergei","family":"Artemenko","sequence":"first","affiliation":[{"name":"University of Haifa, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ronen","family":"Shaltiel","sequence":"additional","affiliation":[{"name":"University of Haifa, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2017,4,27]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-016-0128-9"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539705446950"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804090"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-012-0056-2"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/050641958"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/070691954"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1006\/inco.2000.2885"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/0213053"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00145-006-0438-1"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/1132516.1132615"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22670-0_22"},{"volume-title":"Technical Report TR95--050. 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