E. Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle
{"title":"有时Kolmogorov复杂度投影下的密码硬度","authors":"E. Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle","doi":"10.4230/LIPIcs.ISAAC.2021.54","DOIUrl":null,"url":null,"abstract":"A version of time-bounded Kolmogorov complexity, denoted KT , has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP . Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP -Turing reductions; neither is known to be NP -complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP . In particular, MKTP is hard for DET (a subclass of P ) under nonuniform ≤ NC 0 m reductions. In this paper, we improve this, to show that MKTP is hard for the (apparently larger) class NISZK L under not only ≤ NC 0 m reductions but even under projections. Also MKTP is hard for NISZK under ≤ P / poly m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK L is the non-interactive version of the class SZK L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP , and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP ).","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"23 1","pages":"206-224"},"PeriodicalIF":0.0000,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Cryptographic Hardness under Projections for Time-Bounded Kolmogorov Complexity\",\"authors\":\"E. Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle\",\"doi\":\"10.4230/LIPIcs.ISAAC.2021.54\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A version of time-bounded Kolmogorov complexity, denoted KT , has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP . Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP -Turing reductions; neither is known to be NP -complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP . In particular, MKTP is hard for DET (a subclass of P ) under nonuniform ≤ NC 0 m reductions. In this paper, we improve this, to show that MKTP is hard for the (apparently larger) class NISZK L under not only ≤ NC 0 m reductions but even under projections. Also MKTP is hard for NISZK under ≤ P / poly m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK L is the non-interactive version of the class SZK L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP , and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP ).\",\"PeriodicalId\":23063,\"journal\":{\"name\":\"Theor. Comput. Sci.\",\"volume\":\"23 1\",\"pages\":\"206-224\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theor. Comput. Sci.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.ISAAC.2021.54\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theor. Comput. Sci.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.54","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Cryptographic Hardness under Projections for Time-Bounded Kolmogorov Complexity
A version of time-bounded Kolmogorov complexity, denoted KT , has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP . Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP -Turing reductions; neither is known to be NP -complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP . In particular, MKTP is hard for DET (a subclass of P ) under nonuniform ≤ NC 0 m reductions. In this paper, we improve this, to show that MKTP is hard for the (apparently larger) class NISZK L under not only ≤ NC 0 m reductions but even under projections. Also MKTP is hard for NISZK under ≤ P / poly m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK L is the non-interactive version of the class SZK L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP , and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP ).