Limits on the Hardness of Lattice Problems in ell _p Norms

Chris Peikert
{"title":"Limits on the Hardness of Lattice Problems in ell _p Norms","authors":"Chris Peikert","doi":"10.1109/CCC.2007.12","DOIUrl":null,"url":null,"abstract":"We show that several recent \"positive\" results for lattice problems in the l2 norm also hold in lp norms, for p>2. In particular, for lattices of dimension n: (i) approximating the shortest and closest vector in the lp norm to within O macr(radicn) factors is contained in coNP, (ii) approximating the length of the shortest vector in the lp norm to within O breve(n) factors reduces to the average-case problems studied in related works (Ajtai, STOC 1996; Micciancio and Regev, FOCS 2004; Regev, STOC 2005). These results improve upon prior understanding of lp norms by up to radicn factors. Taken together, they can be viewed as a partial converse to recent reductions from the l2 norm to lp norms (Regev and Rosen, STOC 2006). One of our main technical contributions is a very general analysis of Gaussian distributions over lattices, which may be of independent interest. Our proofs employ analytical techniques of Banaszczyk which, to our knowledge, have yet to be exploited in computer science.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2007-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"36","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Cybersecurity and Cyberforensics Conference","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/CCC.2007.12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 36

Abstract

We show that several recent "positive" results for lattice problems in the l2 norm also hold in lp norms, for p>2. In particular, for lattices of dimension n: (i) approximating the shortest and closest vector in the lp norm to within O macr(radicn) factors is contained in coNP, (ii) approximating the length of the shortest vector in the lp norm to within O breve(n) factors reduces to the average-case problems studied in related works (Ajtai, STOC 1996; Micciancio and Regev, FOCS 2004; Regev, STOC 2005). These results improve upon prior understanding of lp norms by up to radicn factors. Taken together, they can be viewed as a partial converse to recent reductions from the l2 norm to lp norms (Regev and Rosen, STOC 2006). One of our main technical contributions is a very general analysis of Gaussian distributions over lattices, which may be of independent interest. Our proofs employ analytical techniques of Banaszczyk which, to our knowledge, have yet to be exploited in computer science.
ell - p范数格问题的硬度极限
我们证明了最近关于l2范数格问题的几个“正”结果也适用于lp范数,对于p>2。特别地,对于n维的格:(i)将lp范数中最短和最近的向量逼近到O个macr(根数)因子内包含在coNP中,(ii)将lp范数中最短向量的长度逼近到O个breve(n)因子内减少到相关工作中研究的平均情况问题(Ajtai, STOC 1996;Micciancio and Regev, fos 2004;Regev, STOC 2005)。这些结果在先前对lp规范的理解上得到了改善,其主要因素是自由基。总的来说,它们可以被视为最近从l2规范到lp规范的减少的部分相反(Regev和Rosen, STOC 2006)。我们的主要技术贡献之一是对格上高斯分布的非常一般的分析,这可能是独立的兴趣。我们的证明采用了Banaszczyk的分析技术,据我们所知,这些技术还没有在计算机科学中得到利用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信