{"title":"二面隐子群问题","authors":"Imin Chen, David Sun","doi":"10.1515/jmc-2022-0029","DOIUrl":null,"url":null,"abstract":"\n The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical question for post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a finite field. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard” quantum algorithm for HSP on finite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare” as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantum algorithm that uses \n \n \n \n m\n \n m\n \n coset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on the success probability of a quantum algorithm that uses \n \n \n \n m\n \n m\n \n coset samples to solve DCP.","PeriodicalId":43866,"journal":{"name":"Journal of Mathematical Cryptology","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The dihedral hidden subgroup problem\",\"authors\":\"Imin Chen, David Sun\",\"doi\":\"10.1515/jmc-2022-0029\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical question for post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a finite field. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard” quantum algorithm for HSP on finite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare” as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantum algorithm that uses \\n \\n \\n \\n m\\n \\n m\\n \\n coset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on the success probability of a quantum algorithm that uses \\n \\n \\n \\n m\\n \\n m\\n \\n coset samples to solve DCP.\",\"PeriodicalId\":43866,\"journal\":{\"name\":\"Journal of Mathematical Cryptology\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Cryptology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/jmc-2022-0029\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Cryptology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/jmc-2022-0029","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
隐藏子群问题(HSP)是量子计算的基石问题,它捕捉了许多令人感兴趣的问题,并提供了一种基于傅立叶采样的标准框架算法来研究这些问题,傅立叶采样是已知具有量子优势的一类技术,它对某些群成功,但对其他群却不成功。二面体群 HSP 问题的量子硬度是基于有误差学习的后量子密码系统的一个关键问题,也出现在有限域上椭圆曲线间同源性的亚指数算法中。在本文中,我们对有限群上 HSP 的 "标准 "量子算法所处理的二面体隐藏子群问题进行了最新概述,详细说明了强傅里叶采样成功的障碍,并总结了其他已知方法和结果。在处理过程中,我们尽可能 "对比 "了循环和二面情况,以期确定使用 m m coset 样本求解这些群上 HSP 的量子算法的成功概率边界。在最后几节中,我们证明了二面体余集问题(DCP)的一些不成功结果,其动因在于二面体余集问题与量子态克隆之间的联系。这些不成功结果的证明经过调整后,给出了使用 m m coset 样本求解 DCP 的量子算法的成功概率的非难上限。
The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical question for post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a finite field. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard” quantum algorithm for HSP on finite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare” as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantum algorithm that uses
m
m
coset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on the success probability of a quantum algorithm that uses
m
m
coset samples to solve DCP.