计算机科学 ›› 2018, Vol. 45 ›› Issue (6): 1-8.doi: 10.11896/j.issn.1002-137X.2018.06.001
• 综述 • 下一篇
戴文静, 袁家斌
DAI Wen-jing, YUAN Jia-bin
摘要: 在Shor发现大整数因子分解问题的有效量子算法之后,量子计算迫使我们重新审视现有的密码系统。隐含子群问题是量子计算在群结构上的推广,它暗示通过考虑不同的群和函数来解决更困难的问题,以期找到新的指数倍快于其经典对应物的量子算法。有限交换群隐含子群问题的研究已有相对固定的研究框架和方法,而非交换群隐含子群问题的研究一直很活跃。研究表明,二面体群隐含子群问题的有效解决可能攻破基于格的唯一最短向量问题的密码体制,图同构问题可以转化为对称群隐含子群问题。文中对隐含子群问题的研究现状进行综述,希望能够吸引更多研究者对隐含子群问题的注意。最后为隐含子群问题未来的研究方向提出参考意见。
中图分类号:
[1]DIFFIE W,HELLMAN M E.New directions in cryptography[J].IEEE Transactions on Information Theory,1976,22(6):644-654. [2]RIVEST R L,SHAMIR A,ADLEMAN L.A method for obtaining digital signatures and public-key cryptosystems[J].Communications of the Acm,1978,26(2):96-99. [3]ELGAMAL T.A Public-Key Cryptosystem and SignatureSche-me Based on Discrete Logarithms[J].IEEE Transactions on Information Theory,1985,31(4):469-472. [4]MILLER V S.Use of Elliptic Curves in Cryptography[M]//Advances in Cryptology-CRYPTO’85 Proceedings.Springer Berlin Heidelberg,1986:417-426. [5]SHOR P W.Algorithms for Quantum Computation:Discrete Log and Factoring[C]//Proceedings of the 35th Symposium on Foundations of Computer Science.1994:124-134. [6]GROVER L K.A fast quantum mechanical algorithm for database search[C]//ACM Symposium on the Theory of Computing.1996:212-219. [7]WANG H F .Theoretical study on grover quantum search algorithm[D].Harbin:Harbin Institute of Technology,2010.(in Chinese) 王洪福.Grover量子搜索算法理论研究[D].哈尔滨:哈尔滨工业大学,2010. [8]REGEV O.Quantum computation and lattice problems[C]//Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science.2002:520-529. [9]BRIGHT C.From the Shortest Vector Problem to the Dihedral Hidden Subgroup Problem[J/OL].https://cs.uwaterloo.ca/~cbright/reports/cs667proj.pdf. [10]BERNSTEIN D J,BUCHMANN J,DAHMEN E.抗量子计算密码[M].张焕国,王后珍,杨昌,等译.北京:清华大学出版社,2015. [11]DEUTSCH D.Quantum theory,the Church-Turing principle and the universal quantum computer[J].Proceedings of the Royal Society of London A,1985,400(1818):97-117. [12]SIMON D R.On the power of quantum computation[C]//Symposium on Foundations of Computer Science.IEEE Computer Society,1994:116-123. [13]KITAEV A Y.Quantum measurements and the Abelian stabilizer problem[OL].https://arxiv.org/abs/quant-ph/9511026. [14]NIELSEN M A,CHUANG I L.量子计算和量子信息(一)--量子计算部分[M].赵千川,译.北京:清华大学出版社,2004. [15]DAN B,LIPTON R J.Quantum Cryptanalysis of Hidden Linear Functions[C]//International Cryptology Conference on Advances in Cryptology.Springer-Verlag,1995:424-437. [16]BRASSARD G,HØYER P.An Exact Quantum Polynomial-Time Algorithm for Simon’s Problem[C]//Israel Symposium on the Theory of Computing Systems.IEEE Computer Society,1997:12. [17]JOZSA R.Quantum algorithms and the Fourier transform[C]//Proceedings of the Royal Society of London A:Mathematical,Physical and Engineering Sciences.1997:323-337. [18]JOZSA R.Quantum Factoring.Discrete Logarithms,and the Hidden Subgroup Problem[J].Computing in Science & Engineering,2001,3(2):34-43. [19]MOSCA M.Quantum computer algorithms[D].Oxford:University of Oxford,1999. [20]MOSCA M,EKERT A.The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer[M]//Quantum Computing and Quantum Communications.Springer Berlin Heidelberg,1999:174-188. [21]CHEUNG K K H,MOSCA M.Decomposing Finite Abelian Groups[J].Quantum Information & Computation,2001,1(3):26-32. [22]SUN J.Dihedral Hidden Subgroup problem Based on Quantum Computing Algorithms[D].Nanjing:Nanjing University of Aero-nautics and Astronautics,2012.(in Chinese) 孙静.基于量子计算的二面体群隐含子群问题研究[D].南京:南京航空航天大学,2012. [23]EKERT A,JOZSA R.Quantum Algorithms:Entanglement Enhanced Information Processing[J].Philosophical Transactions Mathematical Physical & Engineering Sciences,1998,356(1743):1769-1782. [24]CLEVE R.The query omplexity of order-finding[C]//15th Annual IEEE Conference on Computational Complexity.IEEE,2000:54-59. [25]ETTINGER M,HØYER P.On Quantum Algorithms for Noncommutative Hidden Subgroups[J].Advances in Applied Ma-thematics,1998,25(3):239-251. [26]RÖETTELER M,BETH T.Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups[OL].https://arxiv.org/abs/quant-ph/9812070. [27]PÜSCHEL M,RÖTTELER M,BETH T.Fast Quantum Fourier Transforms for a Class of Non-abelian Groups[J].Transactions of the American Mathematical Society,1999,362(2):1009-1045. [28]BEALS R,BUHRMAN H,CLEVE R,et al.Quantum Lower Bounds by Polynomials[C]//Symposium on Foundations of Computer Science.IEEE Computer Society,1998:352. [29]ETTINGER M,HOYER P,KNILL E.Hidden Subgroup States are Almost Orthogonal[OL].https://arxiv.org/abs/quant-ph/9901034. [30]BERNSTEIN D J,BUCHMANN J,DAHMEN E.Post Quan-tum Cryptography[M].Berlin:Springer Berlin Heidelberg,2008. [31]WANG F.The Hidden Subgroup Problem[OL].https://ar-xiv.org/abs/1008.0010. [32]MURPHY J N.Analysing the quantum fourier transform for finite groups through the hidden subgroup problem[D].Québec:McGill University,2001. [33]KUPERBERG G.A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem[J].Siam Journal on Computing,2003,35(1):170-188. [34]REGEV O.A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space[J].Procee-dings of Annual Symposium on the Foundations of Computer Science,2004,64(1):124-134. [35]KUPERBERG G.Another subexponential-time quantum algo-rithm for the dihedral hidden subgroup problem[OL].https://arxiv.org/abs/1112.3333. [36]KOBAYASHI H,GALL F L.Dihedral Hidden Subgroup Problem :A Survey(Quantum Computation and Information)[J].Information & Media Technologies,2006,1(10):470-477. [37]JIN G L,YUAN J B.Quantum Cloning-based Quantum Algo-rithm for Dihedral Hidden Subgroup Problem[J].ComputerScien-ce,2014,41(8):183-185.(in Chinese) 金广龙,袁家斌.基于量子克隆的二面体群隐含子群问题量子算法的研究[J].计算机科学,2014,41(8):183-185. [38]LOMONACO S J,KAUFFMAN L H.Is Grover’s Algorithm a Quantum Hidden Subgroup Algorithm?[J].Quantum Information Processing,2007,6(6):461-476. [39]BEALS R.Quantum computation of Fourier transforms over symmetric groups[C]//Twenty-Ninth ACM Symposium on the Theory of Computing.1997:48-53. [40]MOORE C,RUSSELL A,SCHULMAN L J.The Symmetric Group Defies Strong Fourier Sampling[C]//IEEE Symposium on Foundations of Computer Science.IEEE Computer Society,2005:479-490. [41]HALLGREN S,MOORE C,RUSSELL A,et al.Limitations of quantum coset states for graph isomorphism[C]//ACM Symposium on Theory of Computing.Seattle,Wa,USA,2006:604-617. [42]KAWANO Y,SEKIGAWA H.Quantum fourier transform over symmetric groups[C]//International Symposium on Symbolic and Algebraic Computation.ACM,2013:227-234. [43]IVANYOS G,MAGNIEZ F,SANTHA M.Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem[J].International Journal of Foundations of Computer Science,2003,14(5):723-739. [44]FRIEDL K,IVANYOS G,MAGNIEZ F,et al.Hidden translation and orbit coset in quantum computing[C]//Proceedings of the thirty-fifth annual ACM symposium on Theory of computing.ACM,2003:1-9. [45]INUI Y,GALL F L,et al.Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups[J].Quantum Information & Computation,2004,7(5):559-570. [46]MOORE C,ROCKMORE D,RUSSELL A,et al.The power of basis selection in Fourier sampling:Hidden subgroup problems in affine groups[C]//Fifteenth Acm-Siam Symposium on Discrete Algorithms(SODA 2004).New Orleans,Louisiana,USA,2004:1113-1122. [47]BACON D,CHILDS A M,DAM W V.From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups[C]//IEEE Symposium on Foundations of Computer Science,2005(FOCS 2005).IEEE Xplore,2005:469-478. [48]MOORE C,ROCKMORE D,RUSSELL A,et al.The power of strong Fourier sampling:Quantum algorithms for affine groups and hidden shifts[J].SIAM Journal on Computing,2007,37(3):938-958. [49]GONCALVES D N,PORTUGAL R.Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups[OL].https://arxiv.org/abs/1104.1361. [50]CHIA N H,HALLGREN S.How hard is deciding trivial versus nontrivial in the dihedral coset problem?[OL].https://arxiv.org/abs/1608.02003. [51]LI F,BAO W,FU X.A quantum algorithm for the dihedral hidden subgroup problem based on lattice basis reduction algorithm[J].Chinese Science Bulletin,2014,59(21):2552-2557. [52]GOGIOSO S,KISSINGER A.Fully graphical treatment of the quantum algorithm for the Hidden Subgroup Problem[OL].https://arxiv.org/abs/1701.08669. [53]CHILDS A M.Lecture notes on quantum algorithms[OL].https://www.cs.umd.edu/~amchilds/qa. |
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