Dihedral Hidden Subgroup Problem: A Survey

After Shor's discovery of an efficient quantum algorithm for integer factoring, hidden subgroup problems play a central role in developing efficient quantum algorithms. In spite of many intensive studies, no efficient quantum algorithms are known for hidden subgroup problems for many non-Abelian groups. Of particular interest are the hidden subgroup problems for the symmetric group and for the dihedral group, because an efficient algorithm for the former implies an efficient solution to the graph isomorphism problem, and that for the latter essentially solves a certain lattice-related problem whose hardness is assumed in cryptography. This paper focuses on the latter case and gives a comprehensive survey of known facts related to the dihedral hidden subgroup problem.

[1]  Alexander Russell,et al.  The Symmetric Group Defies Strong Fourier Sampling: Part I , 2005, ArXiv.

[2]  Robert Beals,et al.  Quantum computation of Fourier transforms over symmetric groups , 1997, STOC '97.

[3]  Mark Ettinger,et al.  On Quantum Algorithms for Noncommutative Hidden Subgroups , 1998, STACS.

[4]  Cynthia Dwork,et al.  A public-key cryptosystem with worst-case/average-case equivalence , 1997, STOC '97.

[5]  O. Regev.  A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space , 2004, quant-ph/0406151.

[6]  Alexei Y. Kitaev,et al.  Quantum measurements and the Abelian Stabilizer Problem , 1995, Electron. Colloquium Comput. Complex..

[7]  Richard J. Lipton,et al.  Quantum Cryptanalysis of Hidden Linear Functions (Extended Abstract) , 1995, CRYPTO.

[8]  Oded Regev,et al.  Quantum computation and lattice problems , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[9]  Oded Regev,et al.  New lattice based cryptographic constructions , 2003, STOC '03.

[10]  Alexander Russell,et al.  The symmetric group defies strong Fourier sampling , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[11]  Alexander Russell,et al.  The power of basis selection in fourier sampling: hidden subgroup problems in affine groups , 2004, SODA '04.

[12]  Emanuel Knill,et al.  The quantum query complexity of the hidden subgroup problem is polynomial , 2004, Inf. Process. Lett..

[13]  Julia Kempe,et al.  The hidden subgroup problem and permutation group theory , 2004, SODA '05.

[14]  Jacques Stern,et al.  Cryptanalysis of the Ajtai-Dwork Cryptosystem , 1998, CRYPTO.

[15]  Greg Kuperberg.  A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem , 2005, SIAM J. Comput..

[16]  Alexander Russell,et al.  For distinguishing conjugate hidden subgroups, the pretty good measurement is as good as it gets , 2007, Quantum Inf. Comput..

[17]  Dave Bacon,et al.  Optimal measurements for the dihedral hidden subgroup problem , 2005, Chic. J. Theor. Comput. Sci..

[18]  William K. Wootters,et al.  A ‘Pretty Good’ Measurement for Distinguishing Quantum States , 1994 .

[19]  Daniel R. Simon,et al.  On the power of quantum computation , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[20]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[21]  Oded Goldreich,et al.  Eliminating Decryption Errors in the Ajtai-Dwork Cryptosystem , 1997, Electron. Colloquium Comput. Complex..

[22]  F. Gall,et al.  An Efficient Algorithm for the Hidden Subgroup Problem over a Class of Semi-direct Product Groups , 2004 .

[23]  Frédéric Magniez,et al.  Hidden translation and orbit coset in quantum computing , 2002, STOC '03.