Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups

Author Martin Roetteler

Thumbnail PDF


  • Filesize: 0.67 MB
  • 16 pages

Document Identifiers

Author Details

Martin Roetteler

Cite AsGet BibTex

Martin Roetteler. Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set and present a general algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special cases of this framework which include a) Paley difference sets based on quadratic residues in finite fields which allow to recover the shifted Legendre function quantum algorithm, b) Hadamard difference sets which allow to recover the shifted bent function quantum algorithm, and c) Singer difference sets which allow us to define instances of the dihedral hidden subgroup problem which can be efficiently solved on a quantum computer.
  • Quantum algorithms
  • hidden shift problem
  • hidden subgroup problem
  • difference sets
  • Fourier transforms


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Dave Bacon, Andrew M. Childs, and Wim van Dam. Optimal measurements for the dihedral hidden subgroup problem. Chicago Journal of Theoretical Computer Science, 2006(2), Oct 2006. quant-ph/0501044. URL: http://cjtcs.cs.uchicago.edu/articles/2006/2/contents.html.
  2. Robert Beals. Quantum computation of Fourier transforms over symmetric groups. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, STOC'97, pages 48-53, New York, NY, USA, 1997. ACM. URL: http://dx.doi.org/10.1145/258533.258548.
  3. Thomas Beth, Dieter Jungnickel, and Hanfried Lenz. Design Theory, volume I. Cambridge University Press, 2nd edition, 1999. Google Scholar
  4. Thomas Beth, Dieter Jungnickel, and Hanfried Lenz. Design Theory, volume II. Cambridge University Press, 2nd edition, 1999. Google Scholar
  5. Dan Boneh and Richard Lipton. Quantum cryptanalysis of hidden linear functions. In Advances in Cryptology — CRYPT0’ 95, volume 963 of Lecture Notes in Computer Science, pages 424-437. Springer, 1995. URL: http://dx.doi.org/10.1007/3-540-44750-4_34.
  6. Gilles Brassard and Peter Høyer. An exact polynomial-time algorithm for Simon’s problem. In Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems, pages 12-33. ISTCS, IEEE Computer Society Press, 1997. arXiv quant-ph/9704027. Google Scholar
  7. Andrew M. Childs, Robin Kothari, Maris Ozols, and Martin Roetteler. Easy and hard functions for the Boolean hidden shift problem. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2013, May 21-23, 2013, Guelph, Canada, pages 50-79, 2013. Google Scholar
  8. Andrew M. Childs, Leonard J. Schulman, and Umesh V. Vazirani. Quantum algorithms for hidden nonlinear structures. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 395-404, 2007. Google Scholar
  9. Andrew M. Childs and Wim van Dam. Quantum algorithms for algebraic problems. Rev. Mod. Phys., 82:1-52, Jan 2010. arXiv: 0812.0380. URL: http://dx.doi.org/10.1103/RevModPhys.82.1.
  10. Andrew M. Childs and Pawel Wocjan. On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems. Quantum Information and Computation, 7(5):504-521, Jul 2007. quant-ph/0510185. URL: http://www.rintonpress.com/journals/qiconline.html#v7n56.
  11. John F. Dillon. A survey of bent functions. The NSA technical journal, pages 191-215, 1972. Google Scholar
  12. Kirsten Eisenträger, Sean Hallgren, Alexei Kitaev, and Fang Song. A quantum algorithm for computing the unit group of an arbitrary degree number field. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 293-302, 2014. Google Scholar
  13. Mark Ettinger and Peter Høyer. A quantum observable for the graph isomorphism problem. quant-ph/9901029, 1999. Google Scholar
  14. Mark Ettinger and Peter Høyer. On quantum algorithms for noncommutative hidden subgroups. Advances in Applied Mathematics, 25(3):239-251, 2000. quant-ph/9807029. URL: http://dx.doi.org/10.1006/aama.2000.0699.
  15. Katalin Friedl, Gábor Ivanyos, Frédéric Magniez, Miklos Santha, and Pranab Sen. Hidden translation and orbit coset in quantum computing. In Proceedings of the 35fth Annual ACM Symposium on Theory of Computing (STOC'03), pages 1-9. ACM, 2002. quant-ph/0211091. URL: http://dx.doi.org/10.1145/780542.780544.
  16. Dmitry Gavinsky, Martin Roetteler, and Jérémie Roland. Quantum algorithm for the boolean hidden shift problem. In Computing and Combinatorics, volume 6842 of Lecture Notes in Computer Science, pages 158-167. Springer Berlin / Heidelberg, 2011. arXiv: 1103.3017. URL: http://dx.doi.org/10.1007/978-3-642-22685-4_14.
  17. Mirmojtaba Gharibi. Reduction from non-injective hidden shift problem to injective hidden shift problem. Quantum Information and Computation, 13(3&4):212-230, 2013. Google Scholar
  18. Sean Hallgren. Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. J. ACM, 54(1):4:1-4:19, Mar 2007. URL: http://dx.doi.org/10.1145/1206035.1206039.
  19. Peter Høyer. Efficient quantum transforms. quant-ph/9702028, 1997. Google Scholar
  20. Bertram Huppert. Endliche Gruppen I. Springer, 1967. Google Scholar
  21. Gábor Ivanyos. On solving systems of random linear disequations. Quantum Information and Computation, 8(6&7):579-594, 2008. arXiv: 0704.2988. URL: http://www.rintonpress.com/journals/qiconline.html#v8n67.
  22. Richard Jozsa. Quantum algorithms and the Fourier transform. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1969):323-337, 1998. quant-ph/9707033. URL: http://dx.doi.org/10.1098/rspa.1998.0163.
  23. Richard Jozsa. Quantum factoring, discrete logarithms, and the hidden subgroup problem. Computing in Science Engineering, 3(2):34-43, Mar/Apr 2001. quant-ph/0012084. URL: http://dx.doi.org/10.1109/5992.909000.
  24. Richard Jozsa. Quantum computation in algebraic number theory: Hallgren’s efficient quantum algorithm for solving Pell’s equation. Annals of Physics, 306(2):241-279, 2003. quant-ph/0302134. URL: http://dx.doi.org/10.1016/S0003-4916(03)00067-8.
  25. Phillip Kaye, Raymond Laflamme, and Michele Mosca. An Introduction to Quantum Computing. Oxford University Press, 2007. Google Scholar
  26. Alexei Yu. Kitaev. Quantum measurements and the abelian stabilizer problem, 1995. quant-ph/9511026. Google Scholar
  27. Alexei Yu. Kitaev, Alexander Shen, and Mikhail N. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. American Mathematical Society, 2002. Google Scholar
  28. Greg Kuperberg. A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. SIAM Journal on Computing, 35(1):170-188, 2005. quant-ph/0302112. URL: http://dx.doi.org/10.1137/S0097539703436345.
  29. Greg Kuperberg. Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem. arXiv: 1112.3333, 2011. URL: http://arxiv.org/abs/1112.3333.
  30. Eric S. Ladner. Symmetric Designs: An Algebraic Approach, volume 74 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1983. Google Scholar
  31. Chris Lomont. The hidden subgroup problem - review and open problems. quant-ph/0411037, 2004. Google Scholar
  32. F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland, Amsterdam, 1977. Google Scholar
  33. Cristopher Moore, Daniel Rockmore, Alexander Russell, and Leonard J. Schulman. The power of strong Fourier sampling: Quantum algorithms for affine groups and hidden shifts. SIAM J. Comput., 37(3):938-958, Jun 2007. quant-ph/0503095. URL: http://dx.doi.org/10.1137/S0097539705447177.
  34. Michele Mosca and Artur Ekert. The hidden subgroup problem and eigenvalue estimation on a quantum computer. In Quantum Computing and Quantum Communications, volume 1509 of Lecture Notes in Computer Science, pages 174-188. Springer, 1999. quant-ph/9903071. URL: http://dx.doi.org/10.1007/3-540-49208-9_15.
  35. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010. Google Scholar
  36. Maris Ozols, Martin Roetteler, and Jérémie Roland. Quantum rejection sampling. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS'12, pages 290-308, New York, NY, USA, 2012. ACM. arXiv: 1103.2774. Google Scholar
  37. Josef Pieprzyk, Thomas Hardjono, and Jennifer Seberry. Fundamentals of computer security. Springer, 2003. Google Scholar
  38. Alexander Pott, Vijay Kumar, Tor Helleseth, and Dieter Jungnickel, editors. Difference sets, sequences, and their correlations, volume 542 of NATO Science Series. Kluwer, 1998. Google Scholar
  39. Oded Regev. New lattice-based cryptographic constructions. J. ACM, 51(6):899-942, 2004. Google Scholar
  40. Oded Regev. Quantum computation and lattice problems. SIAM Journal on Computing, 33(3):738-760, 2004. Google Scholar
  41. Oded Regev. A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space. quant-ph/0406151, 2004. Google Scholar
  42. Martin Roetteler. Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm. In Proceedings of the 34st International Symposium on Mathematical Foundations of Computer Science (MFCS'09), volume 5734 of Lecture Notes in Computer Science, pages 663-674. Springer, 2009. arXiv: 0911.4724. URL: http://dx.doi.org/10.1007/978-3-642-03816-7_56.
  43. Martin Roetteler. Quantum algorithms for highly non-linear Boolean functions. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10), pages 448-457, 2010. arXiv: 0811.3208. URL: http://arxiv.org/abs/0811.3208.
  44. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484-1509, 1997. Preliminary version in FOCS 1994. URL: http://dx.doi.org/10.1137/S0097539795293172.
  45. Douglas R. Stinson. Combinatorial Designs: Constructions and Analysis. Springer, 2003. Google Scholar
  46. Richard J. Turyn. Character sums and difference sets. Pacific Journal of Mathematics, 15(1):319-346, 1965. Google Scholar
  47. Wim van Dam, Sean Hallgren, and Lawrence Ip. Quantum algorithms for some hidden shift problems. SIAM Journal on Computing, 36(3):763-778, 2006. quant-ph/0211140. URL: http://dx.doi.org/10.1137/S009753970343141X.
  48. Wim van Dam and Gadiel Seroussi. Quantum algorithms for estimating Gauss sums. quant-ph/0207131, 2002. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail