Advertisement

The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

  • Kelsey HoranEmail author
  • Delaram Kahrobaei
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)

Abstract

In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum cryptography. We review the relationship between HSP and other computational problems, discuss an optimal solution method, and review results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in the proposed infinite group platforms are not yet known.

Keywords

Hidden Subgroup Problem Quantum computation Post-quantum cryptography Group-based cryptography 

References

  1. 1.
    Childs, A.: Lecture notes on quantum algorithms (2017)Google Scholar
  2. 2.
    Hart, D., et al.: A practical cryptanalysis of WalnutDSA\(^{\text{ TM }}\). In: Abdalla, M., Dahab, R. (eds.) PKC 2018. LNCS, vol. 10769, pp. 381–406. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-76578-5_13CrossRefGoogle Scholar
  3. 3.
    Anshel, I., Atkins, D., Goldfeld, D., Gunnells, P.: WalnutDSA(TM): a quantum resistant group theoretic digital signature algorithm. IACR Cryptology ePrint Archive (2017)Google Scholar
  4. 4.
    Wang, L., Wang, L.: Conjugate searching problem vs. hidden subgroup problem. In: The Third International Workshop on Post-Quantum Cryptography, Recent Results Session (2010)Google Scholar
  5. 5.
    Wang, L., Wang, L., Cao, Z., Yang, Y., Niu, X.: Conjugate adjoining problem in braid groups and new design of braid-based signatures. Sci. China Inf. Sci. 53(3), 524–536 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219 (1996)Google Scholar
  7. 7.
    Wagner, N.R., Magyarik, M.R.: A public-key cryptosystem based on the word problem. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 19–36. Springer, Heidelberg (1985).  https://doi.org/10.1007/3-540-39568-7_3CrossRefGoogle Scholar
  8. 8.
    Flores, R., Kahrobaei, D.: Cryptography with right-angled artin groups. Theoret. Appl. Inform. 28, 8–16 (2016)Google Scholar
  9. 9.
    Flores, R., Kahrobaei, D., Koberda, T.: Algorithmic problems in right-angled artin groups: complexity and applications. arXiv preprint arXiv:1802.04870 (2018)
  10. 10.
    Eick, B., Kahrobaei, D.: Polycyclic groups: a new platform for cryptology? arXiv preprint math/0411077 (2004)
  11. 11.
    Gryak, J., Kahrobaei, D.: The status of polycyclic group-based cryptography: a survey and open problems. Groups Complex. Cryptology 8(2), 171–186 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kahrobaei, D., Koupparis, C.: On-commutative digital signatures using non-commutative groups. Groups Complexity Cryptology, pp. 377–384 (2012)Google Scholar
  13. 13.
    Kahrobaei, D., Khan, B.: A non-commutative generalization of ELGamal key exchange using polycyclic groups. In: IEEE Global Telecommunications Conference 2006, pp. 1–5 (2006)Google Scholar
  14. 14.
    Habeeb, M., Kahrobaei, D., Koupparis, C., Shpilrain, V.: Public key exchange using semidirect product of (semi)groups. In: Jacobson, M., Locasto, M., Mohassel, P., Safavi-Naini, R. (eds.) ACNS 2013. LNCS, vol. 7954, pp. 475–486. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38980-1_30CrossRefGoogle Scholar
  15. 15.
    Shpilrain, V., Ushakov, A.: Thompson’s group and public key cryptography. In: Ioannidis, J., Keromytis, A., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 151–163. Springer, Heidelberg (2005).  https://doi.org/10.1007/11496137_11CrossRefzbMATHGoogle Scholar
  16. 16.
    Chatterji, I., Kahrobaei, D., Lu, N.Y.: Cryptosystems using subgroup distortion. Theoret. Appl. Inform. 29, 14–24 (2017)Google Scholar
  17. 17.
    Shpilrain, V., Zapata, G.: Combinatorial group theory and public key cryptography. Appl. Algebra Eng. Commun. Comput. 17(3–4), 291–302 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kahrobaei, D., Shpilrain, V.: Using semidirect product of (semi)groups in public key cryptography. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds.) CiE 2016. LNCS, vol. 9709, pp. 132–141. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40189-8_14CrossRefzbMATHGoogle Scholar
  19. 19.
    Baumslag, G., Fine, B., Xu, X.: Cryptosystems using linear groups. Appl. Algebra Eng. Commun. Comput. 17(3–4), 205–217 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Petrides, G.: Cryptanalysis of the public key cryptosystem based on the word problem on the Grigorchuk groups. In: Paterson, K.G. (ed.) Cryptography and Coding 2003. LNCS, vol. 2898, pp. 234–244. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-40974-8_19CrossRefGoogle Scholar
  21. 21.
    Grigoriev, D., Ponomarenko, I.: Homomorphic public-key cryptosystems over groups and rings. arXiv preprint cs/0309010 (2003)
  22. 22.
    Kobler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem: Its Structural Complexity. Springer Science & Business Media, New York (2012).  https://doi.org/10.1007/978-1-4612-0333-9CrossRefzbMATHGoogle Scholar
  23. 23.
    Boneh, D., Lipton, R.J.: Quantum cryptanalysis of hidden linear functions. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 424–437. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-44750-4_34CrossRefGoogle Scholar
  24. 24.
    Grigoriev, D.: Testing shift-equivalence of polynomials by deterministic, probabilistic and quantum machines. Theoret. Comput. Sci. 180(1–2), 217–228 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Regev, O.: Quantum computation and lattice problems. SIAM J. Comput. 33(3), 738–760 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Alagic, G., Russell, A.: Quantum-secure symmetric-key cryptography based on hidden shifts. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 65–93. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56617-7_3CrossRefGoogle Scholar
  27. 27.
    Simon, D.R.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kitaev, A.: Quantum computations: algorithms and error correction. Russ. Math. Surv. 52(6), 1191–1249 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Brassard, G., Hoyer, P.: An exact quantum polynomial-time algorithm for Simon’s problem. In: Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems, pp. 12–23. IEEE (1997)Google Scholar
  31. 31.
    Grigni, M., Schulman, L., Vazirani, M., Vazirani, U.: Quantum mechanical algorithms for the nonabelian hidden subgroup problem. In: Proceedings of the thirty-third annual ACM Symposium on Theory of Computing, pp. 68–74 (2001)Google Scholar
  32. 32.
    Gavinsky, D.: Quantum solution to the hidden subgroup problem for poly-near-hamiltonian groups. Quantum Inf. Comput. 4(3), 229–235 (2004)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ivanyos, G., Magniez, F., Santha, M.: Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem. Int. J. Found. Comput. Sci. 14(05), 723–739 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Friedl, K., Ivanyos, G., Magniez, F., Santha, M., Sen, P.: Hidden translation and translating coset in quantum computing. SIAM J. Comput. 43(1), 1–24 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Hallgren, S., Russell, A., Ta-Shma, A.: Normal subgroup reconstruction and quantum computation using group representations. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 627–635 (2000)Google Scholar
  36. 36.
    Childs, A., Ivanyos, G.: Quantum computation of discrete logarithms in semigroups. J. Math. Cryptology 8(4), 405–416 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kuperberg, G.: A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. SIAM J. Comput. 35(1), 170–188 (2005)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Regev, O.: A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space. arXiv preprint quant-ph/0406151 (2004)
  39. 39.
    Roetteler, M., Beth, T.: Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups. arXiv preprint quant-ph/9812070 (1998)
  40. 40.
    Friedl, K., Ivanyos, G., Magniez, F., Santha, M., Sen, P.: Hidden translation and orbit coset in quantum computing. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 1–9 (2003)Google Scholar
  41. 41.
    Moore, C., Rockmore, D., Russell, A., Schulman, L.: The power of basis selection in Fourier sampling: hidden subgroup problems in affine groups. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1113–1122 (2004)Google Scholar
  42. 42.
    Inui, Y., Le Gall, F.: An efficient algorithm for the hidden subgroup problem over a class of semi-direct product groups. Technical report (2004)Google Scholar
  43. 43.
    Gonçalves, D., Portugal, R.: Solution to the hidden subgroup problem for a class of noncommutative groups. arXiv preprint arXiv:1104.1361 (2011)
  44. 44.
    Gonçalves, D., Fernandes, T., Cosme, C.: An efficient quantum algorithm for the hidden subgroup problem over some non-abelian groups. TEMA (São Carlos) 18(2), 215–223 (2017)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Ettinger, M., Høyer, P., Knill, E.: The quantum query complexity of the hidden subgroup problem is polynomial. Inf. Process. Lett. 91(1), 43–48 (2004)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Kissinger, A., Gogioso, S.: Fully graphical treatment of the quantum algorithm for the hidden subgroup problem. arXiv preprint quant-ph 1701.08669 (2017)
  47. 47.
    Eisenträger, K., Hallgren, S., Kitaev, A., Song, F.: A quantum algorithm for computing the unit group of an arbitrary degree number field. In: Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, pp. 293–302 (2014)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Graduate Center, CUNYNew YorkUSA
  2. 2.The Graduate Center and NYCCT, CUNYBrooklynUSA
  3. 3.New York UniversityNew YorkUSA

Personalised recommendations