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Quantum Algorithms for the Subset-Sum Problem

  • Daniel J. Bernstein
  • Stacey Jeffery
  • Tanja Lange
  • Alexander Meurer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7932)

Abstract

This paper introduces a subset-sum algorithm with heuristic asymptotic cost exponent below 0.25. The new algorithm combines the 2010 Howgrave-Graham–Joux subset-sum algorithm with a new streamlined data structure for quantum walks on Johnson graphs.

Keywords

subset sum quantum search quantum walks radix trees decoding SVP CVP 

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References

  1. 1.
    — (no editor): 20th annual symposium on foundations of computer science. IEEE Computer Society, New York (1979). MR 82a:68004. See [32] Google Scholar
  2. 2.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM Journal on Computing 37, 210–239 (2007). http://arxiv.org/abs/quant-ph/0311001. Citations in this document: §3, §3, §3 Google Scholar
  3. 3.
    Becker, A., Coron, J.-S., Joux, A.: Improved generic algorithms for hard knapsacks. In: Eurocrypt 2011 [27] (2011). http://eprint.iacr.org/2011/474. Citations in this document: §1, §1.1, §1, §4, §5
  4. 4.
    Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in 2n/20: how 1 + 1 = 0 improves information set decoding. In: Eurocrypt 2012 [28] (2012). http://eprint.iacr.org/2012/026. Citations in this document: §1,§1,§1, §1, §5
  5. 5.
    Bernstein, D.J.: Cost analysis of hash collisions: Will quantum computers make SHARCS obsolete? In: Workshop Record of SHARCS’09: Special-purpose Hardware for Attacking Cryptographic Systems (2009). http://cr.yp.to/papers.html#collisioncost. Citations in this document: §2
  6. 6.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte Der Physik 46, 493–505 (1998). http://arxiv.org/abs/quant-ph/9605034v1. Citations in this document: §2, §2 Google Scholar
  7. 7.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: [20], pp. 53–74 (2002). http://arxiv.org/abs/quant-ph/0005055. Citations in this document: x4 Quantum Algorithms for the Subset-sum Problem 33
  8. 8.
    Brassard, G., Høyer, G., Tapp, A.: Quantum cryptanalysis of hash and claw-free functions. In: LATIN’98 [21], pp. 163–169 (1998). MR 99g:94013. Citations inthis document: §2, §2 Google Scholar
  9. 9.
    Chang, W.-L., Ren, T.-T., Feng, M., Lu, L.C., Lin, K.W., Guo, M.: Quantum algorithms of the subset-sum problem on a quantum computer. International Conference on Information Engineering 2, 54–57 (2009). Citations in this document: §2 Google Scholar
  10. 10.
    Elsenhans, A.-S., Jahnel, J.: The diophantine equation x4 + 2y4= z4+ 4w4. Mathematics of Computation 75, 935–940 (2006). http://www.uni-math.gwdg.de/jahnel/linkstopaperse.html. Citations in this document: §4 Google Scholar
  11. 11.
    Elsenhans, A.-S., Jahnel, J.: The Diophantine equation x4 + 2y4 = z4 + 4w4— a number of improvements (2006). http://www.uni-math.gwdg.de/jahnel/linkstopaperse.html. Citations in this document: §4
  12. 12.
    Gilbert, H. (ed.): Advances in cryptology—EUROCRYPT 2010, 29th annual international conference on the theory and applications of cryptographic techniques, French Riviera, May 30-June 3, 2010, proceedings. LNCS, vol. 6110. Springer (2010). See [17] Google Scholar
  13. 13.
    Goldwasser, S. (ed.): 35th annual IEEE symposium on the foundations of computer science. Proceedings of the IEEE symposium held in Santa Fe, NM, November 20-22, 1994. IEEE (1994). ISBN 0-8186-6580-7. MR 98h:68008. See [30] Google Scholar
  14. 14.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: [26], pp. 212–219 (1996). MR 1427516. http://arxiv.org/abs/quant-ph/9605043
  15. 15.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters 79, 325–328 (1997). http://arxiv.org/abs/quant-ph/9706033. Citations in this document: §2, §2 Google Scholar
  16. 16.
    Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsackproblem. Journal of the ACM 21, 277–292 (1974). Citations in this document: §2 Google Scholar
  17. 17.
    Howgrave-Graham, N., Joux, A.: New generic algorithms for hard knapsacks. In: Eurocrypt 2010 [12] (2010). http://eprint.iacr.org/2010/189. Citations in this document: §1, §1, §1, §1, §4, §5, §5, §5
  18. 18.
    Johnson, D.S., Feige, U. (eds.): Proceedings of the 39th annual ACM symposium on the theory of computing, San Diego, California, USA, June 11-13, 2007. Association for Computing Machinery (2007). ISBN 978-1-59593-631-8. See [23] Google Scholar
  19. 19.
    Lee, D.H., Wang, X. (eds.): Advances in cryptology—ASIACRYPT 2011, 17th international conference on the theory and application of cryptology and information security, Seoul, South Korea, December 4-8, 2011, proceedings. LNCS,vol. 7073. Springer (2011). ISBN 978-3-642-25384-3. See [24] Google Scholar
  20. 20.
    Lomonaco Jr., S.J., Brandt, H.E. (eds.): Quantum computation and information. Papers from the AMS Special Session held in Washington, DC, January 19-21, 2000. Contemporary Mathematics, vol. 305. American Mathematical Society(2002). ISBN 0-8218-2140-7. MR 2003g:81006. See [7] Google Scholar
  21. 21.
    Lucchesi, C.L., Moura, A.V. (eds.): LATIN’98: theoretical informatics. Proceedings of the 3rd Latin American symposium held in Campinas, April 20-24, 1998. LNCS, vol. 1380. Springer (1998). ISBN 3-540-64275-7. MR 99d:68007. See [8] Google Scholar
  22. 22.
    Lyubashevsky, V., Palacio, A., Segev, G.: Public-key cryptographic primitives provably as secure as subset sum. In: TCC 2010 [25], pp. 382–400 (2010). http://eprint.iacr.org/2009/576. Citations in this document: §1, §1
  23. 23.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk.In: STOC 2007 [18], pp. 575–584 (2007). http://arxiv.org/abs/quant-ph/0608026. Citations in this document: §3
  24. 24.
    May, A., Meurer, A., Thomae, E.: Decoding random linear codes in Õ(20:054n).In: Asiacrypt 2011 [19] (2011). http://www.cits.rub.de/imperia/md/content/may/paper/decoding.pdf. Citations in this document: §1, §1, §1
  25. 25.
    Micciancio, D. (ed.): Theory of cryptography, 7th theory of cryptography conference, TCC 2010, Zurich, Switzerland, February 9-11, 2010, proceedings. LNCS,vol. 5978. Springer (2010). ISBN 978-3-642-11798-5. See [22] Google Scholar
  26. 26.
    Miller, G.L. (ed.): Proceedings of the twenty-eighth annual ACM symposium on the theory of computing, Philadelphia, PA, May 22-24, 1996. Association for Computing Machinery (1996). ISBN 0-89791-785-5. MR 97g:68005. See [14] Google Scholar
  27. 27.
    Paterson, K.G. (ed.): Advances in cryptology—EUROCRYPT 2011, 30th annual international conference on the theory and applications of cryptographic techniques, Tallinn, Estonia, May 15-19, 2011, proceedings. LNCS, vol. 6632. Springer (2011). ISBN 978-3-642-20464-7. See [3] Google Scholar
  28. 28.
    Pointcheval, D., Johansson, T. (eds.): Advances in cryptology—EUROCRYPT2012—31st annual international conference on the theory and applications of cryptographic techniques, Cambridge, UK, April 15-19, 2012, proceedings. LNCS, vol. 7237. Springer (2012). ISBN 978-3-642-29010-7. See [4] Google Scholar
  29. 29.
    Schroeppel, R., Shamir, A.: A T = O(2n/2), S = O(2n/4) algorithm for certain NP-complete problems. SIAM Journal on Computing 10, 456–464 (1981). Citations in this document: §4 Google Scholar
  30. 30.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: [13], pp. 124–134 (1994); see also newer version [31]. MR 1489242 Google Scholar
  31. 31.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26, 1484–1509 (1997); see also older version [30]. MR 98i:11108. http://arxiv.org/abs/quant-ph/9508027
  32. 32.
    Wegman, M.N., Lawrence Carter, J.: New classes and applications of hash functions. In: [1], pp. 175–182 (1979); see also newer version [33] Google Scholar
  33. 33.
    Wegman, M.N., Lawrence Carter, J.: New hash functions and their use in authentication and set equality. Journal of Computer and System Sciences 22, 265–279(1981); see also older version [32]. ISSN 0022-0000. MR 82i:68017. Citations inthis document: §3 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Daniel J. Bernstein
    • 1
    • 2
  • Stacey Jeffery
    • 3
  • Tanja Lange
    • 2
  • Alexander Meurer
    • 4
  1. 1.Department of Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenThe Netherlands
  3. 3.Institute for Quantum ComputingUniversity of WaterlooCanada
  4. 4.Horst Görtz Institute for IT-SecurityRuhr-University BochumGermany

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