A Low-Resource Quantum Factoring Algorithm

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10346)


In this paper, we present a factoring algorithm that, assuming standard heuristics, uses just \((\log N)^{2/3+o(1)}\) qubits to factor an integer N in time \(L^{q+o(1)}\) where \(L = \exp ((\log N)^{1/3}(\log \log N)^{2/3})\) and \(q=\root 3 \of {8/3}\approx 1.387\). For comparison, the lowest asymptotic time complexity for known pre-quantum factoring algorithms, assuming standard heuristics, is \(L^{p+o(1)}\) where \(p>1.9\). The new time complexity is asymptotically worse than Shor’s algorithm, but the qubit requirements are asymptotically better, so it may be possible to physically implement it sooner.


  1. 1.
    Artjuhov, M.M.: Certain criteria for primality of numbers connected with the little Fermat theorem. Acta Arith. 12, 355–364 (1966)MathSciNetGoogle Scholar
  2. 2.
    Barbulescu, R.: Algorithms of discrete logarithm in finite fields. Thesis, Université de Lorraine, December 2013.
  3. 3.
    Beauregard, S.: Circuit for Shor’s algorithm using \(2n+3\) qubits. Quantum Inf. Comput. 3(2), 175–185 (2003)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beckman, D., Chari, A.N., Devabhaktuni, S., Preskill, J.: Efficient networks for quantum factoring. Phys. Rev. A 54, 1034–1063 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bennett, C.H.: Time/space trade-offs for reversible computation. SIAM J. Comput. 18(4), 766–776 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernstein, D.J.: Detecting perfect powers in essentially linear time. Math. Comput. 67(223), 1253–1283 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernstein, D.J.: Circuits for integer factorization: a proposal (2001).
  8. 8.
    Bernstein, D.J.: Factoring into coprimes in essentially linear time. J. Algorithms 54(1), 1–30 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bernstein, D.J., Lenstra Jr., H.W., Pila, J.: Detecting perfect powers by factoring into coprimes. Math. Comput. 76(257), 385–388 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Buhler, J.P., Lenstra Jr., H.W., Pomerance, C.: Factoring integers with the number field sieve. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The development of the number field sieve. LNM, vol. 1554, pp. 50–94. Springer, Heidelberg (1993). doi: 10.1007/BFb0091539 CrossRefGoogle Scholar
  11. 11.
    Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 454. The Royal Society (1998)Google Scholar
  12. 12.
    Coppersmith, D.: Modifications to the number field sieve. J. Cryptol. 6(3), 169–180 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Soc. 22(6), 644–654 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gordon, D.: Discrete logarithms in GF(p) using the number field sieve. SIAM J. Discret. Math. 6, 124–138 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gottesman, D.: Fault-tolerant quantum computation with constant overhead. Quantum Inf. Comput. 14(15–16), 1338–1372 (2014). MathSciNetGoogle Scholar
  16. 16.
    Grosshans, F., Lawson, T., Morain, F., Smith, B.: Factoring safe semiprimes with a single quantum query (2015).
  17. 17.
    Kleinjung, T., Aoki, K., Franke, J., Lenstra, A.K., Thomé, E., Bos, J.W., Gaudry, P., Kruppa, A., Montgomery, P.L., Osvik, D.A., Riele, H., Timofeev, A., Zimmermann, P.: Factorization of a 768-bit RSA modulus. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 333–350. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14623-7_18 CrossRefGoogle Scholar
  18. 18.
    Lenstra, A.K., Lenstra Jr., H.W., Manasse, M.S., Pollard, J.M.: The number field sieve. In: STOC 1990: Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, pp. 564–572. ACM, New York (1990)Google Scholar
  19. 19.
    Lenstra Jr., H.W.: Factoring integers with elliptic curves. Ann. Math. (2) 126(3), 649–673 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Menezes, A., Sarkar, P., Singh, S.: Challenges with assessing the impact of NFS advances on the security of pairing-based cryptography. In: Proceedings of Mycrypt 2016 (2016, to appear).
  21. 21.
    Pollard, J.M.: Factoring with cubic integers. In: Lenstra, A.K., Lenstra Jr., H.W. (eds.) The development of the number field sieve. LNM, vol. 1554, pp. 4–10. Springer, Heidelberg (1993). doi: 10.1007/BFb0091536 CrossRefGoogle Scholar
  22. 22.
    Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schirokauer, O.: Discrete logarithms and local units. Philos. Trans. Phys. Sci. Eng. 345, 409–423 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Seifert, J.-P.: Using fewer qubits in Shor’s factorization algorithm via simultaneous diophantine approximation. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 319–327. Springer, Heidelberg (2001). doi: 10.1007/3-540-45353-9_24 CrossRefGoogle Scholar
  25. 25.
    Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Takahashi, Y., Kunihiro, N.: A quantum circuit for Shor’s factoring algorithm using \(2n+2\) qubits. Quantum Inf. Comput. 6(2), 184–192 (2006)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Vedral, V., Barenco, A., Ekert, A.: Quantum networks for elementary arithmetic operations. Phys. Rev. A 54, 147–153 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenThe Netherlands
  3. 3.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  4. 4.Institute for Quantum Computing and Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  5. 5.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  6. 6.Canadian Institute for Advanced ResearchTorontoCanada

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