Advertisement

Breaking Symmetric Cryptosystems Using Quantum Period Finding

  • Marc Kaplan
  • Gaëtan Leurent
  • Anthony Leverrier
  • María  Naya-Plasencia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9815)

Abstract

Due to Shor’s algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the impact of quantum computing on secret key cryptography is much less understood. In this paper, we consider attacks where an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different states. This model gives a lot of power to the adversary, but recent results show that it is nonetheless possible to build secure cryptosystems in it.

We study applications of a quantum procedure called Simon’s algorithm (the simplest quantum period finding algorithm) in order to attack symmetric cryptosystems in this model. Following previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon’s algorithm: finding a collision requires \(\varOmega (2^{n/2})\) queries in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only O(n) queries in the quantum model.

We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption (e.g. CBC-MAC, PMAC, GMAC, GCM, and OCB) are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD, and Minalpher. This is quite surprising compared to the situation with encryption modes: Anand et al. show that standard modes are secure with a quantum-secure PRF.

Second, we show that Simon’s algorithm can also be applied to slide attacks, leading to an exponential speed-up of a classical symmetric cryptanalysis technique in the quantum model.

Keywords

Post-quantum cryptography Symmetric cryptography Quantum attacks Block ciphers Modes of operation Slide attack 

Notes

Acknowledgements

We would like to thank Thomas Santoli and Christian Schaffner for sharing an early stage manuscript of their work [41], Michele Mosca for discussions and LTCI for hospitality. This work was supported by the Commission of the European Communities through the Horizon 2020 program under project number 645622 PQCRYPTO. MK acknowledges funding through grants ANR-12-PDOC-0022-01 and ESPRC EP/N003829/1.

References

  1. 1.
    Abed, F., Fluhrer, S.R., Forler, C., List, E., Lucks, S., McGrew, D.A., Wenzel, J.: Pipelineable on-line encryption. In: Cid and Rechberger [17], pp. 205–223. http://dx.doi.org/10.1007/978-3-662-46706-0_11
  2. 2.
    Alagic, G., Broadbent, A., Fefferman, B., Gagliardoni, T., Schaffner, C., Jules, M.S.: Computational security of quantum encryption. arXiv preprint (2016). arXiv:1602.01441
  3. 3.
    Anand, M.V., Targhi, E.E., Tabia, G.N., Unruh, D.: Post-quantum security of the CBC, CFB, OFB, CTR, and XTS modes of operation. In: Takagi, T., et al. (eds.) PQCrypto 2016. LNCS, vol. 9606, pp. 44–63. Springer, Heidelberg (2016). http://dx.doi.org/10.1007/978-3-319-29360-8_4 CrossRefGoogle Scholar
  4. 4.
    Andreeva, E., Bogdanov, A., Luykx, A., Mennink, B., Tischhauser, E., Yasuda, K.: Parallelizable and authenticated online ciphers. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part I. LNCS, vol. 8269, pp. 424–443. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/978-3-642-42033-7_22 CrossRefGoogle Scholar
  5. 5.
    Bellare, M., Kilian, J., Rogaway, P.: The security of the cipher block chaining message authentication code. J. Comput. Syst. Sci. 61(3), 362–399 (2000). http://dx.doi.org/10.1006/jcss.1999.1694 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernstei, D.J.: Introduction to post-quantum cryptography. In: Bernstein, D.J., Buchmann, J., Dahmen, E. (eds.) Post-Quantum Cryptography, pp. 1–14. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Biryukov, A., Wagner, D.: Slide attacks. In: Knudsen, L.R. (ed.) FSE 1999. LNCS, vol. 1636, pp. 245–259. Springer, Heidelberg (1999). http://dx.doi.org/10.1007/3-540-48519-8_18 CrossRefGoogle Scholar
  8. 8.
    Biryukov, A., Wagner, D.: Advanced slide attacks. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 589–606. Springer, Heidelberg (2000). http://dx.doi.org/10.1007/3-540-45539-6_41 CrossRefGoogle Scholar
  9. 9.
    Black, J.A., Rogaway, P.: CBC MACs for arbitrary-length messages: the three-key constructions. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 197–215. Springer, Heidelberg (2000). http://dx.doi.org/10.1007/3-540-44598-6_12 CrossRefGoogle Scholar
  10. 10.
    Black, J.A., Rogaway, P.: A block-cipher mode of operation for parallelizable message authentication. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 384–397. Springer, Heidelberg (2002). http://dx.doi.org/10.1007/3-540-46035-7_25 CrossRefGoogle Scholar
  11. 11.
    Boneh, D., Dagdelen, Ö., Fischlin, M., Lehmann, A., Schaffner, C., Zhandry, M.: Random oracles in a quantum world. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 41–69. Springer, Heidelberg (2011). http://dx.doi.org/10.1007/978-3-642-25385-0_3 CrossRefGoogle Scholar
  12. 12.
    Boneh, D., Zhandry, M.: Quantum-secure message authentication codes. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 592–608. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/978-3-642-38348-9_35 CrossRefGoogle Scholar
  13. 13.
    Boneh, D., Zhandry, M.: Secure signatures and chosen ciphertext security in a quantum computing world. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 361–379. Springer, Heidelberg (2013). http://dx.doi.org/10.1007/978-3-642-40084-1_21 CrossRefGoogle Scholar
  14. 14.
    Brassard, G., Høyer, P., Kalach, K., Kaplan, M., Laplante, S., Salvail, L.: Merkle puzzles in a quantum world. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 391–410. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Broadbent, A., Jeffery, S.: Quantum homomorphic encryption for circuits of low t-gate complexity. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 609–629. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  16. 16.
    Carter, L., Wegman, M.N.: Universal classes of hash functions (extended abstract). In: Hopcroft, J.E., Friedman, E.P., Harrison, M.A. (eds.) Proceedings of the 9th Annual ACM Symposium on Theory of Computing, Boulder, Colorado, USA, 4–6 May 1977, pp. 106–112. ACM (1977). http://doi.acm.org/10.1145/800105.803400
  17. 17.
    Cid, C., Rechberger, C. (eds.): FSE 2014. LNCS, vol. 8540. Springer, Heidelberg (2015). http://dx.doi.org/10.1007/978-3-662-46706-0 Google Scholar
  18. 18.
    Cogliani, S., Maimuţ, D., Naccache, D., do Canto, R.P., Reyhanitabar, R., Vaudenay, S., Vizár, D.: OMD: a compression function mode of operation for authenticated encryption. In: Joux, A., Youssef, A. (eds.) SAC 2014. LNCS, vol. 8781, pp. 112–128. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/978-3-319-13051-4_7 CrossRefGoogle Scholar
  19. 19.
    Daemen, J., Rijmen, V.: Probability distributions of correlation and differentials in block ciphers. J. Math. Crypt. 1(3), 221–242 (2007). http://dx.doi.org/10.1515/JMC.2007.011 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Damgård, I., Funder, J., Nielsen, J.B., Salvail, L.: Superposition attacks on cryptographic protocols. In: Padró, C. (ed.) ICITS 2013. LNCS, vol. 8317, pp. 142–161. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/978-3-319-04268-8_9 CrossRefGoogle Scholar
  21. 21.
    Dworkin, M.: Recommendation for block cipher modes of operation: the CMAC mode for authentication. NIST Special Publication 800–38B, National Institute for Standards and Technology, May 2005Google Scholar
  22. 22.
    Even, S., Mansour, Y.: A construction of a cipher from a single pseudorandom permutation. J. Crypt. 10(3), 151–162 (1997). http://dx.doi.org/10.1007/s001459900025 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gagliardoni, T., Hülsing, A., Schaffner, C.: Semantic security and indistinguishability in the quantum world. arXiv preprint (2015). arXiv:1504.05255
  24. 24.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Miller, G.L. (ed.) Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, 22–24 May 1996, pp. 212–219. ACM (1996). http://doi.acm.org/10.1145/237814.237866
  25. 25.
    Hoang, V.T., Krovetz, T., Rogaway, P.: Robust authenticated-encryption AEZ and the problem that it solves. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 15–44. Springer, Heidelberg (2015). http://dx.doi.org/10.1007/978-3-662-46800-5_2 Google Scholar
  26. 26.
    Iwata, T., Kurosawa, K.: OMAC: one-key CBC MAC. In: Johansson, T. (ed.) FSE 2003. LNCS, vol. 2887, pp. 129–153. Springer, Heidelberg (2003). http://dx.doi.org/10.1007/978-3-540-39887-5_11 CrossRefGoogle Scholar
  27. 27.
    Iwata, T., Minematsu, K., Guo, J., Morioka, S.: CLOC: authenticated encryption for short input. In: Cid and Rechberger [17] , pp. 149–167. http://dx.doi.org/10.1007/978-3-662-46706-0_8
  28. 28.
    Kaplan, M.: Quantum attacks against iterated block ciphers. CoRR abs/1410.1434 (2014). http://arxiv.org/abs/1410.1434
  29. 29.
    Kaplan, M., Leurent, G., Leverrier, A., Naya-Plasencia, M.: Quantum differential and linear cryptanalysis. CoRR abs/1510.05836 (2015). http://arxiv.org/abs/1510.05836
  30. 30.
    Krovetz, T., Rogaway, P.: The software performance of authenticated-encryption modes. In: Joux, A. (ed.) FSE 2011. LNCS, vol. 6733, pp. 306–327. Springer, Heidelberg (2011). http://dx.doi.org/10.1007/978-3-642-21702-9_18 CrossRefGoogle Scholar
  31. 31.
    Kuwakado, H., Morii, M.: Quantum distinguisher between the 3-round Feistel cipher and the random permutation. In: 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), June 2010, pp. 2682–2685 (2010)Google Scholar
  32. 32.
    Kuwakado, H., Morii, M.: Security on the quantum-type Even-Mansour cipher. In: 2012 International Symposium on Information Theory and Its Applications (ISITA), October 2012, pp. 312–316 (2012)Google Scholar
  33. 33.
    Liskov, M., Rivest, R.L., Wagner, D.: Tweakable block ciphers. J. Crypt. 24(3), 588–613 (2011). http://dx.doi.org/10.1007/s00145-010-9073-y MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Luby, M., Rackoff, C.: How to construct pseudorandom permutations from pseudorandom functions. SIAM J. Comput. 17(2), 373–386 (1988). http://dx.doi.org/10.1137/0217022 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., Makarov, V.: Hacking commercial quantum cryptography systems by tailored bright illumination. Nat. Photonics 4(10), 686–689 (2010)CrossRefGoogle Scholar
  36. 36.
    McGrew, D.A., Viega, J.: The security and performance of the Galois/Counter Mode (GCM) of operation. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 343–355. Springer, Heidelberg (2004). http://dx.doi.org/10.1007/978-3-540-30556-9_27 CrossRefGoogle Scholar
  37. 37.
    Minematsu, K.: Parallelizable Rate-1 authenticated encryption from pseudorandom functions. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 275–292. Springer, Heidelberg (2014). http://dx.doi.org/10.1007/978-3-642-55220-5_16 CrossRefGoogle Scholar
  38. 38.
    Montanaro, A., de Wolf, R.: A survey of quantum property testing. arXiv preprint (2013). arXiv:1310.2035
  39. 39.
    Rogaway, P.: Efficient instantiations of tweakable blockciphers and refinements to modes OCB and PMAC. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 16–31. Springer, Heidelberg (2004). http://dx.doi.org/10.1007/978-3-540-30539-2_2 CrossRefGoogle Scholar
  40. 40.
    Rogaway, P., Bellare, M., Black, J., Krovetz, T.: OCB: a block-cipher mode of operation for efficient authenticated encryption. In: Reiter, M.K., Samarati, P. (eds.) CCS 2001, Proceedings of the 8th ACM Conference on Computer and Communications Security, Philadelphia, Pennsylvania, USA, 6–8 November 2001, pp. 196–205. ACM (2001). http://doi.acm.org/10.1145/501983.502011
  41. 41.
    Santoli, T., Schaffner, C.: Using simon’s algorithm to attack symmetric-key cryptographic primitives. arXiv preprint (2016). arXiv:1603.07856
  42. 42.
    Sasaki, Y., Todo, Y., Aoki, K., Naito, Y., Sugawara, T., Murakami, Y., Matsui, M., Hirose, S.: Minalpher v1.1. CAESAR submission, August 2015Google Scholar
  43. 43.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). http://dx.doi.org/10.1137/S0097539795293172 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Simon, D.R.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Unruh, D.: Non-interactive zero-knowledge proofs in the quantum random oracle model. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 755–784. Springer, Heidelberg (2015). Preprint on IACR ePrint 2014/587Google Scholar
  46. 46.
    Xu, F., Qi, B., Lo, H.K.: Experimental demonstration of phase-remapping attack in a practical quantum key distribution system. New J. Phys. 12(11), 113026 (2010)CrossRefGoogle Scholar
  47. 47.
    Yuval, G.: Reinventing the travois: Encryption/MAC in 30 ROM bytes. In: Biham, E. (ed.) FSE 1997. LNCS, vol. 1267, pp. 205–209. Springer, Heidelberg (1997). http://dx.doi.org/10.1007/BFb0052347 CrossRefGoogle Scholar
  48. 48.
    Zhandry, M.: How to construct quantum random functions. In: 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, 20–23 October 2012, pp. 679–687. IEEE Computer Society (2012). http://dx.doi.org/10.1109/FOCS.2012.37
  49. 49.
    Zhandry, M.: Secure identity-based encryption in the quantum random oracle model. Int. J. Quan. Inf. 13(04), 1550014 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Zhao, Y., Fung, C.H.F., Qi, B., Chen, C., Lo, H.K.: Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems. Phys. Rev. A 78(4), 042333 (2008)CrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  • Marc Kaplan
    • 1
    • 2
  • Gaëtan Leurent
    • 3
  • Anthony Leverrier
    • 3
  • María  Naya-Plasencia
    • 3
  1. 1.LTCI, Télécom ParisTechParis CEDEX 13France
  2. 2.School of InformaticsUniversity of EdinburghEdinburghUK
  3. 3.Inria ParisParisFrance

Personalised recommendations