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Multi-user Collisions: Applications to Discrete Logarithm, Even-Mansour and PRINCE

  • Pierre-Alain Fouque
  • Antoine Joux
  • Chrysanthi Mavromati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8873)

Abstract

In this paper, we investigate the multi-user setting both in public and in secret-key cryptanalytic applications. In this setting, the adversary tries to recover keys of many users in parallel more efficiently than with classical attacks, i.e., the number of recovered keys multiplied by the time complexity to find a single key, by amortizing the cost among several users. One possible scenario is to recover a single key in a large set of users more efficiently than to recover a key in the classical model. Another possibility is, after some shared precomputation, to be able to learn individual keys very efficiently. This latter model is close to traditional time/memory tradeoff attacks with precomputation. With these goals in mind, we introduce two new algorithmic ideas to improve collision-based attacks in the multi-user setting. Both ideas are derived from the parallelizable collision search as proposed by van Oorschot and Wiener. This collision search uses precomputed chains obtained by iterating some basic function. In our cryptanalytic application, each pair of merging chains can be used to correlate the key of two distinct users. The first idea is to construct a graph, whose vertices are keys and whose edges are these correlations. When the graph becomes connected, we simultaneously recover all the keys. Thanks to random graph analysis techniques, we can show that the number of edges that are needed to make this event occurs is small enough to obtain some improved attacks. The second idea modifies the basic technique of van Oorschot and Wiener: instead of waiting for two chains to merge, we now require that they become parallel.

We first show that, using the first idea alone, we can recover the discrete logarithms of L users in a group of size N in time \(\widetilde{O}(\sqrt{NL})\). We put these two ideas together and we show that in the multi-user Even-Mansour scheme, all the keys of L = N 1/3 users can be found with N 1/3 + ε queries for each user (where N is the domain size). Finally, we consider the PRINCE block cipher (with 128-bit keys and 64-bit blocks) and find the keys of 2 users among a set of 232 users in time 265. We also describe a new generic attack in the classical model for PRINCE.

Keywords

Random Graph Distinguished Point Block Cipher Discrete Logarithm Giant Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Bernstein, D.J., Lange, T.: Computing Small Discrete Logarithms Faster. In: Galbraith, S., Nandi, M. (eds.) INDOCRYPT 2012. LNCS, vol. 7668, pp. 317–338. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J., Lange, T.: Non-uniform Cracks in the Concrete: The Power of Free Precomputation. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part II. LNCS, vol. 8270, pp. 321–340. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Biham, E.: How to decrypt or even substitute DES-encrypted messages in 228 steps. Inf. Process. Lett. 84(3), 117–124 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Biryukov, A., Mukhopadhyay, S., Sarkar, P.: Improved Time-Memory Trade-Offs with Multiple Data. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 110–127. Springer, Heidelberg (2006), http://dx.doi.org/10.1007/11693383_8 CrossRefGoogle Scholar
  5. 5.
    Biryukov, A., Shamir, A.: Cryptanalytic Time/Memory/Data Tradeoffs for Stream Ciphers. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 1–13. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge studies in advanced mathematics (2001)Google Scholar
  7. 7.
    Borghoff, J., Canteaut, A., Güneysu, T., Kavun, E.B., Knezevic, M., Knudsen, L.R., Leander, G., Nikov, V., Paar, C., Rechberger, C., Rombouts, P., Thomsen, S.S., Yalçın, T.: PRINCE – A Low-Latency Block Cipher for Pervasive Computing Applications. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 208–225. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Canteaut, A., Fuhr, T., Gilbert, H., Naya-Plasencia, M., Reinhard, J.-R.: Multiple Differential Cryptanalysis of Round-Reduced PRINCE (Full version). IACR Cryptology ePrint Archive, 2014:89 (2014)Google Scholar
  9. 9.
    Canteaut, A., Naya-Plasencia, M., Vayssière, B.: Sieve-in-the-middle: Improved MITM attacks. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 222–240. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Chatterjee, S., Menezes, A., Sarkar, P.: Another Look at Tightness. In: Miri, A., Vaudenay, S. (eds.) SAC 2011. LNCS, vol. 7118, pp. 293–319. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Costello, K.P., Vu, V.H.: The rank of random graphs. Random Structures & Algorithms 33(3), 269–285 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Daemen, J.: Limitations of the Even-Mansour Construction. In: Matsumoto, T., Imai, H., Rivest, R.L. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 495–498. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  13. 13.
    Dunkelman, O., Keller, N., Shamir, A.: Minimalism in Cryptography: The Even-Mansour Scheme Revisited. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 336–354. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Even, S., Mansour, Y.: A construction of a cipher from a single pseudorandom permutation. In: Imai, H., Rivest, R., Matsumoto, T. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 210–224. Springer, Heidelberg (1993)Google Scholar
  15. 15.
    Jean, J., Nikolic, I., Peyrin, T., Wang, L.: S. Wu Security Analysis of PRINCE. In: Fast Software Encryption - 20th International Workshop, FSE 2013, Singapore, March 11-13, 2013. Revised Selected Papers, pp. 92–111 (2013)Google Scholar
  16. 16.
    Kilian, J., Rogaway, P.: How to Protect DES Against Exhaustive Key Search (an Analysis of DESX). J. Cryptology 14(1), 17–35 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kuhn, F., Struik, R.: Random walks revisited: Extensions of pollard’s rho algorithm for computing multiple discrete logarithms. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 212–229. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Lampe, R., Patarin, J., Seurin, Y.: An Asymptotically Tight Security Analysis of the Iterated Even-Mansour Cipher. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 278–295. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Lee, H.T., Cheon, J.H., Hong, J.: Accelerating ID-based Encryption based on Trapdoor DL using Pre-computation. Cryptology ePrint Archive, Report 2011/187 (2011), http://eprint.iacr.org/
  20. 20.
    Menezes, A.: Another Look at Provable Security. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 8–8. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Pollard, J.M.: Kangaroos, Monopoly and Discrete Logarithms. J. Cryptology 13(4), 437–447 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Quisquater, J.-J., Delescaille, J.-P.: How Easy Is Collision Search. New Results and Applications to DES. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 408–413. Springer, Heidelberg (1990)Google Scholar
  23. 23.
    Soleimany, H., Blondeau, C., Yu, X., Wu, W., Nyberg, K., Zhang, H., Zhang, L., Wang, Y.: Reflection Cryptanalysis of PRINCE-Like Ciphers. In: Fast Software Encryption - 20th International Workshop, FSE 2013, Singapore, March 11-13, 2013. Revised Selected Papers, pp. 71–91 (2013)Google Scholar
  24. 24.
    van Oorschot, P.C., Wiener, M.J.: Parallel Collision Search with Cryptanalytic Applications. J. Cryptology 12(1), 1–28 (1999)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  • Pierre-Alain Fouque
    • 1
  • Antoine Joux
    • 2
  • Chrysanthi Mavromati
    • 3
  1. 1.Université Rennes 1 and Institut Universitaire de FranceFrance
  2. 2.CryptoExperts, France and Chaire de Cryptologie de la Fondation de l’UPMC, Laboratoire d’Informatique de Paris 6UPMC Sorbonne UniversitésFrance
  3. 3.Sogeti/ESEC R&D Lab, France, Université de Versailles Saint-Quentin-en-YvelinesFrance

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