Fast Algorithms for the Free Riders Problem in Broadcast Encryption

  • Zulfikar Ramzan
  • David P. Woodruff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4117)

Abstract

We provide algorithms to solve the free riders problem in broadcast encryption. In this problem, the broadcast server is allowed to choose some small subset F of the revoked set R of users to allow to decrypt the broadcast, despite having been revoked. This may allow the server to significantly reduce network traffic while only allowing a small set of non-privileged users to decrypt the broadcast.

Although there are worst-case instances of broadcast encryption schemes where the free riders problem is difficult to solve (or even approximate), we show that for many specific broadcast encryption schemes, there are efficient algorithms. In particular, for the complete subtree method [25] and some other schemes in the subset-cover framework, we show how to find the optimal assignment of free riders in O(|R||F|) time, which is independent of the total number of users. We also define an approximate version of this problem, and study specific distributions of R for which this relaxation yields even faster algorithms.

Along the way we develop the first approximation algorithms for the following problem: given two integer sequences a 1a 2 ≥⋯≥a n and b 1b 2 ≥⋯≥b n , output for all i, an integer j′ for which a j + b \(_{i--{\it j}\prime}\) ≤(1+ε) min j (a j + b \(_{i--{\it j}}\)). We show that if the differences a i a i + 1, b i b i + 1 are bounded, then there is an O(n 4/3/ε 2/3)-time algorithm for this problem, improving upon the O(n 2) time of the naive algorithm.

Keywords

free riders broadcast encryption applications 

References

  1. 1.
    Abdalla, M., Shavitt, Y., Wool, A.: Key Management for Restricted Multicast Using Broadcast Encryption. ACM Trans. on Networking 8(4) (2000)Google Scholar
  2. 2.
    Aiello, W., Lodha, S.P., Ostrovsky, R.: Fast digital identity revocation. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, p. 137. Springer, Heidelberg (1998)Google Scholar
  3. 3.
    Anzai, J., Matsuzaki, N., Matsumoto, T.: A Quick Group Key Distribution Scheme with “Entity Revocation”. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 333–347. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Asano, T.: A Revocation Scheme with Minimal Storage at Receivers. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 433–450. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Azuma, K.: Weighted Sums of Certain Dependent Random Variables. Tôhoku Math. Journ. 19, 357–367 (1967)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Berkovits, S.: How to Broadcast a Secret. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 535–541. Springer, Heidelberg (1991)Google Scholar
  7. 7.
    Canetti, R., Malkin, T.G., Nissim, K.: Efficient Communication-Storage Tradeoffs for Multicast Encryption. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, p. 459. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Chan, T.M.: All-pairs shortest paths with real weights in O(n 3/logn) time. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 318–324. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. The MIT Press/McGraw-Hill (1990)Google Scholar
  10. 10.
    Content Protection for Pre-recorded Media Specification and Content Protection for Recordable Media Specification, available from: http://www.4centity.com/tech/cprm
  11. 11.
    van Emde Boas, P.: Preserving order in a forest in less than logarithmic time and linear space. Inform. Process. Lett. 6(3), 80–82 (1977)MATHCrossRefGoogle Scholar
  12. 12.
    van Emde Boas, P., Kaas, R., Zijlstra, E.: Design and implementation of an efficient priority queue. Math Systems Theory 10(2) (1976/1977) (Also FOCS 1975)Google Scholar
  13. 13.
  14. 14.
    Fiat, A., Naor, M.: Broadcast encryption. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 480–491. Springer, Heidelberg (1994)Google Scholar
  15. 15.
    Gentry, C., Ramzan, Z.: RSA Accumulator Based Broadcast Encryption. In: Proc. Information Security Conference (2004)Google Scholar
  16. 16.
    Han, Y.: Deterministic sorting in O(n loglogn) time and linear space. Journal of Algorithms, 96–105 (2004)Google Scholar
  17. 17.
    Halevy, D., Shamir, A.: The LSD Broadcast Encryption Scheme. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 47–60. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Han, Y., Thorup, M.: Sorting in \(O(n \sqrt{\log \log n})\) expected time and linear space. In: FOCS, pp. 135–144 (2002)Google Scholar
  19. 19.
    Kim, Y., Perrig, A., Tsudik, G.: Simple and Fault-Tolerant Key Agreement for Dynamic Collaborative Groups. In: Proc. Of ACM CCS (2000)Google Scholar
  20. 20.
    Kumar, R., Rajagopalan, S., Sahai, A.: Coding Constructions for Blacklisting Problems without Computational Assumptions. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 609–623. Springer, Heidelberg (1999)Google Scholar
  21. 21.
    Luby, M., Staddon, J.: Combinatorial Bounds for Broadcast Encryption. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 512–526. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  22. 22.
    Matsuzaki, N., Anzai, J., Matsumoto, T.: Light Weight Broadcast Exclusion Using Secret Sharing. In: Clark, A., Boyd, C., Dawson, E.P. (eds.) ACISP 2000. LNCS, vol. 1841. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  23. 23.
    Mitra Iolus, S.: A Framework for Scalable Secure Multicasting. In: Proc. of ACM SIGCOMM (September 1997)Google Scholar
  24. 24.
    McGrew, D.A., Sherman, A.T.: Key Establishment in Large Dynamic Groups Using One-Way Function Trees. Manuscript available at: http://www.csee.umbc.edu/~Sherman/Papers/itse.ps
  25. 25.
    Naor, D., Naor, M., Lotspiech, J.: Revocation and Tracing Schemes for Stateless Receivers. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, p. 41. Springer, Heidelberg (2001) (Full version: ECCC Report No. 43, 2002)CrossRefGoogle Scholar
  26. 26.
    Panconesi, A., Srinivasan, A.: Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds. In: SICOMP (1997)Google Scholar
  27. 27.
    Poovendran, R., Baras, J.S.: An Information Theoretic Analysis of Rooted-Tree Based Secure Multicast Key Distribution Scheme. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, p. 624. Springer, Heidelberg (1999)Google Scholar
  28. 28.
    Ramzan, Z., Woodruff, D.P.: Fast Algorithms for the Free Riders Problem in Broadcast Encryption. IACR ePrint Archive (2006), http://eprint.iacr.org
  29. 29.
    Wallner, D.M., Harder, E.J., Agee, R.C.: Key Management for Multicast: Issues and Architectures. Internet Draft (September 1998)Google Scholar
  30. 30.
    Wong, C.K., Gouda, M., Lam, S.S.: Secure Group Communications Using Key Graphs. In: Proc. of SIGCOMM (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zulfikar Ramzan
    • 1
  • David P. Woodruff
    • 2
    • 3
  1. 1.Symantec, Inc. 
  2. 2.MIT 
  3. 3.Tsinghua University 

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