Efficient Message Authentication Codes with Combinatorial Group Testing

  • Kazuhiko MinematsuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9326)


Message authentication code, MAC for short, is a symmetric-key cryptographic function for authenticity. A standard MAC verification only tells whether the message is valid or invalid, and thus we can not identify which part is corrupted in case of invalid message. In this paper we study a class of MAC functions that enables to identify the part of corruption, which we call group testing MAC (GTM). This can be seen as an application of a classical (non-adaptive) combinatorial group testing to MAC. Although the basic concept of GTM (or its keyless variant) has been proposed in various application areas, such as data forensics and computer virus testing, they rather treat the underlying MAC function as a black box, and exact computation cost for GTM seems to be overlooked. In this paper, we study the computational aspect of GTM, and show that a simple yet non-trivial extension of parallelizable MAC (PMAC) enables \(O(m+t)\) computation for m data items and t tests, irrespective of the underlying test matrix we use, under a natural security model. This greatly improves efficiency from naively applying a black-box MAC for each test, which requires O(mt) time. Based on existing group testing methods, we also present experimental results of our proposal and observe that ours runs as fast as taking single MAC tag, with speed-up from the conventional method by factor around 8 to 15 for \(m=10^4\) to \(10^5\) items.


Message authentication code Combinatorial group testing Data corruption Provable security 



The author would like to thank Kengo Mori, Jun Furukawa and Toshihiko Okamura for fruitful discussions, and Hiroyasu Kubo for initial-stage implementation, and anonymous reviewers for helpful comments.


  1. 1.
    CAESAR : competition for authenticated encryption: security, applicability, and robustness.
  2. 2.
    Recommendation for block cipher modes of operation: the CMAC mode for authentication. NIST special publication 800–38B (2005), national institute of standards and technologyGoogle Scholar
  3. 3.
    Bellare, M., Desai, A., Jokipii, E., Rogaway, P.: A concrete security treatment of symmetric encryption. In: FOCS 1997, pp. 394–403. IEEE Computer Society (1997).
  4. 4.
    Bellare, M., Goldreich, O., Mityagin, A.: The Power of verification queries in message authentication and authenticated encryption. Cryptology ePrint Archive, Report 2004/309 (2004).
  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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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) CrossRefGoogle Scholar
  7. 7.
    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). CrossRefGoogle Scholar
  8. 8.
    De Bonis, A., Di Crescenzo, G.: Combinatorial group testing for corruption localizing hashing. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 579–591. Springer, Heidelberg (2011). CrossRefGoogle Scholar
  9. 9.
    Cheraghchi, M.: Noise-resilient group testing: limitations and constructions. Discrete Appl. Math. 161(1–2), 81–95 (2013). MathSciNetCrossRefGoogle Scholar
  10. 10.
    Di Crescenzo, G., Arce, G.: Data forensics constructions from cryptographic hashing and coding. In: Shi, Y.Q., Kim, H.-J., Perez-Gonzalez, F. (eds.) IWDW 2011. LNCS, vol. 7128, pp. 494–509. Springer, Heidelberg (2012). CrossRefGoogle Scholar
  11. 11.
    Di Crescenzo, G.D., Ge, R., Arce, G.R.: Design and analysis of DBMAC, an error localizing message authentication code. In: GLOBECOM 2004, pp. 2224–2228. IEEE (2004).
  12. 12.
    Di Crescenzo, G., Jiang, S., Safavi-Naini, R.: Corruption-localizing hashing. In: Backes, M., Ning, P. (eds.) ESORICS 2009. LNCS, vol. 5789, pp. 489–504. Springer, Heidelberg (2009). CrossRefGoogle Scholar
  13. 13.
    Di Crescenzo, G.D., Vakil, F.: Cryptographic hashing for virus localization. In: Jahanian, F. (ed.) WORM 2006. pp. 41–48. ACM Press (2006).
  14. 14.
    Dorfman, R.: The detection of defective members of large populations. Ann. Math. Stat. 14(4), 436–440 (1943)CrossRefGoogle Scholar
  15. 15.
    Du, D., Hwang, F.: Combinatorial Group Testing and Its Applications: Series on Applied Mathematics. World Scientific, Singapore (2000). Google Scholar
  16. 16.
    Eppstein, D., Goodrich, M.T., Hirschberg, D.S.: Improved combinatorial group testing algorithms for real-world problem sizes. SIAM J. Comput. 36(5), 1360–1375 (2007). MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fang, J., Jiang, L.Z., Yiu, S., Hui, L.C.: Hard disk integrity check by hashing with combinatorial group testing. In: CSA 2009, pp. 1–6 (2009).
  18. 18.
    Ferguson, N., Lucks, S., Schneier, B., Whiting, D., Bellare, M., Kohno, T., Callas, J., Walker, J.: Skein hash function. SHA-3 Submission (2008).
  19. 19.
    Goldreich, O.: Modern Cryptography, Probabilistic Proofs and Pseudorandomness. Algorithms and Combinatorics. Springer, Heidelberg (1998) Google Scholar
  20. 20.
    Goodrich, M.T., Atallah, M.J., Tamassia, R.: Indexing information for data forensics. In: Ioannidis, J., Keromytis, A.D., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 206–221. Springer, Heidelberg (2005). CrossRefGoogle Scholar
  21. 21.
    Iwata, T., Kurosawa, K.: OMAC: one-key CBC MAC. In: Johansson, T. (ed.) FSE 2003. LNCS, vol. 2887, pp. 129–153. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  22. 22.
    Jean, J., Nikolic, I., Peyrin, T.: Tweaks and keys for block ciphers: the TWEAKEY framework. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8874, pp. 274–288. Springer, Heidelberg (2014). Google Scholar
  23. 23.
    Liskov, M., Rivest, R.L., Wagner, D.: Tweakable block ciphers. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 31–46. Springer, Heidelberg (2002). CrossRefGoogle Scholar
  24. 24.
    Ngo, H.Q., Du, D.Z.: A Survey on combinatorial group testing algorithms with applications to DNA library screening. DIMACS Series in Discrete Mathematics and Theoretical Computer Science (2000)Google Scholar
  25. 25.
    Ngo, H.Q., Porat, E., Rudra, A.: Efficiently decodable error-correcting list disjunct matrices and applications (Extended Abstract). In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 557–568. Springer, Heidelberg (2011). CrossRefGoogle Scholar
  26. 26.
    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). CrossRefGoogle Scholar
  27. 27.
    Thierry-Mieg, N.: A new pooling strategy for high-throughput screening: the shifted transversal design. BMC Bioinform. 7, 28 (2006).
  28. 28.
    Thierry-Mieg, N., Bailly, G.: Interpool: interpreting smart-pooling results. Bioinformatics 24(5), 696–703 (2008)CrossRefGoogle Scholar
  29. 29.
    Wagner, D.: A generalized birthday problem. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 288–304. Springer, Heidelberg (2002). CrossRefGoogle Scholar
  30. 30.
    Zaverucha, G.M., Stinson, D.R.: Group testing and batch verification. In: Kurosawa, K. (ed.) ICITS 2009. LNCS, vol. 5973, pp. 140–157. Springer, Heidelberg (2010). CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.NEC CorporationKawasakiJapan

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