Advertisement

The Chaining Lemma and Its Application

  • Ivan Damgård
  • Sebastian Faust
  • Pratyay Mukherjee
  • Daniele Venturi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9063)

Abstract

We present a new information-theoretic result which we call the Chaining Lemma. It considers a so-called “chain” of random variables, defined by a source distribution X (0) with high min-entropy and a number (say, t in total) of arbitrary functions (T 1,...,T t ) which are applied in succession to that source to generate the chain X (0) \(\underrightarrow{T_1}\) X (1) \(\underrightarrow{T_2}\) X (2)... \(\underrightarrow{T_t}\) X (t) . Intuitively, the Chaining Lemma guarantees that, if the chain is not too long, then either (i) the entire chain is “highly random”, in that every variable has high min-entropy; or (ii) it is possible to find a point j (1 ≤ j ≤ t) in the chain such that, conditioned on the end of the chain i.e. X (j) \(\underrightarrow{T_{j+1}}\) X (j + 1) ... \(\underrightarrow{T_t}\) X (t), the preceding part X (0) \(\underrightarrow{T_{1}}\) X (1) ... \(\underrightarrow{T_j}\) X (j) remains highly random. We think this is an interesting information-theoretic result which is intuitive but nevertheless requires rigorous case-analysis to prove.

We believe that the above lemma will find applications in cryptography. We give an example of this, namely we show an application of the lemma to protect essentially any cryptographic scheme against memory-tampering attacks. We allow several tampering requests, the tampering functions can be arbitrary, however, they must be chosen from a bounded size set of functions that is fixed a priori.

Keywords

Hash Function Source Distribution Target Distribution Conditional Probability Distribution Pseudorandom Function 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggarwal, D., Dodis, Y., Lovett, S.: Non-malleable codes from additive combinatorics. Electronic Colloquium on Computational Complexity (ECCC) 20, 81 (2013), To appear in STOC 2014Google Scholar
  2. 2.
    Agrawal, S., Gupta, D., Maji, H.K., Pandey, O., Prabhakaran, M.: Explicit non-malleable codes resistant to permutations. IACR Cryptology ePrint Archive, 2014:316 (2014)Google Scholar
  3. 3.
    Agrawal, S., Gupta, D., Maji, H.K., Pandey, O., Prabhakaran, M.: Explicit non-malleable codes resistant to permutations and perturbations. IACR Cryptology ePrint Archive, 2014:841 (2014)Google Scholar
  4. 4.
    Agrawal, S., Gupta, D., Maji, H.K., Pandey, O., Prabhakaran, M.: Explicit optimal-rate non-malleable codes against bit-wise tampering and permutations. IACR Cryptology ePrint Archive, 2014:842 (2014)Google Scholar
  5. 5.
    Anderson, R., Kuhn, M.: Tamper resistance: A cautionary note. In: WOEC 1996: Proceedings of the 2nd Conference on Proceedings of the Second USENIX Workshop on Electronic Commerce, pp. 1–1. USENIX Association, Berkeley (1996)Google Scholar
  6. 6.
    Applebaum, B., Harnik, D., Ishai, Y.: Semantic security under related-key attacks and applications. In: ICS, pp. 45–60 (2011)Google Scholar
  7. 7.
    Bellare, M., Cash, D.: Pseudorandom functions and permutations provably secure against related-key attacks. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 666–684. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Bellare, M., Cash, D., Miller, R.: Cryptography secure against related-key attacks and tampering. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 486–503. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Bellare, M., Kohno, T.: A theoretical treatment of related-key attacks: RkA-PRPs, RkA-PRFs, and applications. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 491–506. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Bellare, M., Paterson, K.G., Thomson, S.: RKA security beyond the linear barrier: IBE, encryption and signatures. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 331–348. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Bhattacharyya, R., Roy, A.: Secure message authentication against related key attack. In: Moriai, S. (ed.) FSE 2013. LNCS, vol. 8424, pp. 305–324. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  12. 12.
    Boneh, D., DeMillo, R.A., Lipton, R.J.: On the importance of eliminating errors in cryptographic computations. J. Cryptology 14(2), 101–119 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Cheraghchi, M., Guruswami, V.: Capacity of non-malleable codes. In: ITCS, pp. 155–168 (2014)Google Scholar
  14. 14.
    Cheraghchi, M., Guruswami, V.: Non-malleable coding against bit-wise and split-state tampering. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 440–464. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  15. 15.
    Choi, S.G., Kiayias, A., Malkin, T.: BiTR: Built-in tamper resilience. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 740–758. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Coretti, S., Dodis, Y., Tackmann, B., Venturi, D.: Self-destruct non-malleability. IACR Cryptology ePrint Archive, 2014:866 (2014)Google Scholar
  17. 17.
    Coretti, S., Maurer, U., Tackmann, B., Venturi, D.: From single-bit to multi-bit public-key encryption via non-malleable codes. IACR Cryptology ePrint Archive, 2014:324 (2014)Google Scholar
  18. 18.
    Dachman-Soled, D., Kalai, Y.T.: Securing circuits against constant-rate tampering. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 533–551. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    Dachman-Soled, D., Kalai, Y.T.: Securing circuits and protocols against 1/poly(k) tampering rate. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 540–565. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  20. 20.
    Damgård, I., Faust, S., Mukherjee, P., Venturi, D.: Bounded tamper resilience: How to go beyond the algebraic barrier. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part II. LNCS, vol. 8270, pp. 140–160. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Damgård, I., Faust, S., Mukherjee, P., Venturi, D.: The chaining lemma and its application. IACR Cryptology ePrint Archive, 2014:979 (2014)Google Scholar
  22. 22.
    Dodis, Y., Ostrovsky, R., Reyzin, L., Smith, A.: Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. SIAM J. Comput. 38(1), 97–139 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Dziembowski, S., Pietrzak, K., Wichs, D.: Non-malleable codes. In: ICS, pp. 434–452 (2010)Google Scholar
  24. 24.
    Faust, S., Mukherjee, P., Nielsen, J.B., Venturi, D.: Continuous non-malleable codes. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 465–488. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  25. 25.
    Faust, S., Mukherjee, P., Nielsen, J.B., Venturi, D.: A tamper and leakage resilient von Neumann architecture. IACR Cryptology ePrint Archive, 2014:338 (2014)Google Scholar
  26. 26.
    Faust, S., Mukherjee, P., Venturi, D., Wichs, D.: Efficient non-malleable codes and key-derivation for poly-size tampering circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 111–128. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  27. 27.
    Faust, S., Pietrzak, K., Venturi, D.: Tamper-proof circuits: How to trade leakage for tamper-resilience. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 391–402. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  28. 28.
    Gennaro, R., Lysyanskaya, A., Malkin, T., Micali, S., Rabin, T.: Algorithmic tamper-proof (ATP) security: Theoretical foundations for security against hardware tampering. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 258–277. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  29. 29.
    Goyal, V., O’Neill, A., Rao, V.: Correlated-input secure hash functions. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 182–200. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  30. 30.
    Ishai, Y., Prabhakaran, M., Sahai, A., Wagner, D.: Private circuits II: Keeping secrets in tamperable circuits. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 308–327. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  31. 31.
    Jafargholi, Z., Wichs, D.: Tamper detection and continuous non-malleable codes. Cryptology ePrint Archive, Report 2014/956 (2014), http://eprint.iacr.org/
  32. 32.
    Kalai, Y.T., Kanukurthi, B., Sahai, A.: Cryptography with tamperable and leaky memory. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 373–390. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  33. 33.
    Lucks, S.: Ciphers secure against related-key attacks. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 359–370. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  34. 34.
    Pietrzak, K.: Subspace LWE. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 548–563. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  35. 35.
    Qin, B., Liu, S., Yuen, T.H., Deng, R.H., Chen, K.: Continuous non-malleable key derivation and its application to related-key security. Cryptology ePrint Archive, Report 2015/003 (2015), http://eprint.iacr.org/
  36. 36.
    Wee, H.: Public key encryption against related key attacks. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 262–279. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ivan Damgård
    • 1
  • Sebastian Faust
    • 2
  • Pratyay Mukherjee
    • 1
  • Daniele Venturi
    • 3
  1. 1.Aarhus UniversityAarhusDenmark
  2. 2.EPFLLausanneSwitzerland
  3. 3.Sapienza University of RomeRomaItaly

Personalised recommendations