Advertisement

Obfuscated Fuzzy Hamming Distance and Conjunctions from Subset Product Problems

  • Steven D. GalbraithEmail author
  • Lukas Zobernig
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11891)

Abstract

We consider the problem of obfuscating programs for fuzzy matching (in other words, testing whether the Hamming distance between an n-bit input and a fixed n-bit target vector is smaller than some predetermined threshold). This problem arises in biometric matching and other contexts. We present a virtual-black-box (VBB) secure and input-hiding obfuscator for fuzzy matching for Hamming distance, based on certain natural number-theoretic computational assumptions. In contrast to schemes based on coding theory, our obfuscator is based on computational hardness rather than information-theoretic hardness, and can be implemented for a much wider range of parameters. The Hamming distance obfuscator can also be applied to obfuscation of matching under the \(\ell _1\) norm on \(\mathbb {Z}^n\).

We also consider obfuscating conjunctions. Conjunctions are equivalent to pattern matching with wildcards, which can be reduced in some cases to fuzzy matching. Our approach does not cover as general a range of parameters as other solutions, but it is much more compact. We study the relation between our obfuscation schemes and other obfuscators and give some advantages of our solution.

Notes

Acknowledgements

We thank Trey Li for several corrections and comments. We thank the Marsden Fund of the Royal Society of New Zealand for funding this research, and the reviewers for suggestions.

References

  1. 1.
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, New York (1992)zbMATHGoogle Scholar
  2. 2.
    Barak, B., Bitansky, N., Canetti, R., Kalai, Y.T., Paneth, O., Sahai, A.: Obfuscation for evasive functions. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 26–51. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-54242-8_2CrossRefGoogle Scholar
  3. 3.
    Barak, B., et al.: On the (im)possibility of obfuscating programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 1–18. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44647-8_1CrossRefGoogle Scholar
  4. 4.
    Bartusek, J., Lepoint, T., Ma, F., Zhandry, M.: New techniques for obfuscating conjunctions. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11478, pp. 636–666. Springer, Cham (2019).  https://doi.org/10.1007/978-3-030-17659-4_22CrossRefGoogle Scholar
  5. 5.
    Bellare, M., Stepanovs, I.: Point-function obfuscation: a framework and generic constructions. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9563, pp. 565–594. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49099-0_21CrossRefGoogle Scholar
  6. 6.
    Bishop, A., Kowalczyk, L., Malkin, T., Pastro, V., Raykova, M., Shi, K.: A simple obfuscation scheme for pattern-matching with wildcards. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 731–752. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96878-0_25CrossRefGoogle Scholar
  7. 7.
    Bitansky, N., et al.: The impossibility of obfuscation with auxiliary input or a universal simulator. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 71–89. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44381-1_5CrossRefGoogle Scholar
  8. 8.
    Bitansky, N., Paneth, O.: Point obfuscation and 3-round zero-knowledge. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 190–208. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28914-9_11CrossRefGoogle Scholar
  9. 9.
    Brakerski, Z., Rothblum, G.N.: Obfuscating conjunctions. J. Cryptol. 30(1), 289–320 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brakerski, Z., Vaikuntanathan, V., Wee, H., Wichs, D.: Obfuscating conjunctions under entropic ring LWE. In: 2016 ACM Conference on Innovations in Theoretical Computer Science, pp. 147–156. ACM (2016)Google Scholar
  11. 11.
    Bringer, J., Chabanne, H., Cohen, G., Kindarji, B., Zemor, G.: Theoretical and practical boundaries of binary secure sketches. IEEE Trans. Inf. Forensics Secur. 3(4), 673–683 (2008)CrossRefGoogle Scholar
  12. 12.
    Canetti, R., Rothblum, G.N., Varia, M.: Obfuscation of hyperplane membership. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 72–89. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-11799-2_5CrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, C., Genise, N., Micciancio, D., Polyakov, Y., Rohloff, K.: Implementing token-based obfuscation under (ring) LWE. Cryptology ePrint Archive, Report 2018/1222 (2018). https://eprint.iacr.org/2018/1222
  14. 14.
    Chernoff, H., et al.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23(4), 493–507 (1952)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Contini, S., Lenstra, A.K., Steinfeld, R.: VSH, an efficient and provable collision-resistant hash function. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 165–182. Springer, Heidelberg (2006).  https://doi.org/10.1007/11761679_11CrossRefGoogle Scholar
  16. 16.
    Coster, M.J., Joux, A., LaMacchia, B.A., Odlyzko, A.M., Schnorr, C.P., Stern, J.: Improved low-density subset sum algorithms. Comput. Complex. 2(2), 111–128 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dixon, J.D.: The number of steps in the Euclidean algorithm. J. Number Theory 2(4), 414–422 (1970)MathSciNetCrossRefGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dodis, Y., Reyzin, L., Smith, A.: Fuzzy extractors: how to generate strong keys from biometrics and other noisy data. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 523–540. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_31CrossRefGoogle Scholar
  20. 20.
    Dodis, Y., Smith, A.: Correcting errors without leaking partial information. In: STOC 2005, pp. 654–663. ACM (2005)Google Scholar
  21. 21.
    Ducas, L., Pierrot, C.: Polynomial time bounded distance decoding near minkowski’s bound in discrete logarithm lattices. Des. Codes Crypt. 87(8), 1737–1748 (2019)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fielder, D.C., Alford, C.O.: Pascal’s triangle: top gun or just one of the gang? In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds.) Applications of Fibonacci Numbers, pp. 77–90. Springer, Dordrecht (1991).  https://doi.org/10.1007/978-94-011-3586-3_10CrossRefzbMATHGoogle Scholar
  23. 23.
    Fuller, B., Reyzin, L., Smith, A.: When are fuzzy extractors possible? In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 277–306. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53887-6_10CrossRefGoogle Scholar
  24. 24.
    Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. SIAM J. Comput. 45(3), 882–929 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Goldwasser, S., Kalai, Y., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: Reusable garbled circuits and succinct functional encryption. In: STOC 2013, pp. 555–564. ACM (2013)Google Scholar
  26. 26.
    Goyal, R., Koppula, V., Waters, B.: Lockable obfuscation. In: FOCS 2017, pp. 612–621. IEEE (2017)Google Scholar
  27. 27.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, Oxford (1975)Google Scholar
  28. 28.
    Hensley, D.: The number of steps in the Euclidean algorithm. J. Number Theory 49(2), 142–182 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hoffstein, J., Pipher, J., Silverman, J.H.: An Introduction to Mathematical Cryptography. UTM. Springer, New York (2014).  https://doi.org/10.1007/978-1-4939-1711-2CrossRefzbMATHGoogle Scholar
  30. 30.
    Hurwitz, A.: Über die angenäherte darstellung der irrationalzahlen durch rationale brüche. Math. Ann. 39(2), 279–284 (1891)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Impagliazzo, R., Naor, M.: Efficient cryptographic schemes provably as secure as subset sum. J. Cryptol. 9(4), 199–216 (1996)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Jain, A.K., Nandakumar, K., Nagar, A.: Biometric template security. EURASIP J. Adv. Signal Process. 2008, 113 (2008)CrossRefGoogle Scholar
  33. 33.
    Karabina, K., Canpolat, O.: A new cryptographic primitive for noise tolerant template security. Pattern Recogn. Lett. 80, 70–75 (2016)CrossRefGoogle Scholar
  34. 34.
    Lagarias, J.C., Odlyzko, A.M.: Solving low-density subset sum problems. J. ACM 32(1), 229–246 (1985)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Li, Q., Sutcu, Y., Memon, N.: Secure sketch for biometric templates. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 99–113. Springer, Heidelberg (2006).  https://doi.org/10.1007/11935230_7CrossRefGoogle Scholar
  37. 37.
    Lynn, B., Prabhakaran, M., Sahai, A.: Positive results and techniques for obfuscation. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 20–39. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24676-3_2CrossRefGoogle Scholar
  38. 38.
    Micciancio, D., Mol, P.: Pseudorandom knapsacks and the sample complexity of LWE search-to-decision reductions. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 465–484. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_26CrossRefGoogle Scholar
  39. 39.
    Brier, E., Coron, J.-S., Géraud, R., Maimuţ, D., Naccache, D.: A number-theoretic error-correcting code. In: Bica, I., Naccache, D., Simion, E. (eds.) SECITC 2015. LNCS, vol. 9522, pp. 25–35. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-27179-8_2CrossRefGoogle Scholar
  40. 40.
    Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge University Press, New York (2009)zbMATHGoogle Scholar
  41. 41.
    Sutcu, Y., Li, Q., Memon, N.: Protecting biometric templates with sketch: theory and practice. IEEE Trans. Inf. Forensics Secur. 2(3), 503–512 (2007)CrossRefGoogle Scholar
  42. 42.
    Tuyls, P., Akkermans, A.H.M., Kevenaar, T.A.M., Schrijen, G.-J., Bazen, A.M., Veldhuis, R.N.J.: Practical biometric authentication with template protection. In: Kanade, T., Jain, A., Ratha, N.K. (eds.) AVBPA 2005. LNCS, vol. 3546, pp. 436–446. Springer, Heidelberg (2005).  https://doi.org/10.1007/11527923_45CrossRefGoogle Scholar
  43. 43.
    Wee, H.: On obfuscating point functions. In: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, pp. 523–532. ACM, New York (2005)Google Scholar
  44. 44.
    Wichs, D., Zirdelis, G.: Obfuscating compute-and-compare programs under LWE. In: FOCS 2017, pp. 600–611. IEEE (2017)Google Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

Personalised recommendations