Advertisement

Bisimilarity Distances for Approximate Differential Privacy

  • Dmitry Chistikov
  • Andrzej S. Murawski
  • David PurserEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11138)

Abstract

Differential privacy is a widely studied notion of privacy for various models of computation. Technically, it is based on measuring differences between probability distributions. We study \(\epsilon ,\delta \)-differential privacy in the setting of labelled Markov chains. While the exact differences relevant to \(\epsilon ,\delta \)-differential privacy are not computable in this framework, we propose a computable bisimilarity distance that yields a sound technique for measuring \(\delta \), the parameter that quantifies deviation from pure differential privacy. We show this bisimilarity distance is always rational, the associated threshold problem is in NP, and the distance can be computed exactly with polynomially many calls to an NP oracle.

Keywords

Bisimilarity distances Kantorovich metric Differential privacy Labelled Markov chains Bisimulation Analysis of probabilistic systems 

Notes

Acknowledgment

David Purser gratefully acknowledges funding by the UK Engineering and Physical Sciences Research Council (EP/L016400/1), the EPSRC Centre for Doctoral Training in Urban Science. Andrzej Murawski is supported by a Royal Society Leverhulme Trust Senior Research Fellowship and the International Exchanges Scheme (IE161701).

References

  1. 1.
    Albarghouthi, A., Hsu, J.: Synthesizing coupling proofs of differential privacy. Proc. ACM Program. Lang. 2, 58:1–58:30 (2018)CrossRefGoogle Scholar
  2. 2.
    Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: On-the-fly exact computation of bisimilarity distances. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 1–15. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36742-7_1CrossRefzbMATHGoogle Scholar
  3. 3.
    Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)Google Scholar
  4. 4.
    Barthe, G., Espitau, T., Grégoire, B., Hsu, J., Stefanesco, L., Strub, P.-Y.: Relational reasoning via probabilistic coupling. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 387–401. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-48899-7_27CrossRefzbMATHGoogle Scholar
  5. 5.
    Barthe, G., Köpf, B., Olmedo, F., Zanella Béguelin, S.: Probabilistic relational reasoning for differential privacy. In: POPL, pp. 97–110. ACM (2012)Google Scholar
  6. 6.
    Billingsley, P.: Probability and Measure, 2nd edn. Wiley, New York (1986)Google Scholar
  7. 7.
    van Breugel, F.: Probabilistic bisimilarity distances. ACM SIGLOG News 4(4), 33–51 (2017)Google Scholar
  8. 8.
    van Breugel, F., Sharma, B., Worrell, J.: Approximating a behavioural pseudometric without discount for probabilistic systems. In: Seidl, H. (ed.) FoSSaCS 2007. LNCS, vol. 4423, pp. 123–137. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-71389-0_10CrossRefGoogle Scholar
  9. 9.
    van Breugel, F., Worrell, J.: An algorithm for quantitative verification of probabilistic transition systems. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 336–350. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44685-0_23CrossRefGoogle Scholar
  10. 10.
    van Breugel, F., Worrell, J.: The complexity of computing a bisimilarity pseudometric on probabilistic automata. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. A Tribute to Prakash Panangaden. LNCS, vol. 8464, pp. 191–213. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-06880-0_10CrossRefGoogle Scholar
  11. 11.
    Cardinal, J.: Computational geometry column 62. SIGACT News 46(4), 69–78 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chatzikokolakis, K., Gebler, D., Palamidessi, C., Xu, L.: Generalized bisimulation metrics. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 32–46. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44584-6_4CrossRefGoogle Scholar
  13. 13.
    Chaum, D.: The dining cryptographers problem: Unconditional sender and recipient untraceability. J. Cryptol. 1(1), 65–75 (1988)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, D., van Breugel, F., Worrell, J.: On the complexity of computing probabilistic bisimilarity. In: Birkedal, L. (ed.) FoSSaCS 2012. LNCS, vol. 7213, pp. 437–451. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28729-9_29CrossRefzbMATHGoogle Scholar
  15. 15.
    Deng, Y., Du, W.: The Kantorovich metric in computer science: a brief survey. Electron Notes Theor. Comput. Sci. 253(3), 73–82 (2009)CrossRefGoogle Scholar
  16. 16.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theor. Comput. Sci. 318(3), 323–354 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Desharnais, J., Jagadeesan, R., Gupta, V., Panangaden, P.: The metric analogue of weak bisimulation for probabilistic processes. In: LICS, pp. 413–422. IEEE (2002)Google Scholar
  18. 18.
    Dwork, C., McSherry, F., Nissim, K., Smith, A.: Calibrating noise to sensitivity in private data analysis. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 265–284. Springer, Heidelberg (2006).  https://doi.org/10.1007/11681878_14CrossRefGoogle Scholar
  19. 19.
    Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM J. Comput. 39(6), 2531–2597 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics, vol. 2. Springer, Berlin (1988)CrossRefGoogle Scholar
  21. 21.
    Kantorovich, L.V.: On the translocation of masses. Doklady Akademii Nauk SSSR 37(7–8), 227–229 (1942)MathSciNetGoogle Scholar
  22. 22.
    Kiefer, S.: On computing the total variation distance of hidden Markov models. In: ICALP, pp. 130:1–130:13 (2018)Google Scholar
  23. 23.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schaefer, M., Stefankovic, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60(2), 172–193 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sontag, E.D.: Real addition and the polynomial hierarchy. IPL 20(3), 115–120 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tang, Q., van Breugel, F.: Computing probabilistic bisimilarity distances via policy iteration. In: CONCUR, pp. 22:1–22:15. Leibniz-Zentrum (2016)Google Scholar
  27. 27.
    Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5(2), 285–309 (1955)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tschantz, M.C., Kaynar, D., Datta, A.: Formal verification of differential privacy for interactive systems. ENTCS 276, 61–79 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Vadhan, S.P.: The complexity of differential privacy. In: Tutorials on the Foundations of Cryptography, pp. 347–450. Springer, Berlin (2017)CrossRefGoogle Scholar
  30. 30.
    Xu, L.: Formal verification of differential privacy in concurrent systems. Ph.D. thesis, Ecole Polytechnique (Palaiseau, France) (2015)Google Scholar
  31. 31.
    Xu, L., Chatzikokolakis, K., Lin, H.: Metrics for differential privacy in concurrent systems. In: Ábrahám, E., Palamidessi, C. (eds.) FORTE 2014. LNCS, vol. 8461, pp. 199–215. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-43613-4_13CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Dmitry Chistikov
    • 1
  • Andrzej S. Murawski
    • 2
  • David Purser
    • 1
    Email author
  1. 1.Centre for Discrete Mathematics and its Applications (DIMAP) and Department of Computer ScienceUniversity of WarwickCoventryUK
  2. 2.Department of Computer ScienceUniversity of OxfordOxfordUK

Personalised recommendations