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Bisimilarity Distances for Approximate Differential Privacy

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Automated Technology for Verification and Analysis (ATVA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,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.

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Notes

  1. 1.

    A pseudometric must satisfy \(m(x,x)=0\), \(m(x,y)=m(y,x)\) and \(m(x,z)\le m(x,y)+m(y,z)\). For metrics, one additionally requires that \(m(x,y)=0\) should imply \(x=y\).

  2. 2.

    More precisely, the existence of such a procedure would be a breakthrough in the computational complexity theory, showing that \(\mathbf {NP} = \exists \mathbb R\). This would imply that a multitude of problems in computational geometry could be solved using SAT solvers [11, 24]. Unlike for \( bd _\alpha \), variable assignments in these problems may need to be irrational, even if all numbers in the input data are integer or rational.

References

  1. Albarghouthi, A., Hsu, J.: Synthesizing coupling proofs of differential privacy. Proc. ACM Program. Lang. 2, 58:1–58:30 (2018)

    Article  Google Scholar 

  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_1

    Chapter  MATH  Google Scholar 

  3. Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)

    Google Scholar 

  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_27

    Chapter  MATH  Google Scholar 

  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. Billingsley, P.: Probability and Measure, 2nd edn. Wiley, New York (1986)

    Google Scholar 

  7. van Breugel, F.: Probabilistic bisimilarity distances. ACM SIGLOG News 4(4), 33–51 (2017)

    Google Scholar 

  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_10

    Chapter  Google Scholar 

  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_23

    Chapter  Google Scholar 

  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_10

    Chapter  Google Scholar 

  11. Cardinal, J.: Computational geometry column 62. SIGACT News 46(4), 69–78 (2015)

    Article  MathSciNet  Google Scholar 

  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_4

    Chapter  Google Scholar 

  13. Chaum, D.: The dining cryptographers problem: Unconditional sender and recipient untraceability. J. Cryptol. 1(1), 65–75 (1988)

    Article  MathSciNet  Google Scholar 

  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_29

    Chapter  MATH  Google Scholar 

  15. Deng, Y., Du, W.: The Kantorovich metric in computer science: a brief survey. Electron Notes Theor. Comput. Sci. 253(3), 73–82 (2009)

    Article  Google Scholar 

  16. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theor. Comput. Sci. 318(3), 323–354 (2004)

    Article  MathSciNet  Google Scholar 

  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. 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_14

    Chapter  Google Scholar 

  19. Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM J. Comput. 39(6), 2531–2597 (2010)

    Article  MathSciNet  Google Scholar 

  20. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics, vol. 2. Springer, Berlin (1988)

    Book  Google Scholar 

  21. Kantorovich, L.V.: On the translocation of masses. Doklady Akademii Nauk SSSR 37(7–8), 227–229 (1942)

    MathSciNet  Google Scholar 

  22. Kiefer, S.: On computing the total variation distance of hidden Markov models. In: ICALP, pp. 130:1–130:13 (2018)

    Google Scholar 

  23. Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)

    Article  MathSciNet  Google Scholar 

  24. Schaefer, M., Stefankovic, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60(2), 172–193 (2017)

    Article  MathSciNet  Google Scholar 

  25. Sontag, E.D.: Real addition and the polynomial hierarchy. IPL 20(3), 115–120 (1985)

    Article  MathSciNet  Google Scholar 

  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. Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5(2), 285–309 (1955)

    Article  MathSciNet  Google Scholar 

  28. Tschantz, M.C., Kaynar, D., Datta, A.: Formal verification of differential privacy for interactive systems. ENTCS 276, 61–79 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Vadhan, S.P.: The complexity of differential privacy. In: Tutorials on the Foundations of Cryptography, pp. 347–450. Springer, Berlin (2017)

    Chapter  Google Scholar 

  30. Xu, L.: Formal verification of differential privacy in concurrent systems. Ph.D. thesis, Ecole Polytechnique (Palaiseau, France) (2015)

    Google Scholar 

  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_13

    Chapter  Google Scholar 

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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).

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Authors and Affiliations

  1. Centre for Discrete Mathematics and its Applications (DIMAP) and Department of Computer Science, University of Warwick, Coventry, UK

    Dmitry Chistikov & David Purser

  2. Department of Computer Science, University of Oxford, Oxford, UK

    Andrzej S. Murawski

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  1. Dmitry Chistikov
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  2. Andrzej S. Murawski
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  3. David Purser
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Correspondence to David Purser .

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  1. Microsoft Research, Redmond, WA, USA

    Shuvendu K. Lahiri

  2. University of Southern California, Los Angeles, CA, USA

    Chao Wang

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Chistikov, D., Murawski, A.S., Purser, D. (2018). Bisimilarity Distances for Approximate Differential Privacy. In: Lahiri, S., Wang, C. (eds) Automated Technology for Verification and Analysis. ATVA 2018. Lecture Notes in Computer Science(), vol 11138. Springer, Cham. https://doi.org/10.1007/978-3-030-01090-4_12

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  • DOI: https://doi.org/10.1007/978-3-030-01090-4_12

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