Advertisement

Information-Theoretic Privacy Through Chaos Synchronization and Optimal Additive Noise

  • Carlos MurguiaEmail author
  • Iman Shames
  • Farhad Farokhi
  • Dragan Nešić
Chapter
First Online:

Abstract

We study the problem of maximizing privacy of data sets by adding random vectors generated via synchronized chaotics oscillators. In particular, we consider the setup where information about data sets, queries, is sent through public (unsecured) communication channels to a remote station. To hide private features (specific entries) within the data set, we corrupt the response to queries by adding random vectors. We send the distorted query (the sum of the requested query and the random vector) through the public channel. The distribution of the additive random vector is designed to minimize the mutual information (our privacy metric) between private entries of the data set and the distorted query. We cast the synthesis of this distribution as a convex program in the probabilities of the additive random vector. Once we have the optimal distribution, we propose an algorithm to generate pseudorandom realizations from this distribution using trajectories of a chaotic oscillator. At the other end of the channel, we have a second chaotic oscillator, which we use to generate realizations from the same distribution. Note that if we obtain the same realizations on both sides of the channel, we can simply subtract the realization from the distorted query to recover the requested query. To generate equal realizations, we need the two chaotic oscillators to be synchronized, i.e., we need them to generate exactly the same trajectories on both sides of the channel synchronously in time. We force the two chaotic oscillators into exponential synchronization using a driving signal. Simulations are presented to illustrate our results.

This is a preview of subscription content, log in to check access.

References

  1. 1.
    Alvarez G, Li S, Montoya F, Pastor G, Romera M (2005) Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos, Solitons Fractals 24:775–783CrossRefGoogle Scholar
  2. 2.
    Akyol E, Langbort C, Basar T (2015) Privacy constrained information processing. In: 2015 54th IEEE conference on decision and control (CDC), pp 4511–4516Google Scholar
  3. 3.
    Angeli D (2000) A Lyapunov approach to incremental stability properties. IEEE Trans Automat Contr 47:410–421MathSciNetCrossRefGoogle Scholar
  4. 4.
    Anishchenko VS, Astakhov V, Neiman A, Vadivasova T, Schimansky-Geier L (2007) Nonlinear dynamics of chaotic and stochastic systems: tutorial and modern developments (Springer Series in Synergetics). Springer, Berlin, HeidelbergGoogle Scholar
  5. 5.
    Arcak M, Angeli D, Sontag E (2002) A unifying integral iss framework for stability of nonlinear cascades. SIAM J Control Optim 40:1888–1904MathSciNetCrossRefGoogle Scholar
  6. 6.
    Belykh VN, Belykh I, Mosekilde E (2005) Hyperbolic plykin attractor can exist in neuron models. I. J Bifurc Chaos 15:3567–3578MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blake C, Merz C (1998) UCI machine learning repository databases. http://archive.ics.uci.edu/ml
  8. 8.
    Boyd S, Vandenberghe L (2004) convex optimization. Cambridge University Press, New York, NY, USAGoogle Scholar
  9. 9.
    Calmon F, Fawaz N (2012) Privacy against statistical inference. In: 2012 50th annual Allerton conference on communication, control, and computing (Allerton), pp 1401–1408Google Scholar
  10. 10.
    Chaillet A, Angeli D, Ito H (2014) Combining iiss and iss with respect to small inputs: the strong iiss property. IEEE Trans Autom Control 59:2518–2524MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cover TM, Thomas JA (1991) Elements of information theory. Wiley-Interscience, New York, NY, USAGoogle Scholar
  12. 12.
    Demidovich B (1967) Lectures on stability theory. Russian, MoscowGoogle Scholar
  13. 13.
    Dwork C (2008) Differential privacy: A survey of results. In: Theory and applications of models of computation, pp 1–19. Springer, Berlin, HeidelbergGoogle Scholar
  14. 14.
    Dwork C, Roth A (2014) The algorithmic foundations of differential privacy. Found Trends Theor Comput Sci 9:211–407MathSciNetCrossRefGoogle Scholar
  15. 15.
    Farokhi F, Nair G (2016) Privacy-constrained communication. IFAC-PapersOnLine 49:43–48MathSciNetCrossRefGoogle Scholar
  16. 16.
    Farokhi F, Sandberg H (2017) Optimal privacy-preserving policy using constrained additive noise to minimize the fisher information. In: 2017 IEEE 56th annual conference on decision and control (CDC) (2017)Google Scholar
  17. 17.
    Farokhi F, Sandberg H, Shames I, Cantoni M (2015) Quadratic Gaussian privacy games. In: 2015 54th IEEE conference on decision and control (CDC), pp 4505–4510 (2015)Google Scholar
  18. 18.
    Geng Q, Viswanath P (2014) The optimal mechanism in differential privacy. In: 2014 IEEE international symposium on information theory, pp 2371–2375Google Scholar
  19. 19.
    Gottwald GA, Melbourne I (2004) A new test for chaos in deterministic systems. In: Proceedings of the royal society of London. Series A: Mathematical, Physical and Engineering Sciences, vol 460, pp 603–611MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grzybowski J, Rafikov M, Balthazar J (2009) Synchronization of the unified chaotic system and application in secure communication. Commun Nonlinear Sci Numer Simul 14:2793–2806MathSciNetCrossRefGoogle Scholar
  21. 21.
    Han S, Topcu U, Pappas GJ (2014) Differentially private convex optimization with piecewise affine objectives. In: 53rd IEEE conference on decision and control (2014)Google Scholar
  22. 22.
    Hoh B, Xiong H, Gruteser M, Alrabady A (2006) Enhancing security and privacy in traffic-monitoring systems. IEEE Pervasive Computing 5:38–46CrossRefGoogle Scholar
  23. 23.
    Huang Z, Wang Y, Mitra S, Dullerud GE (2014) On the cost of differential privacy in distributed control systems. In: Proceedings of the 3rd international conference on high confidence networked systems, pp 105–114Google Scholar
  24. 24.
    Kapitaniak T, Wojewoda J, Brindley J (2000) Synchronization and desynchronization in quasi-hyperbolic chaotic systems. Phys Lett A 210:283–289MathSciNetCrossRefGoogle Scholar
  25. 25.
    Keuninckx L, Soriano M, Fischer I, Mirasso C, Nguimdo R, van der Sande G (2017) Encryption key distribution via chaos synchronization. Sci Reports 7:1–15CrossRefGoogle Scholar
  26. 26.
    Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  27. 27.
    Kocarev L, Halle K, Eckert K, Chua L, Parlitz U (1992) Experimental demonstration of secure communications via chaotic synchronization. Chua’s Circuit: A Paradigm for Chaos 371–378CrossRefGoogle Scholar
  28. 28.
    Kovacic I, Brennan M (2011) The Duffing equation: nonlinear oscillators and their behaviour. WileyGoogle Scholar
  29. 29.
    Kuznetsov S (2012) Hyperbolic chaos: a physicist View. Springer, Berlin HeidelbergCrossRefGoogle Scholar
  30. 30.
    Kuznetsov SP, Kruglov VP (2017) On some simple examples of mechanical systems with hyperbolic chaos. In: Proceedings of the Steklov Institute of Mathematics, 297Google Scholar
  31. 31.
    Kuznetsov SP, Pikovsky A (2007) Autonomous coupled oscillators with hyperbolic strange attractors. Physica D 232:87–102MathSciNetCrossRefGoogle Scholar
  32. 32.
    Liu X, Chen T (2008) Boundedness and synchronization of y-coupled lorenz systems with or without controllers. Physica D 237:630–639MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lohmiller W, Slotine J (1998) On contraction analysis for nonlinear systems. Automatica 34:683–695MathSciNetCrossRefGoogle Scholar
  34. 34.
    Lu J, Wu X, Lu J (2002) Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 305:365–370MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mackey M, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:287–289CrossRefGoogle Scholar
  36. 36.
    Murguia C, Shames I, Farokhi F, Nešić D (2018) On privacy of quantized sensor measurements through additive noise. In: proceedings of the 57th IEEE conference on decision and control (CDC) (2018)Google Scholar
  37. 37.
    Ny JL, Pappas GJ (2014) Differentially private filtering. IEEE Trans Autom Control 59:341–354MathSciNetCrossRefGoogle Scholar
  38. 38.
    Pavlov A, Pogromsky A, van de Wouw N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst Control Lett 52:257MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824MathSciNetCrossRefGoogle Scholar
  40. 40.
    Pogromsky A (1998) Passivity based design of synchronizing systems. Int J Bifurcation Chaos 8:295–319MathSciNetCrossRefGoogle Scholar
  41. 41.
    Pol Van der B, der Mark V (1927) Frequency demultiplication. Nature 120:363–364CrossRefGoogle Scholar
  42. 42.
    Rajagopalan SR, Sankar L, Mohajer S, Poor HV (2011) Smart meter privacy: a utility-privacy framework. In: 2011 IEEE international conference on smart grid communications (SmartGridComm), pp 190–195 (2011)Google Scholar
  43. 43.
    Robert CP, Casella G (2005) Monte carlo statistical methods (Springer Texts in Statistics). Springer, Berlin, HeidelbergGoogle Scholar
  44. 44.
    Ross M (2006) Introduction to probability models, 9th edn. Academic Press Inc, Orlando, FL, USAGoogle Scholar
  45. 45.
    Salamatian S, Zhang A, du Pin Calmon F, Bhamidipati S, Fawaz N, Kveton B, Oliveira P, Taft N (2015) Managing your private and public data: bringing down inference attacks against your privacy. IEEE J Sel Topics Signal Process 9:1240–1255CrossRefGoogle Scholar
  46. 46.
    Scardovi L, Sepulchre R (2010) Synchronization in networks of identical linear systems. IEEE Trans Automat Contr 57:2132–2143zbMATHGoogle Scholar
  47. 47.
    Sontag E, Wang Y (1995) On characterizations of the input-to-state stability property. Syst Control Lett 24:351–359MathSciNetCrossRefGoogle Scholar
  48. 48.
    Soria-Comas J, Domingo-Ferrer J (2013) Optimal data-independent noise for differential privacy. Inf Sci 250:200–214MathSciNetCrossRefGoogle Scholar
  49. 49.
    Steur E, Tyukin I, Nijmeijer H (2009) Semi-passivity and synchronization of diffusively coupled neuronal oscillators. Physica D 238:2119–2128MathSciNetCrossRefGoogle Scholar
  50. 50.
    Strogatz SH (2000) Nonlinear dynamics and chaos: with applications to physics. Biology Chemist EngGoogle Scholar
  51. 51.
    Tan O, Gunduz D, Poor HV (2013) Increasing smart meter privacy through energy harvesting and storage devices. IEEE J Sel Areas Commun 31:1331–1341CrossRefGoogle Scholar
  52. 52.
    Turukina L, Pikovsky A (2011) Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators. Phys Lett A 11:1407–1411MathSciNetCrossRefGoogle Scholar
  53. 53.
    Wang Y, Huang Z, Mitra S, Dullerud GE (2014) Entropy-minimizing mechanism for differential privacy of discrete-time linear feedback systems. In: 53rd IEEE conference on decision and control, pp 2130–2135 (2014)Google Scholar
  54. 54.
    Weber RH (2010) Internet of things as new security and privacy challenges. Comput Law Secur Rev 26:23–30MathSciNetCrossRefGoogle Scholar
  55. 55.
    Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos. Texts in Applied Mathematics. Springer, New YorkGoogle Scholar
  56. 56.
    Wu C, Chua L (1995) Synchronization in an array of linearly coupled dynamical systems. IEEE Trans Circuit Sys I 42:430–447MathSciNetCrossRefGoogle Scholar
  57. 57.
    Yang T, Wu C, Chua L (1997) Cryptography based on chaotic systems. IEEE Trans Circuit Syst I: Fund Theory Appl 44:469–472CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Carlos Murguia
    • 1
    Email author
  • Iman Shames
    • 1
  • Farhad Farokhi
    • 1
    • 2
  • Dragan Nešić
    • 1
  1. 1.Department of Electrical and Electronic EngineeringUniversity of MelbourneMelbourneAustralia
  2. 2.The Commonwealth Scientific and Industrial Research Organisation (CSIRO), Data61CanberraAustralia

Personalised recommendations

Cite chapter

Buy options