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.
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References
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 24:775–783
Akyol E, Langbort C, Basar T (2015) Privacy constrained information processing. In: 2015 54th IEEE conference on decision and control (CDC), pp 4511–4516
Angeli D (2000) A Lyapunov approach to incremental stability properties. IEEE Trans Automat Contr 47:410–421
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, Heidelberg
Arcak M, Angeli D, Sontag E (2002) A unifying integral iss framework for stability of nonlinear cascades. SIAM J Control Optim 40:1888–1904
Belykh VN, Belykh I, Mosekilde E (2005) Hyperbolic plykin attractor can exist in neuron models. I. J Bifurc Chaos 15:3567–3578
Blake C, Merz C (1998) UCI machine learning repository databases. http://archive.ics.uci.edu/ml
Boyd S, Vandenberghe L (2004) convex optimization. Cambridge University Press, New York, NY, USA
Calmon F, Fawaz N (2012) Privacy against statistical inference. In: 2012 50th annual Allerton conference on communication, control, and computing (Allerton), pp 1401–1408
Chaillet A, Angeli D, Ito H (2014) Combining iiss and iss with respect to small inputs: the strong iiss property. IEEE Trans Autom Control 59:2518–2524
Cover TM, Thomas JA (1991) Elements of information theory. Wiley-Interscience, New York, NY, USA
Demidovich B (1967) Lectures on stability theory. Russian, Moscow
Dwork C (2008) Differential privacy: A survey of results. In: Theory and applications of models of computation, pp 1–19. Springer, Berlin, Heidelberg
Dwork C, Roth A (2014) The algorithmic foundations of differential privacy. Found Trends Theor Comput Sci 9:211–407
Farokhi F, Nair G (2016) Privacy-constrained communication. IFAC-PapersOnLine 49:43–48
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)
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)
Geng Q, Viswanath P (2014) The optimal mechanism in differential privacy. In: 2014 IEEE international symposium on information theory, pp 2371–2375
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 603–611
Grzybowski J, Rafikov M, Balthazar J (2009) Synchronization of the unified chaotic system and application in secure communication. Commun Nonlinear Sci Numer Simul 14:2793–2806
Han S, Topcu U, Pappas GJ (2014) Differentially private convex optimization with piecewise affine objectives. In: 53rd IEEE conference on decision and control (2014)
Hoh B, Xiong H, Gruteser M, Alrabady A (2006) Enhancing security and privacy in traffic-monitoring systems. IEEE Pervasive Computing 5:38–46
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 105–114
Kapitaniak T, Wojewoda J, Brindley J (2000) Synchronization and desynchronization in quasi-hyperbolic chaotic systems. Phys Lett A 210:283–289
Keuninckx L, Soriano M, Fischer I, Mirasso C, Nguimdo R, van der Sande G (2017) Encryption key distribution via chaos synchronization. Sci Reports 7:1–15
Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice-Hall, Englewood Cliffs, NJ
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 371–378
Kovacic I, Brennan M (2011) The Duffing equation: nonlinear oscillators and their behaviour. Wiley
Kuznetsov S (2012) Hyperbolic chaos: a physicist View. Springer, Berlin Heidelberg
Kuznetsov SP, Kruglov VP (2017) On some simple examples of mechanical systems with hyperbolic chaos. In: Proceedings of the Steklov Institute of Mathematics, 297
Kuznetsov SP, Pikovsky A (2007) Autonomous coupled oscillators with hyperbolic strange attractors. Physica D 232:87–102
Liu X, Chen T (2008) Boundedness and synchronization of y-coupled lorenz systems with or without controllers. Physica D 237:630–639
Lohmiller W, Slotine J (1998) On contraction analysis for nonlinear systems. Automatica 34:683–695
Lu J, Wu X, Lu J (2002) Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 305:365–370
Mackey M, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:287–289
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)
Ny JL, Pappas GJ (2014) Differentially private filtering. IEEE Trans Autom Control 59:341–354
Pavlov A, Pogromsky A, van de Wouw N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst Control Lett 52:257
Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824
Pogromsky A (1998) Passivity based design of synchronizing systems. Int J Bifurcation Chaos 8:295–319
Pol Van der B, der Mark V (1927) Frequency demultiplication. Nature 120:363–364
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)
Robert CP, Casella G (2005) Monte carlo statistical methods (Springer Texts in Statistics). Springer, Berlin, Heidelberg
Ross M (2006) Introduction to probability models, 9th edn. Academic Press Inc, Orlando, FL, USA
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 9:1240–1255
Scardovi L, Sepulchre R (2010) Synchronization in networks of identical linear systems. IEEE Trans Automat Contr 57:2132–2143
Sontag E, Wang Y (1995) On characterizations of the input-to-state stability property. Syst Control Lett 24:351–359
Soria-Comas J, Domingo-Ferrer J (2013) Optimal data-independent noise for differential privacy. Inf Sci 250:200–214
Steur E, Tyukin I, Nijmeijer H (2009) Semi-passivity and synchronization of diffusively coupled neuronal oscillators. Physica D 238:2119–2128
Strogatz SH (2000) Nonlinear dynamics and chaos: with applications to physics. Biology Chemist Eng
Tan O, Gunduz D, Poor HV (2013) Increasing smart meter privacy through energy harvesting and storage devices. IEEE J Sel Areas Commun 31:1331–1341
Turukina L, Pikovsky A (2011) Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators. Phys Lett A 11:1407–1411
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)
Weber RH (2010) Internet of things as new security and privacy challenges. Comput Law Secur Rev 26:23–30
Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos. Texts in Applied Mathematics. Springer, New York
Wu C, Chua L (1995) Synchronization in an array of linearly coupled dynamical systems. IEEE Trans Circuit Sys I 42:430–447
Yang T, Wu C, Chua L (1997) Cryptography based on chaotic systems. IEEE Trans Circuit Syst I: Fund Theory Appl 44:469–472
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Murguia, C., Shames, I., Farokhi, F., Nešić, D. (2020). Information-Theoretic Privacy Through Chaos Synchronization and Optimal Additive Noise. In: Farokhi, F. (eds) Privacy in Dynamical Systems. Springer, Singapore. https://doi.org/10.1007/978-981-15-0493-8_6
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