Discrete Event Dynamic Systems

, Volume 28, Issue 1, pp 83–108 | Cite as

Opacity for linear constraint Markov chains

  • Béatrice Bérard
  • Olga Kouchnarenko
  • John MullinsEmail author
  • Mathieu Sassolas
Part of the following topical collections:
  1. Special Issue on Performance Analysis and Optimization of Discrete Event Systems


On a partially observed system, a secret φ is opaque if an observer cannot ascertain that its trace belongs to φ. We consider specifications given as Constraint Markov Chains (CMC), which are underspecified Markov chains where probabilities on edges are required to belong to some set. The nondeterminism is resolved by a scheduler, and opacity on this model is defined as a worst case measure over all implementations obtained by scheduling. This measures the information obtained by a passive observer when the system is controlled by the smartest scheduler in coalition with the observer. When restricting to the subclass of Linear CMC, we compute (or approximate) this measure and prove that refinement of a specification can only improve opacity.


Opacity Markov models Specification Refinement 



Partially supported by a grant from Coopération France-Québec, Service Coopération et Action Culturelle 2012/26/SCAC (French Government), the NSERC Discovery Individual grant No. 13321 (Government of Canada), the FQRNT Team grant No. 167440 (Quebec’s Government) and the CFQCU France-Quebec Cooperative grant No. 167671 (Quebec’s Government). This research has been partially performed while the third author was visiting the LIP6, Université Pierre & Marie Curie.


  1. Alur R, Černý P, Zdancewic S (2006) Preserving secrecy under refinement. In: Proc. ICALP’06, LNCS, vol 4052. Springer, pp 107–118Google Scholar
  2. Baier C, Katoen JP (2008) Principles of model checking (representation and mind series). The MIT PressGoogle Scholar
  3. Baier C, Katoen JP, Hermanns H, Wolf V (2005) Comparative branching-time semantics for Markov chains. Inf Comput 200:149–214MathSciNetCrossRefzbMATHGoogle Scholar
  4. Benedikt M, Lenhardt R, Worrell J (2013) LTL model checking of interval Markov chains. In: Proc. TACAS’13, LNCS, vol 7795. Springer, pp 32–46Google Scholar
  5. Bérard B, Mullins J, Sassolas M (2010) Quantifying opacity. In: Ciardo G, Segala R (eds) Proc. QEST’10. IEEE Computer Society, pp 263–272Google Scholar
  6. Bérard B, Chatterjee K, Sznajder N (2015a) Probabilistic opacity for Markov decision processes. Inf Process Lett 115(1):52–59MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bérard B, Mullins J, Sassolas M (2015b) Quantifying opacity. Math Struct Comput Sci 25(2):361–403MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bérard B, Kouchnarenko O, Mullins J, Sassolas M (2016) Preserving opacity on interval Markov chains under simulation. In: Cassandras CG, Giua A, Li Z (eds) Proceedings of 13th international workshop on discrete event systems, WODES’16. IEEE, pp 319–324Google Scholar
  9. Bhargava M, Palamidessi C (2005) Probabilistic anonymity. In: Abadi M, de Alfaro L (eds) Proc. CONCUR’05, LNCS, vol 3653, pp 171–185Google Scholar
  10. Billingsley P (1995) Probability and measure, 3rd edn. WileyGoogle Scholar
  11. Biondi F, Legay A, Nielsen BF, Wa̧sowski A (2014) Maximizing entropy over Markov processes. J Logic Algebr Methods Programm 83(5–6):384–399MathSciNetCrossRefzbMATHGoogle Scholar
  12. Bryans JW, Koutny M, Mazaré L, Ryan PYA (2008) Opacity generalised to transition systems. Int J Inf Secur 7(6):421–435CrossRefGoogle Scholar
  13. Caillaud B, Delahaye B, Larsen KG, Legay A, Pedersen ML, Wasowski A (2011) Constraint Markov chains. Theor Comput Sci 412(34):4373–4404MathSciNetCrossRefzbMATHGoogle Scholar
  14. Chatterjee K, Henzinger T, Sen K (2008) Model-checking omega-regular properties of interval Markov chains. In Amadio RM (ed) Proc. FoSSaCS’08, pp 302–317Google Scholar
  15. Chaum D (1988) The dining cryptographers problem: unconditional sender and recipient untraceability. J Cryptol 1:65–75MathSciNetCrossRefzbMATHGoogle Scholar
  16. Clarkson MR, Schneider FB (2010) Hyperproperties. J Comput Secur 18(6):1157–1210CrossRefGoogle Scholar
  17. Delahaye B (2015) Consistency for parametric interval Markov chains. In: André É, Frehse G (eds) Proc SynCoP’15, OASICS, vol 44. Schloss Dagstuhl - LZI, pp 17–32Google Scholar
  18. Jonsson B, Larsen KG (1991) Specification and refinement of probabilistic processes. In: Proceedings LICS’91. IEEE Computer Society, , pp 266–277Google Scholar
  19. Mazaré L. (2005) Decidability of opacity with non-atomic keys. In: Proceedings FAST’04, international federation for information processing, vol 173. Springer, pp 71–84Google Scholar
  20. Piterman N (2007) From nondeterministic Büchi and Streett automata to deterministic parity automata. Logic Methods Comput Sci 3(3)Google Scholar
  21. Roos C, Terlaky T, Vial JP (1997) Theory and algorithms for linear optimization. An interior point approach. John Wiley & Sons Ltd, Wiley-IntersciencezbMATHGoogle Scholar
  22. Saboori A, Hadjicostis CN (2014) Current-state opacity formulations in probabilistic finite automata. IEEE Trans Autom Control 59(1):120–133MathSciNetCrossRefzbMATHGoogle Scholar
  23. Segala R (1995) Modeling and verification of randomized distributed real-time systems. Ph.D. thesis, MIT Department of Electrical Engineering and Computer ScienceGoogle Scholar
  24. Sen K, Viswanathan M, Agha G (2006) Model-checking Markov chains in the presence of uncertainties. In: Hermanns H, Palsberg J (eds) Proceedings of 12th international conference on tools and algorithms for the construction and analysis of systems, TACAS’06, LNCS, vol 3920. Springer, pp 394–410Google Scholar
  25. Vardi MY (1985) Automatic verification of probabilistic concurrent finite-state programs. In: Proceedings 26th annual symposium on foundations of computer science (FOCS’85). IEEE Computer Society, pp 327–338Google Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Sorbonne UniversitéParisFrance
  2. 2.INRIA and LSV, CNRS and ENS CachanUniversité Paris-SaclaySaclayFrance
  3. 3.Université Bourgogne Franche-ComtéFEMTO-ST, CNRS UMR 6174BesançonFrance
  4. 4.Department of Computer & Software EngineeringÉcole Polytechnique de MontréalMontrealCanada
  5. 5.Université Paris-Est, LACLCréteilFrance

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