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Comparing the notions of opacity for discrete-event systems

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Abstract

Opacity is an information flow property characterizing whether a system reveals its secret to a passive observer. Several notions of opacity have been introduced in the literature. We study the notions of language-based opacity, current-state opacity, initial-state opacity, initial-and-final-state opacity, K-step opacity, and infinite-step opacity. Comparing the notions is a natural question that has been investigated and summarized by Wu and Lafortune, who provided transformations among current-state opacity, initial-and-final-state opacity, and language-based opacity, and, for prefix-closed languages, also between language-based opacity and initial-state opacity. We extend these results by showing that all the discussed notions of opacity are transformable to each other. Besides a deeper insight into the differences among the notions, the transformations have applications in complexity results. In particular, the transformations are computable in polynomial time and preserve the number of observable events and determinism, and hence the computational complexities of the verification of the notions coincide. We provide a complete and improved complexity picture of the verification of the discussed notions of opacity, and improve the algorithmic complexity of deciding language-based opacity, infinite-step opacity, and K-step opacity.

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References

  • Alur R, Černý P, Zdancewic S (2006) Preserving secrecy under refinement. In: International colloquium on automata, languages and programming (ICALP), pp 107–118

  • Arora S, Barak B (2009) Computational complexity – A modern approach. Cambridge University Press

  • Asveld PRJ, Nijholt A (2000) The inclusion problem for some subclasses of context-free languages. Theor Comput Sci 230(1-2):247–256

    Article  MathSciNet  Google Scholar 

  • Badouel E, Bednarczyk M, Borzyszkowski A, Caillaud B, Darondeau P (2007) Concurrent secrets. Discret Event Dyn Syst 17(4):425–446

    Article  MathSciNet  Google Scholar 

  • Balun J, Masopust T (2020) On opacity verification for discrete-event systems. In: IFAC World congress, pp 2105–2110

  • Bryans JW, Koutny M, Mazarė L, Ryan PYA (2008) Opacity generalised to transition systems. Int J Inf Secur 7(6):421–435

    Article  Google Scholar 

  • Bryans JW, Koutny M, Ryan PYA (2005) Modelling opacity using Petri nets. Electron Notes Theor Comput Sci 121:101–115

    Article  Google Scholar 

  • Cassandras CG, Lafortune S (2008) Introduction to Discrete Event Systems, 2nd edn. Springer

  • Cassez F, Dubreil J, Marchand H (2012) Synthesis of opaque systems with static and dynamic masks. Formal Methods Syst Des 40(1):88–115

    Article  Google Scholar 

  • Domaratzki M, Salomaa K (2007) Transition complexity of language operations. Theor Comput Sci 387(2):147–154

    Article  MathSciNet  Google Scholar 

  • Dubreil J, Darondeau P, Marchand H (2008) Opacity enforcing control synthesis. In: Workshop on discrete event systems (WODES), pp 28–35

  • Focardi R, Gorrieri R (1994) A taxonomy of trace-based security properties for ccs. In: Computer security foundations workshop VII, pp 126–136

  • Hadj-Alouane NB, Lafrance S, Lin F, Mullins J, Yeddes MM (2005) On the verification of intransitive noninterference in mulitlevel security. IEEE Trans Syst Man Cybern Part B 35(5):948–958

    Article  Google Scholar 

  • Holzer M, Kutrib M (2011) Descriptional and computational complexity of finite automata—A survey. Inf Comput 209(3):456–470

    Article  MathSciNet  Google Scholar 

  • Hopcroft JE, Ullman JD (1979) Introduction to automata theory, languages and computation. Addison-Wesley

  • Immerman N (1988) Nondeterministic space is closed under complementation. SIAM J Comput 17:935–938

    Article  MathSciNet  Google Scholar 

  • Jacob R, Lesage J, Faure J (2016) Overview of discrete event systems opacity: models, validation, and quantification. Annu Rev Control 41:135–146

    Article  Google Scholar 

  • Jirȧskovȧ G, Masopust T (2012) On a structural property in the state complexity of projected regular languages. Theor Comput Sci 449:93–105

    Article  MathSciNet  Google Scholar 

  • Jones ND (1975) Space-bounded reducibility among combinatorial problems. J Comput Syst Sci 11(1):68–85

    Article  MathSciNet  Google Scholar 

  • Lin F (2011) Opacity of discrete event systems and its applications. Automatica 47(3):496–503

    Article  MathSciNet  Google Scholar 

  • Mazarė L. (2004) Decidability of opacity with non-atomic keys. In: Formal aspects in security and trust, pp 71–84

  • Saboori A (2011) Verification and enforcement of state-based notions of opacity in discrete event systems. Ph.D. thesis University of Illinois at Urbana-Champaign

  • Saboori A, Hadjicostis CN (2007) Notions of security and opacity in discrete event systems. In: Conference on decision and control (CDC), pp 5056–5061

  • Saboori A, Hadjicostis CN (2008) Opacity-enforcing supervisory strategies for secure discrete event systems. In: Conference on decision and control. IEEE

  • Saboori A, Hadjicostis CN (2011) Verification of K-step opacity and analysis of its complexity. IEEE Trans Autom Sci Eng 8(3):549–559

    Article  Google Scholar 

  • Saboori A, Hadjicostis CN (2012) Verification of infinite-step opacity and complexity considerations. IEEE Trans Autom Control 57(5):1265–1269

    Article  MathSciNet  Google Scholar 

  • Schneider S, Sidiropoulos A (1996) CSP and anonymity. In: European symposium on research in computer security (ESORICS), LNCS, vol 1146, pp 198–218

  • Stockmeyer LJ, Meyer AR (1973) Word problems requiring exponential time: Preliminary report. In: ACM Symposium on theory of computing (STOC), pp 1–9

  • Szelepcsėnyi R (1988) The method of forced enumeration for nondeterministic automata. Acta Inf 26:279–284

    Article  MathSciNet  Google Scholar 

  • Wong K (1998) On the complexity of projections of discrete-event systems. In: Workshop on discrete event systems (WODES), pp 201–206

  • Wu YC, Lafortune S (2013) Comparative analysis of related notions of opacity in centralized and coordinated architectures. Discrete Event Dyn Syst 23 (3):307–339

    Article  MathSciNet  Google Scholar 

  • Yin X, Lafortune S (2017) A new approach for the verification of infinite-step and K-step opacity using two-way observers. Automatica 80:162–171

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We gratefully acknowledge suggestions and comments of the anonymous referees.

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Correspondence to Jiří Balun.

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This article belongs to the Topical Collection: Topical Collection on Cybersecurity

Guest Editors: Rong Su and Carlos Basilio

Partially supported by the Ministry of Education, Youth and Sports under the INTER-EXCELLENCE project LTAUSA19098 and by the University projects IGA PrF 2020 019 and IGA PrF 2021 022.

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Balun, J., Masopust, T. Comparing the notions of opacity for discrete-event systems. Discrete Event Dyn Syst 31, 553–582 (2021). https://doi.org/10.1007/s10626-021-00344-2

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