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CryptHOL: game-based proofs in higher-order logic. (English) Zbl 1455.94121
Summary: Game-based proofs are a well-established paradigm for structuring security arguments and simplifying their understanding. We present a novel framework, CryptHOL, for rigorous game-based proofs that is supported by mechanical theorem proving. CryptHOL is based on a new semantic domain with an associated functional programming language for expressing games. We embed our framework in the Isabelle/HOL theorem prover and, using the theory of relational parametricity, we tailor Isabelle’s existing proof automation to game-based proofs. By basing our framework on a conservative extension of higher-order logic and providing automation support, the resulting proofs are trustworthy and comprehensible, and the framework is extensible and widely applicable. We evaluate our framework by formalising different game-based proofs from the literature and comparing the results with existing formal-methods tools.

MSC:
94A60 Cryptography
94A62 Authentication, digital signatures and secret sharing
68V15 Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.)
91A99 Game theory
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[1] G. Asharov, A. Beimel, N. Makriyannis, E. Omri, Complete characterization of fairness in secure two-party computation of boolean functions, in Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9014, (Springer, 2015), pp. 199-228 · Zbl 1354.94020
[2] Audebaud, P.; Paulin-Mohring, C., Proofs of randomized algorithms in Coq, Sci. Comput. Program., 74, 8, 568-589 (2009) · Zbl 1178.68667
[3] Baader, F.; Nipkow, T., Term Rewriting and All That (1998), Cambridge: Cambridge University Press, Cambridge
[4] M. Backes, M. Berg, D. Unruh, A formal language for cryptographic pseudocode, in LPAR 2008. LNCS, vol. 5330, (Springer, 2008), pp. 353-376 · Zbl 1182.94035
[5] G. Barthe, C. Fournet, B. Grégoire, P.Y. Strub, N. Swamy, S. Zanella Béguelin, Probabilistic relational verification for cryptographic implementations. in POPL 2014. (ACM, 2014) pp. 193-205 · Zbl 1284.68380
[6] G. Barthe, B. Grégoire, S. Heraud, S. Zanella Béguelin, Computer-aided security proofs for the working cryptographer. in CRYPTO 2011. LNCS, vol. 6841, (Springer 2011), pp. 71-90 · Zbl 1287.94048
[7] G. Barthe, B. Grégoire, J. Hsu, P.Y. Strub, Coupling proofs are probabilistic product programs. in POPL 2017. (ACM, 2017), pp. 161-174 · Zbl 1380.68267
[8] G. Barthe, B. Grégoire, S. Zanella Béguelin, Formal certification of code-based cryptographic proofs. in POPL 2009. (ACM, 2009), pp. 90-101 · Zbl 1315.68081
[9] Basin, D.; Kaufmann, M.; Huet, G.; Plotkin, G., The Boyer-Moore prover and Nuprl: An experimental comparison, Logical Frameworks, 89-119 (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0799.68169
[10] M. Bellare, A. Boldyreva, S. Micali, Public-key encryption in a multi-user setting: Security proofs and improvements. in Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, (Springer, 2000), pp. 259-274 · Zbl 1082.94504
[11] M. Bellare, P. Rogaway, Optimal asymmetric encryption. in Workshop on the Theory and Application of Cryptographic Techniques. (Springer, 1994), pp. 92-111 · Zbl 0881.94010
[12] M. Bellare, P. Rogaway, Code-based game-playing proofs and the security of triple encryption. Cryptology ePrint Archive, Report 2004/331 (2004), http://eprint.iacr.org/2004/331 · Zbl 1140.94321
[13] M. Bellare, P. Rogaway, The security of triple encryption and a framework for code-based game-playing proofs. in EUROCRYPT 2006. LNCS, vol. 4004, (Springer, 2006), pp. 409-426 · Zbl 1140.94321
[14] Bengtson, J.; Bhargavan, K.; Fournet, C.; Gordon, AD; Maffeis, S., Refinement types for secure implementations, ACM Trans. Program. Lang. Syst., 33, 2, 8:1-8:45 (2011)
[15] M. Berg, Formal verification of cryptographic security proofs. Ph.D. thesis, Universität des Saarlandes (2013)
[16] S. Berghofer, M. Wenzel, Logic-free reasoning in Isabelle/Isar. in Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) CICM 2008. LNCS, vol. 5144, (Springer, 2008), pp. 355-369 · Zbl 1166.68337
[17] K. Bhargavan, C. Fournet, M. Kohlweiss, A. Pironti, P.Y. Strub, Implementing TLS with verified cryptographic security. in S&P 2013. (IEEE, 2013), pp. 445-459
[18] Blanchet, B., A computationally sound mechanized prover for security protocols, IEEE Trans. Dependable Secure Comput., 5, 4, 193-207 (2008)
[19] J.C. Blanchette, A. Bouzy, A. Lochbihler, A. Popescu, D. Traytel, Friends with benefits: Implementing corecursion in foundational proof assistants. in Yang, H. (ed.) ESOP 2017. LNCS, (Springer 2017), pp. 111-140 · Zbl 06721319
[20] J.C. Blanchette, J. Hölzl, A. Lochbihler, L. Panny, Popescu, A., D. Traytel, Truly modular (co)datatypes for Isabelle/HOL. in ITP 2014. LNCS, vol. 8558, (Springer, 2014), pp. 93-110 · Zbl 1416.68151
[21] D. Butler, D. Aspinall, A. Gascon, How to simulate it in Isabelle: Towards formal proof for secure multi-party computation (2017), accepted at ITP 2017 · Zbl 06821847
[22] D. Butler, D. Aspinall, A. Gascón, On the formalisation of \(\varSigma \)-protocols and commitment schemes. in Nielson, F., Sands, D. (eds.) POST 2019. LNCS, vol. 11426, (Springer, 2019), pp. 175-196
[23] R. Canetti, Universally composable security: A new paradigm for cryptographic protocols. in Proceedings of 42nd IEEE Symposium on Foundations of Computer Science, 2001. (IEEE, 2001), pp. 136-145
[24] Church, A., A formulation of the simple theory of types, J. Symb. Log., 5, 2, 56-68 (1940) · JFM 66.1192.06
[25] R. Cohen, S. Coretti, J. Garay, V. Zikas, Probabilistic termination and composability of cryptographic protocols. in Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, (Springer ,2016), pp. 240-269 · Zbl 1406.94040
[26] Easycrypt: Reference manual. https://www.easycrypt.info/documentation/refman.pdf (2018), version 1.x, 19 February 2018
[27] Elgamal, T., A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inf. Theory, 31, 4, 469-472 (1985) · Zbl 0571.94014
[28] Goldwasser, S.; Micali, S., Probabilistic encryption, J. Comput. Syst. Sci., 28, 2, 270-299 (1984) · Zbl 0563.94013
[29] Gordon, SD; Hazay, C.; Katz, J.; Lindell, Y., Complete fairness in secure two-party computation, J. ACM, 58, 6, 24:1-24:37 (2011) · Zbl 1281.94081
[30] O. Grumberg, N. Francez, S. Katz, Fair termination of communicating processes. in PODC 1984. (ACM, 1984), pp. 254-265
[31] S. Halevi, A plausible approach to computer-aided cryptographic proofs. Cryptology ePrint Archive, Report 2005/181 (2005)
[32] M. Hofmann, A. Karbyshev, H. Seidl, What is a pure functional? in Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, (Springer 2010), pp. 199-210
[33] J. Hölzl, A. Lochbihler, D. Traytel, A formalized hierarchy of probabilistic system types. in ITP 2015. LNCS, vol. 9236, (Springer, 2015), pp. 203-220 · Zbl 06481864
[34] B. Huffman, O. Kunčar, Lifting and Transfer: A modular design for quotients in Isabelle/HOL. in CPP 2013. LNCS, vol. 8307, (Springer, 2013), pp. 131-146 · Zbl 1426.68284
[35] J. Hurd, A formal approach to probabilistic termination. in TPHOLs 2002. LNCS, vol. 2410, (Springer, 2002), pp. 230-245 · Zbl 1013.68193
[36] Kilian, J.; Rogaway, P., How to protect DES against exhaustive key search (an analysis of DESX), J. Cryptol., 14, 1, 17-35 (2001) · Zbl 1068.94531
[37] Knuth, DE; Yao, AC; Traub, JF, The complexity of nonuniform random number generation, Algorithms and Complexity-New Directions and Recent Results, 357-428 (1976), New York: Academic Press Inc, New York
[38] Koblitz, N.; Menezes, AJ, Another look at “provable security”, J. Cryptol., 20, 1, 3-37 (2007) · Zbl 1115.68078
[39] A. Krauss, Automating Recursive Definitions and Termination Proofs in Higher-Order Logic. Ph.D. thesis, Technische Universität München (2009)
[40] A. Krauss, Recursive definitions of monadic functions. in PAR 2010. EPTCS, vol. 43, pp. 1-13 (2010)
[41] O. Kunčar, A. Popescu, A consistent foundation for Isabelle/HOL. in Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, (Springer, 2015), pp. 234-252 · Zbl 1433.68556
[42] O. Kunčar, A. Popescu, Comprehending Isabelle/HOL’s consistency. in Yang, H. (ed.) ESOP 2017. LNCS, vol. 10201, (Springer, 2017), pp. 724-749 · Zbl 06721341
[43] O. Kunčar, A. Popescu, Safety and conservativity of definitions in HOL and Isabelle/HOL. in POPL 2018. Proc. ACM Program. Lang., vol. 2, (ACM, 2017), pp. 24:1-24:26
[44] P. Lammich, Automatic data refinement. in ITP 2013. LNCS, vol. 7998, (Springer, 2013), pp. 84-99 · Zbl 1317.68216
[45] Larsen, KG; Skou, A., Bisimulation through probabilistic testing, Inf. Comp., 94, 1, 1-28 (1991) · Zbl 0756.68035
[46] Lindvall, T., Lectures on the Coupling Method (2002), New York: Dover Publications, Inc., New York · Zbl 1013.60001
[47] A. Lochbihler, A formal proof of the max-flow min-cut theorem for countable networks. Archive of Formal Proofs (2016), http://isa-afp.org/entries/MFMC_Countable.shtml, Formal proof development
[48] A. Lochbihler, Probabilistic functions and cryptographic oracles in higher order logic. in Thiemann, P. (ed.) Programming Languages and Systems (ESOP 2016). LNCS, vol. 9632, (Springer, 2016), pp. 503-531 · Zbl 1335.68033
[49] A. Lochbihler, CryptHOL. Archive of Formal Proofs (2017), http://isa-afp.org/entries/CryptHOL.shtml, Formal proof development
[50] A. Lochbihler, Probabilistic while loop. Archive of Formal Proofs (2017), http://isa-afp.org/entries/Probabilistic_While.html, Formal proof development
[51] A. Lochbihler, S.R. Sefidgar, A tutorial introduction to CryptHOL. Cryptology ePrint Archive, Report 2018/941 (2018), https://eprint.iacr.org/2018/941
[52] A. Lochbihler, S.R. Sefidgar, D.A. Basin, U. Maurer, Formalizing constructive cryptography using CryptHOL. in CSF 2019. (IEEE Computer Society, 2019), pp. 152-166
[53] A. Lochbihler, S.R. Sefidgar, B. Bhatt, Game-based cryptography in HOL. Archive of Formal Proofs (2017), http://isa-afp.org/entries/Game_Based_Crypto.shtml, Formal proof development
[54] A. Lochbihler, M. Züst, Programming TLS in Isabelle/HOL. Isabelle Workshop 2014 (2014)
[55] J. Lumbroso, Optimal discrete uniform generation from coin flips, and applications. CoRR arXiv:1304.1916 (2013)
[56] U. Maurer, Constructive cryptography – a new paradigm for security definitions and proofs. in Moedersheim, S., Palamidessi, C. (eds.) Theory of Security and Applications (TOSCA 2011). LNCS, vol. 6993, (Springer, 2011), pp. 33-56 · Zbl 1378.94055
[57] D. Micciancio, S. Tessaro, An equational approach to secure multi-party computation. in ITCS 2013. (ACM, 2013), pp. 355-372 · Zbl 1362.68080
[58] Milner, R.; Rose, HE; Shepherdson, J., Processes: A mathematical model of computing agents, Logic Colloquium 1973, Studies in Logic and the Foundations of Mathematics, 157-173 (1975), New York: Elsevier, New York
[59] Milner, R., A theory of type polymorphism in programming, J. Comput. Syst. Sci., 17, 3, 348-375 (1978) · Zbl 0388.68003
[60] J.C. Mitchell, Representation independence and data abstraction. in POPL 1986. (ACM, 1986), pp. 263-276
[61] Nipkow, T.; Klein, G., Concrete Semantics (2014), New York: Springer, New York · Zbl 1410.68004
[62] Nipkow, T.; Paulson, LC; Wenzel, M., Isabelle/HOL: A Proof Assistant for Higher-Order Logic, LNCS (2002), New York: Springer, New York · Zbl 0994.68131
[63] R. Pass, E. Shi, F. Tramer, Formal abstractions for attested execution secure processors. Cryptology ePrint Archive, Report 2016/1027 (2016), http://eprint.iacr.org/2016/1027 · Zbl 1411.94082
[64] A. Petcher, G. Morrisett, The foundational cryptography framework. in POST 2015. LNCS, vol. 9036, (Springer, 2015), pp. 53-72
[65] A. Petcher, G. Morrisett, A mechanized proof of security for searchable symmetric encryption. in CSF 2015. (IEEE 2015), pp. 481-494
[66] M. Piróg, J. Gibbons, The coinductive resumption monad. in Jacobs, B., Silva, A., Staton, S. (eds.) MFPS 2014. ENTCS, vol. 308, (2014), pp. 273-288 · Zbl 1337.68189
[67] Pitts, AM; Gordon, MJC; Melham, TF, The HOL logic, Introduction to HOL: a theorem proving environment for higher order logic, 191-232 (1993), Cambridge: Cambridge University Press, Cambridge
[68] N. Ramsey, A. Pfeffer, Stochastic lambda calculus and monads of probability distributions. in POPL 2002. (ACM, 2002), pp. 154-165 · Zbl 1323.68150
[69] J.C. Reynolds, Types, abstraction and parametric polymorphism. in IFIP 1983. Information Processing, vol. 83, (North-Holland/IFIP, 1983), pp. 513-523
[70] J. Sack, L. Zhang, A general framework for probabilistic characterizing formulae. in VMCAI 2012. LNCS, vol. 7148, (Springer, 2012), pp. 396-411 · Zbl 1326.68176
[71] N. Schirmer, M. Wenzel, State spaces – the locale way. in Huuck, R., Klein, G., Schlich, B. (eds.) SSV 2009. Electronic Notes in Theoretical Computer Science, vol. 254, (2009), pp. 161-179
[72] R. Segala, Modeling and Verification of Randomized Distributed Real-Time Systems. Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1995)
[73] V. Shoup, OAEP reconsidered. in Annual International Cryptology Conference. (Springer, 2001), pp. 239-259
[74] V. Shoup, Sequences of games: A tool for taming complexity in security proofs. Cryptology ePrint Archive, Report 2004/332 (2004), http://eprint.iacr.org/2004/332
[75] N.P. Smart, Cryptography Made Simple. Information Security and Cryptography, Springer (2016) · Zbl 1401.94002
[76] A. Sokolova, Coalgebraic Analysis of Probabilistic Systems. Ph.D. thesis, Technische Universiteit Eindhoven (2005)
[77] J. Stern, D. Pointcheval, J. Malone-Lee, N.P. Smart, Flaws in applying proof methodologies to signature schemes. in Annual International Cryptology Conference. (Springer, 2002), pp. 93-110 · Zbl 1026.94550
[78] P.Y. Strub, Some questions. Easycrypt Mailing list, post 383. https://lists.gforge.inria.fr/pipermail/easycrypt-club/2016-March/000383.html (2016)
[79] Swamy, N.; Chen, J.; Fournet, C.; Strub, PY; Bhargavan, K.; Yang, J., Secure distributed programming with value-dependent types, J. Funct. Program., 23, 4, 402-451 (2013) · Zbl 1290.68033
[80] P. Wadler, Theorems for free! in FPCA 1989. (ACM, 1989), pp. 347-359
[81] P. Wadler, The essence of functional programming. in POPL 1992. (ACM, 1992), pp. 1-14
[82] F. Wiedijk, A synthesis of the procedural and declarative styles of interactive theorem proving. Logical Methods in Computer Science 8(1:30), (2012) · Zbl 1238.68147
[83] L. Xi, K. Yang, Z. Zhang, D. Feng, DAA-related APIs in TPM 2.0 revisited. in (International Conference on Trust and Trustworthy Computing). (Springer, 2014), pp. 1-18
[84] A.C. Yao, Theory and application of trapdoor functions. in FOCS 1982. (IEEE Computer Society, 1982), pp. 80-91
[85] S. Zanella Béguelin, Formal Certification of Game-Based Cryptographic Proofs. Ph.D. thesis, École Nationale Supérieure des Mines de Paris (2010) · Zbl 1315.68081
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