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International Conference on Foundations of Software Science and Computation Structures

FoSSaCS 2022: Foundations of Software Science and Computation Structures pp 161–183Cite as

Categorical composable cryptography

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13242)

Abstract

We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in cryptography. Our model is able to incorporate computational security, set-up assumptions and various attack models such as colluding or independently acting subsets of adversaries in a modular, flexible fashion. We conclude by using string diagrams to rederive the security of the one-time pad and no-go results concerning the limits of bipartite and tripartite cryptography, ruling out e.g., composable commitments and broadcasting.

Keywords

  • Cryptography
  • composable security
  • quantum cryptography
  • category theory

This work was supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0375, Canada’s NFRF and NSERC, an Ontario ERA, and the University of Ottawa’s Research Chairs program.

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References

  1. Abramsky, S., Barbosa, R.S., Karvonen, M., Mansfield, S.: A comonadic view of simulation and quantum resources. In: 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE (2019). https://doi.org/10.1109/LICS.2019.8785677

  2. Backes, M., Pfitzmann, B., Waidner, M.: A general composition theorem for secure reactive systems. In: 1st Theory of Cryptography Conference—TCC 2004. pp. 336–354 (2004). https://doi.org/10.1007/978-3-540-24638-1_19

  3. Backes, M., Pfitzmann, B., Waidner, M.: The reactive simulatability (RSIM) framework for asynchronous systems. Information and Computation 205(12), 1685–1720 (2007). https://doi.org/10.1016/j.ic.2007.05.002

  4. Ben-Or, M., Canetti, R., Goldreich, O.: Asynchronous secure computation. In: Proceedings of the twenty-fifth annual ACM symposium on Theory of computing. pp. 52–61 (1993). https://doi.org/10.1145/167088.167109

  5. Ben-Or, M., Horodecki, M., Leung, D.W., Mayers, D., Oppenheim, J.: The universal composable security of quantum key distribution. In: 2nd Theory of Cryptography Conference—TCC 2005. pp. 386–406 (2005). https://doi.org/10.1007/978-3-540-30576-7_21

  6. Ben-Or, M., Mayers, D.: General security definition and composability for quantum & classical protocols (2004), https://arxiv.org/abs/quant-ph/0409062

  7. Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: International Conference on Computers, Systems and Signal Processing. pp. 175–179 (1984)

    Google Scholar 

  8. Biham, E., Boyer, M., Boykin, P.O., Mor, T., Roychowdhury, V.: A proof of the security of quantum key distribution (extended abstract). In: 32nd Annual ACM Symposium on Theory of Computing—STOC 2000. pp. 715 – 724 (2000). https://doi.org/10.1145/335305.335406

  9. Breiner, S., Kalev, A., Miller, C.A.: Parallel self-testing of the GHZ state with a proof by diagrams. In: Proceedings of QPL 2018. Electronic Proceedings in Theoretical Computer Science, vol. 287, pp. 43–66 (2018). https://doi.org/10.4204/eptcs.287.3

  10. Breiner, S., Miller, C.A., Ross, N.J.: Graphical methods in device-independent quantum cryptography. Quantum 3,  146 (2019). https://doi.org/10.22331/q-2019-05-27-146

  11. Broadbent, A., Fitzsimons, J., Kashefi, E.: Universal blind quantum computation. In: 50th Annual Symposium on Foundations of Computer Science—FOCS 2009. pp. 517–526 (2009). https://doi.org/10.1109/FOCS.2009.36

  12. Camenisch, J., Küsters, R., Lysyanskaya, A., Scafuro, A.: Practical Yet Composably Secure Cryptographic Protocols (Dagstuhl Seminar 19042). Dagstuhl Reports 9(1), 88–103 (2019). https://doi.org/10.4230/DagRep.9.1.88

  13. Canetti, R.: Universally composable security: A new paradigm for cryptographic protocols. In: 42nd Annual Symposium on Foundations of Computer Science—FOCS 2001. pp. 136–145 (2001). https://doi.org/10.1109/SFCS.2001.959888

  14. Canetti, R., Fischlin, M.: Universally composable commitments. In: Advances in cryptology—CRYPTO 2001. pp. 19–40. Springer (2001). https://doi.org/10.1007/3-540-44647-8_2

  15. Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification. Physical Review A 81(6) (Jun 2010). https://doi.org/10.1103/physreva.81.062348

  16. Chitambar, E., Gour, G.: Quantum resource theories. Reviews of Modern Physics 91(2), 025001 (2019). https://doi.org/10.1103/revmodphys.91.025001

  17. Chitambar, E., Leung, D., Mančinska, L., Ozols, M., Winter, A.: Everything you always wanted to know about LOCC (but were afraid to ask). Communications in Mathematical Physics 328(1), 303–326 (2014). https://doi.org/10.1007/s00220-014-1953-9

  18. Clairambault, P., De Visme, M., Winskel, G.: Game semantics for quantum programming. Proceedings of the ACM on Programming Languages 3(POPL), 1–29 (2019). https://doi.org/10.1145/3290345

  19. Clairambault, P., de Visme, M., Winskel, G.: Concurrent quantum strategies. In: International Conference on Reversible Computation. pp. 3–19. Springer (2019). https://doi.org/10.1007/978-3-030-21500-2_1

  20. Coecke, B., Fritz, T., Spekkens, R.W.: A mathematical theory of resources. Information and Computation 250, 59–86 (2016). https://doi.org/10.1016/j.ic.2016.02.008

  21. Coecke, B., Perdrix, S.: Environment and classical channels in categorical quantum mechanics. Logical Methods in Computer Science Volume 8, Issue 4 (2012). https://doi.org/10.2168/LMCS-8(4:14)2012

  22. Coecke, B., Wang, Q., Wang, B., Wang, Y., Zhang, Q.: Graphical calculus for quantum key distribution (extended abstract). Electronic Notes in Theoretical Computer Science 270(2), 231–249 (2011). https://doi.org/10.1016/j.entcs.2011.01.034

  23. Cruttwell, G., Gavranović, B., Ghani, N., Wilson, P., Zanasi, F.: Categorical foundations of gradient-based learning (2021), https://arxiv.org/abs/2103.01931

  24. Cunningham, O., Heunen, C.: Purity through factorisation. In: Proceedings of QPL 2017. Electronic Proceedings in Theoretical Computer Science, vol. 266, pp. 315–328 (2017). https://doi.org/10.4204/EPTCS.266.20

  25. Datta, A., Derek, A., Mitchell, J.C., Pavlovic, D.: A derivation system for security protocols and its logical formalization. In: 16th IEEE Computer Security Foundations Workshop, 2003. Proceedings. pp. 109–125. IEEE (2003). https://doi.org/10.1109/csfw.2003.1212708

  26. Datta, A., Derek, A., Mitchell, J.C., Pavlovic, D.: Secure protocol composition. Electronic Notes in Theoretical Computer Science 83, 201–226 (2003). https://doi.org/10.1016/s1571-0661(03)50011-1

  27. Datta, A., Derek, A., Mitchell, J.C., Pavlovic, D.: A derivation system and compositional logic for security protocols. Journal of Computer Security 13(3), 423–482 (Aug 2005). https://doi.org/10.3233/JCS-2005-13304

  28. Datta, A., Derek, A., Mitchell, J.C., Roy, A.: Protocol composition logic (PCL). Electronic Notes in Theoretical Computer Science 172, 311–358 (Apr 2007). https://doi.org/10.1016/j.entcs.2007.02.012

  29. Durgin, N., Mitchell, J., Pavlovic, D.: A compositional logic for protocol correctness. In: Proceedings. 14th IEEE Computer Security Foundations Workshop, 2001. IEEE (2001). https://doi.org/10.1109/csfw.2001.930150

  30. Durgin, N., Mitchell, J., Pavlovic, D.: A compositional logic for proving security properties of protocols. Journal of Computer Security 11(4), 677–721 (Oct 2003). https://doi.org/10.3233/JCS-2003-11407

  31. Fong, B., Spivak, D., Tuyeras, R.: Backprop as functor: A compositional perspective on supervised learning. In: 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (2019). https://doi.org/10.1109/lics.2019.8785665

  32. Fritz, T.: Resource convertibility and ordered commutative monoids. Mathematical Structures in Computer Science 27(6), 850–938 (2015). https://doi.org/10.1017/s0960129515000444

  33. Gavranović, B.: Compositional deep learning (2019), https://arxiv.org/abs/1907.08292

  34. Goldwasser, S., Micali, S.: Probabilistic encryption. Journal of Computer and System Sciences 28(2), 270–299 (1984). https://doi.org/10.1016/0022-0000(84)90070-9

  35. Heunen, C.: Compactly accessible categories and quantum key distribution. Logical Methods in Computer Science 4(4) (2008). https://doi.org/10.2168/lmcs-4(4:9)2008

  36. Hillebrand, A.: Superdense coding with GHZ and quantum key distribution with W in the ZX-calculus. In: Proceedings of QPL 2011. Electronic Proceedings in Theoretical Computer Science, vol. 95, pp. 103–121 (2011). https://doi.org/10.4204/EPTCS.95.10

  37. Hines, P.M.: A diagrammatic approach to information flow in encrypted communication (2020). https://doi.org/10.1007/978-3-030-62230-5_9

  38. Hofheinz, D., Shoup, V.: GNUC: A new universal composability framework. Journal of Cryptology 28(3), 423–508 (2015). https://doi.org/10.1007/s00145-013-9160-y

  39. Horodecki, M., Oppenheim, J.: (Quantumness in the context of) Resource Theories. International Journal of Modern Physics B 27(01n03), 1345019 (2013). https://doi.org/10.1142/s0217979213450197

  40. Katz, J., Maurer, U., Tackmann, B., Zikas, V.: Universally composable synchronous computation. In: Theory of Cryptography, pp. 477–498. Springer (2013). https://doi.org/10.1007/978-3-642-36594-2_27

  41. Kissinger, A., Tull, S., Westerbaan, B.: Picture-perfect quantum key distribution (2017), https://arxiv.org/abs/1704.08668

  42. König, R., Renner, R., Bariska, A., Maurer, U.: Small accessible quantum information does not imply security. Physical Review Letters 98(14), 140502 (2007). https://doi.org/10.1103/PhysRevLett.98.140502

  43. Küsters, R., Tuengerthal, M., Rausch, D.: The IITM model: a simple and expressive model for universal composability. Journal of Cryptology 33(4), 1461–1584 (2020). https://doi.org/10.1007/s00145-020-09352-1

  44. Liao, K., Hammer, M.A., Miller, A.: ILC: a calculus for composable, computational cryptography. In: Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation. pp. 640–654. ACM (Jun 2019). https://doi.org/10.1145/3314221.3314607

  45. Lo, H.K., Chau, H.F.: Is quantum bit commitment really possible? Physical Review Letters 78(17), 3410–3413 (1997). https://doi.org/10.1103/PhysRevLett.78.3410

  46. Matt, C., Maurer, U., Portmann, C., Renner, R., Tackmann, B.: Toward an algebraic theory of systems. Theoretical Computer Science 747, 1–25 (2018). https://doi.org/10.1016/j.tcs.2018.06.001

  47. Maurer, U.: Constructive cryptography–a new paradigm for security definitions and proofs. In: Joint Workshop on Theory of Security and Applications—TOSCA 2011. pp. 33–56 (2011). https://doi.org/10.1007/978-3-642-27375-9_3

  48. Maurer, U., Renner, R.: Abstract cryptography. In: Innovations in Computer Science—ICS 2011 (2011)

    Google Scholar 

  49. Mayers, D.: The trouble with quantum bit commitment (1996), http://arxiv.org/abs/quant-ph/9603015

  50. Mayers, D.: Unconditional security in quantum cryptography. Journal of the ACM 48(3), 351–406 (2001). https://doi.org/10.1145/382780.382781

  51. Micciancio, D., Tessaro, S.: An equational approach to secure multi-party computation. In: 4th Conference on Innovations in Theoretical Computer Science—ITCS 2013. pp. 355–372 (2013). https://doi.org/10.1145/2422436.2422478

  52. Mifsud, A., Milner, R., Power, J.: Control structures. In: Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science. pp. 188–198. IEEE (1995). https://doi.org/10.1109/lics.1995.523256

  53. Moeller, J., Vasilakopoulou, C.: Monoidal Grothendieck construction. Theory and Applications of Categories 35(31), 1159–1207 (2020)

    Google Scholar 

  54. Müller-Quade, J., Renner, R.: Composability in quantum cryptography. New Journal of Physics 11(8), 085006 (2009). https://doi.org/10.1088/1367-2630/11/8/085006

  55. Pavlovic, D.: Categorical logic of names and abstraction in action calculi. Mathematical Structures in Computer Science 7(6), 619–637 (1997). https://doi.org/10.1017/S0960129597002296

  56. Pavlovic, D.: Tracing the man in the middle in monoidal categories. In: Coalgebraic Methods in Computer Science. pp. 191–217. Springer (2012). https://doi.org/10.1007/978-3-642-32784-1_11

  57. Pavlovic, D.: Chasing diagrams in cryptography. In: Casadio, C., Coecke, B., Moortgat, M., Scott, P. (eds.) Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday, pp. 353–367. Springer Berlin Heidelberg, Berlin, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54789-8_19

  58. Pfitzmann, B., Waidner, M.: A model for asynchronous reactive systems and its application to secure message transmission. In: 2001 IEEE Symposium on Security and Privacy—S&P 2001. pp. 184–200 (2000). https://doi.org/10.1109/SECPRI.2001.924298

  59. Portmann, C., Matt, C., Maurer, U., Renner, R., Tackmann, B.: Causal boxes: quantum information-processing systems closed under composition. IEEE Transactions on Information Theory 63(5), 3277–3305 (2017). https://doi.org/10.1109/TIT.2017.2676805

  60. Portmann, C., Renner, R.: Cryptographic security of quantum key distribution (2014), https://arxiv.org/abs/1409.3525

  61. Prabhakaran, M., Rosulek, M.: Cryptographic complexity of multi-party computation problems: Classifications and separations. In: Advances in Cryptology—CRYPTO 2008. pp. 262–279 (2008). https://doi.org/10.1007/978-3-540-85174-5_15

  62. Renner, R.: Security of quantum key distribution. International Journal of Quantum Information 06(01), 1–127 (2005). https://doi.org/10.1142/S0219749908003256

  63. Selinger, P.: A survey of graphical languages for monoidal categories. In: New structures for physics, pp. 289–355. Springer (2010). https://doi.org/10.1007/978-3-642-12821-9_4

  64. Shor, P.W., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Physical Review Letters 85(2), 441–444 (2000). https://doi.org/10.1103/physrevlett.85.441

  65. Stay, M., Vicary, J.: Bicategorical semantics for nondeterministic computation. In: Proceedings of the Twenty-ninth Conference on the Mathematical Foundations of Programming Semantics, MFPS XXIX. Electronic Notes in Theoretical Computer Science, vol. 298, pp. 367 – 382 (2013). https://doi.org/10.1016/j.entcs.2013.09.022

  66. Sun, X., He, F., Wang, Q.: Impossibility of quantum bit commitment, a categorical perspective. Axioms 9(1),  28 (2020). https://doi.org/10.3390/axioms9010028

  67. Tomamichel, M., Lim, C.C.W., Gisin, N., Renner, R.: Tight finite-key analysis for quantum cryptography. Nature Communications 3,  634 (2012). https://doi.org/10.1038/ncomms1631

  68. Unruh, D.: Universally composable quantum multi-party computation. In: Advances in Cryptology—EUROCRYPT 2010. pp. 486–505 (2010). https://doi.org/10.1007/978-3-642-13190-5_25

  69. Winskel, G.: Distributed probabilistic and quantum strategies. Electronic Notes in Theoretical Computer Science 298, 403–425 (2013). https://doi.org/10.1016/j.entcs.2013.09.024

  70. Wolf, S., Wullschleger, J.: New monotones and lower bounds in unconditional two-party computation. IEEE Transactions on Information Theory 54(6), 2792–2797 (2008). https://doi.org/10.1109/tit.2008.921674

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    Anne Broadbent & Martti Karvonen

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  1. Anne Broadbent
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Broadbent, A., Karvonen, M. (2022). Categorical composable cryptography. In: Bouyer, P., Schröder, L. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2022. Lecture Notes in Computer Science, vol 13242. Springer, Cham. https://doi.org/10.1007/978-3-030-99253-8_9

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