Abstract
Interactive behaviors are ubiquitous in modern cryptography, but are also present in \(\lambda \)-calculi, in the form of higher-order constructions. Traditionally, however, typed \(\lambda \)-calculi simply do not fit well into cryptography, being both deterministic and too powerful as for the complexity of functions they can express. We study interaction in a \(\lambda \)-calculus for probabilistic polynomial time computable functions. In particular, we show how notions of context equivalence and context metric can both be characterized by way of traces when defined on linear contexts. We then give evidence on how this can be turned into a proof methodology for computational indistinguishability, a key notion in modern cryptography. We also hint at what happens if a more general notion of a context is used.
This work is partially supported by the ANR project 12IS02001 PACE.
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Notes
- 1.
Following the literature on the subject, this stands for any function \(\delta :A\times A\rightarrow \mathbb {R}\) such that \(\delta (x,y)=\delta (y,x)\), \(\delta (x,x)=0\) and \(\delta (x,y)+\delta (y,z)\ge \delta (x,z)\).
- 2.
A negligible function is a function which tends to 0 faster than any inverse polynomial (see [9] for more details).
References
Abramsky, S.: The lazy lambda calculus. In: Turner, D. (ed.) Research Topics in Functional Programming, pp. 65–117. Addison Wesley, Boston (1990)
Cappai, A., Dal Lago, U.: On equivalences, metrics, and polynomial time (long version) (2015) http://arxiv.org/abs/1506.03710
Crubillé, R., Dal Lago, U.: On probabilistic applicative bisimulation and call-by-value \(\lambda \)-calculi. In: Shao, Z. (ed.) ESOP 2014 (ETAPS). LNCS, vol. 8410, pp. 209–228. Springer, Heidelberg (2014)
Dal Lago, U., Sangiorgi, D., Alberti, M.: On coinductive equivalences for higher-order probabilistic functional programs. In: POPL (2014)
Dal Lago, U., Parisen Toldin, P.: A higher-order characterization of probabilistic polynomial time. In: Peña, R., van Eekelen, M., Shkaravska, O. (eds.) FOPARA 2011. LNCS, vol. 7177, pp. 1–18. Springer, Heidelberg (2012)
Dal Lago, U., Zuppiroli, S., Gabbrielli, M.: Probabilistic recursion theory and implicit computational complexity. Sci. Ann. Comp. Sci. 24(2), 177–216 (2014)
Deng, Y., Zhang, Y.: Program equivalence in linear contexts. CoRR, abs/1106.2872 (2011)
Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labeled markov systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 258–273. Springer, Heidelberg (1999)
Goldreich, O.: The Foundations of Cryptography. Basic Techniques, vol. 1. Cambridge University Press, New York (2001)
Goldreich, O., Sudan, M.: Computational indistinguishability: a sample hierarchy. In: CCC, pp. 24–33 (1998)
Hofmann, M.: A mixed modal/linear lambda calculus with applications to bellantoni-cook safe recursion. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 275–294. Springer, Heidelberg (1997)
Hofmann, M.: Safe recursion with higher types and bck-algebra. Ann. Pure Appl. Logic 104(1–3), 113–166 (2000)
Howe, D.J.: Proving congruence of bisimulation in functional programming languages. Inf. Comput. 124(2), 103–112 (1996)
Katz, J., Lindell, Y.: Introduction to Modern Cryptography. Chapman and Hall/CRC Press, Boca Raton (2007)
Mitchell, J.C., Mitchell, M., Scedrov, A.: A linguistic characterization of bounded oracle computation and probabilistic polynomial time. In: FOCS, pp. 725–733 (1998)
Nowak, D., Zhang, Y.: A calculus for game-based security proofs. In: Heng, S.-H., Kurosawa, K. (eds.) ProvSec 2010. LNCS, vol. 6402, pp. 35–52. Springer, Heidelberg (2010)
Zhang, Y.: The computational SLR: a logic for reasoning about computational indistinguishability. Mathematical Structures in Computer Science 20(5), 951–975 (2010)
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Cappai, A., Dal Lago, U. (2015). On Equivalences, Metrics, and Polynomial Time. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_24
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