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Nonuniform Indistinguishability and Unpredictability Hardcore Lemmas: New Proofs and Applications to Pseudoentropy

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Information Theoretic Security (ICITS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9063))

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Abstract

Hardcore lemmas are results in complexity theory which state that average-case hardness must have a very hard “kernel”, that is a subset of instances where the given problem is extremely hard. They find important applications in hardness amplification. In this paper we revisit the following two fundamental results:

  1. 1

    The hardcore lemma for unpredictability, due to Impagliazzo (FOCS ’95). It states that if a boolean function f is “moderately” hard to predict on average, then there must be a set of noticeable size on which f is “extremely” hard to predict.

  2. 2

    The hardcore lemma for indistinguishability, proved by Maurer and Tesaro (TCC’10), states that for two random variables X and Y which are ε-computationally close, there are events A and B of probability 1 − ε such that the distributions of X|A and Y|B are “computationally” identical.

Using only the standard min-max theorem and some basic facts about convex approximations in L p spaces, we provide alternative modular proofs and some generalizations of these results in the nonuniform setting, achieving best possible bounds for (a) and slightly improving the known bounds for (b). As an interesting application, we show a strengthening of the transformation between two most popular pseudoentropy variants: HILL and Metric Entropy, and apply it to show how to extract pseudorandomness from a sequence of metric-entropy sources of poor quality. In this case we significantly improve security parameters, comparing to the best known techniques.

A preliminary version of this work appeared in the Proceedings of Student Research Forum Papers and Posters at SOFSEM 2015.

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References

  1. Barak, B., Hardt, M., Kale, S.: The uniform hardcore lemma via approximate bregman projections. In: SODA 2009. Society for Industrial and Applied Mathematics, pp. 1193–1200 (2009)

    Google Scholar 

  2. Barak, B., Shaltiel, R., Wigderson, A.: Computational analogues of entropy. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 200–215. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Donahue, M.J., Darken, C., Gurvits, L., Sontag, E.: Rates of convex approximation in non-hilbert spaces. Constructive Approximation 13(2), 187–220 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dziembowski, S., Pietrzak, K.: Leakage-resilient cryptography. In: FOCS 2008, pp. 293–302. IEEE Computer Society (2008)

    Google Scholar 

  5. Fuller, B., O’Neill, A., Reyzin, L.: A unified approach to deterministic encryption: New constructions and a connection to computational entropy. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 582–599. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  6. Goldreich, O., Nisan, N., Wigderson, A.: On Yao’s XOR-lemma. In: Goldreich, O. (ed.) Studies in Complexity and Cryptography. LNCS, vol. 6650, pp. 273–301. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  7. Hastad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Holenstein, T.: Key agreement from weak bit agreement. In: STOC 2005, pp. 664–673. ACM (2005)

    Google Scholar 

  9. Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: FOCS 1995, pp. 538–545. IEEE Computer Society (1995)

    Google Scholar 

  10. Klivans, A.R., Servedio, R.A.: Boosting and hard-core sets. In: FOCS 1999, pp. 624–633. IEEE Computer Society (1999)

    Google Scholar 

  11. Klivans, A.R., Servedio, R.A.: Boosting and hard-core set construction. Mach. Learn. 51(3), 217–238 (2003)

    Article  MATH  Google Scholar 

  12. Lin, H., Tessaro, S.: Amplification of chosen-ciphertext security. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 503–519. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  13. Lu, C.-J., Tsai, S.-C., Wu, H.-L.: On the complexity of hard-core set constructions. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 183–194. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  14. Maurer, U., Tessaro, S.: A hardcore lemma for computational indistinguishability. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 237–254. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. Vadhan, S., Zheng, C.J.: Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In: STOC 2012, pp. 817–836. ACM (2012)

    Google Scholar 

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Correspondence to Maciej Skorski .

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Skorski, M. (2015). Nonuniform Indistinguishability and Unpredictability Hardcore Lemmas: New Proofs and Applications to Pseudoentropy. In: Lehmann, A., Wolf, S. (eds) Information Theoretic Security. ICITS 2015. Lecture Notes in Computer Science(), vol 9063. Springer, Cham. https://doi.org/10.1007/978-3-319-17470-9_8

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  • DOI: https://doi.org/10.1007/978-3-319-17470-9_8

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-17469-3

  • Online ISBN: 978-3-319-17470-9

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