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A Modular Framework for Quantum-Proof Randomness Extractors

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Abstract

A quantum-proof extractor is a function that is used to extract randomness from any weakly random source X in the presence of prior quantum information about X. It is known that some constructions are quantum-proof, such as Trevisan’s construction. However, these extractors are generally restrictive for applications on the one-bit output construction and the weak design. Here, we give a modular framework to combine multi-bit output extractors (not only one-bit) with pseudorandom transform, and show that it is sound in the presence of quantum side information. Then combined with the theory of operator spaces, we improve previous theoretical proofs, and discuss the security of two-bit output extractor by giving a tighter bound for it.

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References

  1. Calude, B.C.S.: Information and Randomness: An Algorithmic Perspective, 2nd edn, Revised and Extended. Siam Review (2010)

  2. Binder, K.: Applications of monte carlo methods to statistical physics. Reports Progress Phys. 60(60), 487–559(73) (1997)

    Article  ADS  Google Scholar 

  3. Ferrenberg, A.M., Landau, D.P., Wong, Y.J.: Monte carlo simulations: Hidden errors from “good” random number generators. Phys. Rev. Lett. 69(23), 3382–3384 (1992)

    Article  ADS  Google Scholar 

  4. Shaltiel, R.: Recent developments in explicit constructions of extractors. Bull. Eur. Assoc. Theor. Comput. Sci. Eatcs 77(77), 67–95 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Lu, C.J., Reingold, O., Vadhan, S., et al.: Extractors: Optimal up to constant factors. Stoc, 602–611 (2003)

  6. Trevisan, L., Vadhan, S.: Extracting randomness from samplable distributions. In: Annual Symposium on Foundations of Computer Science, pp. 32–42 (2000)

  7. Mauerer, W., Portmann, C., Scholz, V.B.: A modular framework for randomness extraction based on Trevisan’s construction. Comput. Sci. (2012)

  8. Bennett, C.H., Brassard, G., Robert, J.M.: Privacy amplification by public discussion. SIAM J. Comput. 17, 210–229 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Renner, R.: Security of quantum key distribution. Int. J. Quantum Inf. 6(1), 1–127 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Tomamichel, M., Schaffner, C., Smith, A., Renner, R.: Leftover hashing against quantum side information. IEEE Trans. Inf. Theory 57(8), 2703–2707 (2010)

    MathSciNet  Google Scholar 

  11. Barak, B., Shaltiel, R., Tromer, E.: True random number generators secure in a changing environment. Lecture Notes Comput. Sci. 2779, 166–180 (1970)

    Article  MATH  Google Scholar 

  12. Trevisan, L.: Extractors and pseudorandom generators. J. Acm. 48(4), 860–879 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hayashi, M., Tsurumaru, T.: More efficient privacy amplification with lessrandom seeds via dual universal hash function (2015)

  14. Ma, X., Xu, F., Xu, H., et al.: Postprocessing for quantum random number generators: Entropy evaluation and randomness extraction. Phys. Rev. A 87 (6), 944–948 (2012)

    Google Scholar 

  15. Mo, L.I., Zhang, C.M., Yin, Z.Q., et al: An overview on the post-processing procedure in quantum key distribution. J. Cryptol. Res. (2015)

  16. Vazirani, U., Vidick, T.: Certifiable quantum dice. R. Soc. London Philosop. Trans. 370(1971), 3432–3448 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. König, R.T., Terhal, B.M.: The bounded-storage model in the presence of a quantum adversary. IEEE Trans. Inf. Theory 54(2), 749–762 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. De, A., Portmann, C., Vidick, T., et al.: Trevisan’s extractor in the presence of quantum side information. Siam J. Comput. 41(4) (2009)

  19. Ta-Shma, A., Zuckerman, D., Safra, S.: Extractors from reedmuller codes. J. Comput. Syst. Sci. 72, 786–812 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shaltiel, R., Umans, C.: Simple extractors for all min-entropies and a new pseudorandom generator. J. Acm 52(2), 172–216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gavinsky, D., Kempe, J., Wolf, R.D.: Exponential separation of quantum and classical one-way communication complexity for a boolean function. Electron Colloquium Comput Complex (2006)

  22. Berta, M., Fawzi, O., Scholz, V.B.: Quantum-proof randomness extractors via operator space theory. Physics (2015)

  23. König, R., Renner, R., Schaffner, C.: The operational meaning of min- and max-entropy. IEEE Trans. Inf. Theory 55(9), 4337–4347 (2009)

    Article  MathSciNet  Google Scholar 

  24. Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. König, R, Renner, R.: Sampling of min-entropy relative to quantum knowledge. IEEE Trans. Inf. Theory 57(7), 4760–4787 (2007)

    Article  MathSciNet  Google Scholar 

  26. De, A., Vidick T.: Near-optimal extractors against quantum storage. In: Proceedings of the Annual Acm Symposium on Theory of Computing, pp. 161–170 (2010)

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Acknowledgments

This work is funded by the China Postdoctoral Science Foundation (Grants No. 2014M562582).

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Authors and Affiliations

  1. Zhengzhou Information Science and Technology Institute, Zhengzhou, China

    Yipeng Liu, JianSheng Guo & Jingyi Cui

  2. Science and Technology on Information Assurance Laboratory, Beijing, China

    JianSheng Guo

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  1. Yipeng Liu
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  2. JianSheng Guo
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  3. Jingyi Cui
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Correspondence to JianSheng Guo.

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Liu, Y., Guo, J. & Cui, J. A Modular Framework for Quantum-Proof Randomness Extractors. Int J Theor Phys 55, 5157–5171 (2016). https://doi.org/10.1007/s10773-016-3137-1

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  • DOI: https://doi.org/10.1007/s10773-016-3137-1

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