On the Complexity of Non-adaptively Increasing the Stretch of Pseudorandom Generators

  • Eric Miles
  • Emanuele Viola
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6597)


We study the complexity of black-box constructions of linear-stretch pseudorandom generators starting from a 1-bit stretch oracle generator G. We show that there is no construction which makes non-adaptive queries to G and then just outputs bits of the answers. The result extends to constructions that both work in the non-uniform setting and are only black-box in the primitive G (not the proof of correctness), in the sense that any such construction implies NP/poly \(\ne\) P/poly. We then argue that not much more can be obtained using our techniques: via a modification of an argument of Reingold, Trevisan, and Vadhan (TCC ’04), we prove in the non-uniform setting that there is a construction which only treats the primitive G as black-box, has polynomial stretch, makes non-adaptive queries to the oracle G, and outputs an affine function (i.e., parity or its complement) of the oracle query answers.


  1. 1.
    Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC 0. SIAM J. Comput. 36(4), 845–888 (2006)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bronson, J., Juma, A., Papakonstantinou, P.A.: Limits on the stretch of non-adaptive constructions of pseudo-random generators. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 522–539. Springer, Heidelberg (2011)Google Scholar
  3. 3.
    Gennaro, R., Gertner, Y., Katz, J., Trevisan, L.: Bounds on the efficiency of generic cryptographic constructions. SIAM J. Comput. 35(1), 217–246 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Goldreich, O.: Foundations of Cryptography. Basic Tools, vol. 1. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  5. 5.
    Goldreich, O., Krawczyk, H., Luby, M.: On the existence of pseudorandom generators. SIAM J. Comput. 22(6), 1163–1175 (1993)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Goldreich, O., Levin, L.: A hard-core predicate for all one-way functions. In: 21st Annual ACM Symposium on Theory of Computing (STOC), pp. 25–32 (1989)Google Scholar
  7. 7.
    Haitner, I., Reingold, O., Vadhan, S.P.: Efficiency improvements in constructing pseudorandom generators from one-way functions. In: 42nd ACM Symposium on Theory of Computing (STOC), pp. 437–446 (2010)Google Scholar
  8. 8.
    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999) (electronic) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Impagliazzo, R.: Very strong one-way functions and pseudo-random generators exist relative to a random oracle (1996) (manuscript)Google Scholar
  10. 10.
    Lu, C.-J.: On the complexity of parallel hardness amplification for one-way functions. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 462–481. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Nisan, N., Wigderson, A.: Hardness vs randomness. J. Computer & Systems Sciences 49(2), 149–167 (1994)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Reingold, O., Trevisan, L., Vadhan, S.: Notions of Reducibility between Cryptographic Primitives. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 1–20. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Viola, E.: On constructing parallel pseudorandom generators from one-way functions. In: 20th Annual Conference on Computational Complexity (CCC), pp. 183–197. IEEE, Los Alamitos (2005)CrossRefGoogle Scholar
  14. 14.
    Yao, A.: Theory and applications of trapdoor functions. In: 23rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 80–91. IEEE, Los Alamitos (1982)CrossRefGoogle Scholar
  15. 15.
    Zimand, M.: Efficient privatization of random bits. In: Randomized Algorithms Satellite Workshop of the 23rd International Symposium on Mathematical Foundations of Computer Science (1998), http://triton.towson.edu/~mzimand/pub/rand-privat.ps

Copyright information

© International Association for Cryptologic Research 2011

Authors and Affiliations

  • Eric Miles
    • 1
  • Emanuele Viola
    • 1
  1. 1.Northeastern UniversityUSA

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