We show that proving results such as \(\mathcal{BPP}=\mathcal{P}\) essentially necessitate the construction of suitable pseudorandom generators (i.e., generators that suffice for such derandomization results). In particular, the main incarnation of this equivalence refers to the standard notion of uniform derandomization and to the corresponding pseudorandom generators (i.e., the standard uniform notion of “canonical derandomizers”). This equivalence bypasses the question of which hardness assumptions are required for establishing such derandomization results, which has received considerable attention in the last decade or so (starting with Impagliazzo and Wigderson [JCSS, 2001]).

We also identify a natural class of search problems that can be solved by deterministic polynomial-time reductions to \(\mathcal{BPP}\). This result is instrumental to the construction of the aforementioned pseudorandom generators (based on the assumption \(\mathcal{BPP}=\mathcal{P}\)), which is actually a reduction of the “construction problem” to \(\mathcal{BPP}\).

Caveat: Throughout the text, we abuse standard notation by letting \(\mathcal{BPP},\mathcal{P}\) etc denote classes of promise problems. We are aware of the possibility that this choice may annoy some readers, but believe that promise problem actually provide the most adequate formulation of natural decisional problems.


BPP derandomization pseudorandom generators promise problems search problems FPTAS randomized constructions 


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  1. 1.
    Aydinlioglu, B., Gutfreund, D., Hitchcock, J.M., Kawachi, A.: Derandomizing Arthur-Merlin Games and Approximate Counting Implies Exponential-Size Lower Bounds. Computational Complexity (to appear)Google Scholar
  2. 2.
    Blum, M., Micali, S.: How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits. In: SICOMP, vol. 13, pp. 850–864 (1984); Preliminary version in 23rd FOCS, pp. 80–91 (1982)Google Scholar
  3. 3.
    Chor, B., Goldreich, O.: On the Power of Two–Point Based Sampling. Jour. of Complexity 5, 96–106 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Even, S., Selman, A.L., Yacobi, Y.: The Complexity of Promise Problems with Applications to Public-Key Cryptography. Inform. and Control 61, 159–173 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fortnow, L.: Comparing Notions of Full Derandomization. In: 16th CCC, pp. 28–34 (2001)Google Scholar
  6. 6.
    Friedman, J.: A Proof of Alon’s Second Eigenvalue Conjecture. In: 35th STOC, pp. 720–724 (2003)Google Scholar
  7. 7.
    Gauss, C.F.: Untersuchungen Über Höhere Arithmetik, 2nd edn. Chelsea publishing company, New York (1981) (reprinted)Google Scholar
  8. 8.
    Goldreich, O.: Foundation of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Goldreich, O.: Computational Complexity: A Conceptual Perspective. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Goldreich, O., Wigderson, A.: On Pseudorandomness with respect to Deterministic Observers. In: RANDOM 2000, Proceedings of the Satellite Workshops of the 27th ICALP. Carleton Scientific (Proc. in Inform. 8), pp. 77–84 (2000); See also ECCC, TR00-056Google Scholar
  11. 11.
    Goldwasser, S., Micali, S.: Probabilistic Encryption. JCSS 28(2), 270–299 (1984); Preliminary version in 14th STOC (1982) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Grollmann, J., Selman, A.L.: Complexity Measures for Public-Key Cryptosystems. In: SICOMP, vol. 17(2), pp. 309–335 (1988)Google Scholar
  13. 13.
    Hochbaum, D. (ed.): Approximation Algorithms for NP-Hard Problems. PWS (1996)Google Scholar
  14. 14.
    Huxley, M.N.: On the Difference Between Consecutive Primes. Invent. Math. 15, 164–170 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Impagliazzo, R., Kabanets, V., Wigderson, A.: In Search of an Easy Witness: Exponential Time vs Probabilistic Polynomial Time. JCSS 65(4), 672–694 (2002); Preliminary version in 16th CCC (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Impagliazzo, R., Wigderson, A.: P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma. In: 29th STOC, pp. 220–229 (1997)Google Scholar
  17. 17.
    Impagliazzo, R., Wigderson, A.: Randomness vs. Time: De-randomization under a uniform assumption. JCSS 63(4), 672–688 (2001); Preliminary version in 39th FOCS (1998)zbMATHGoogle Scholar
  18. 18.
    Jerrum, M., Valiant, L., Vazirani, V.V.: Random Generation of Combinatorial Structures from a Uniform Distribution. In: TCS, vol. 43, pp. 169–188 (1986)Google Scholar
  19. 19.
    Kabanets, V., Impagliazzo, R.: Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds. Computational Complexity 13, 1–46 (2003); Preliminary version in 35th STOC (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nisan, N., Wigderson, A.: Hardness vs Randomness. JCSS 49(2), 149–167 (1994); Preliminary version in 29th FOCS (1988)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ostrovsky, R., Wigderson, A.: One-Way Functions are Essential for Non-Trivial Zero-Knowledge. In: 2nd Israel Symp. on Theory of Computing and Systems, pp. 3–17. IEEE Comp. Soc. Press, Los Alamitos (1993)CrossRefGoogle Scholar
  22. 22.
    Reingold, O., Trevisan, L., Vadhan, S.: Pseudorandom walks on regular digraphs and the RL vs. L problem. In: 38th STOC, pp. 457–466 (2006); See details in ECCC, TR05-022Google Scholar
  23. 23.
    Shaltiel, R., Umans, C.: Low-end Uniform Hardness vs Randomness Tradeoffs for AM. SICOMP 39(3), 1006–1037 (2009); Preliminary version in 39th STOC (2007)CrossRefzbMATHGoogle Scholar
  24. 24.
    Trevisan, L., Vadhan, S.: Pseudorandomness and Average-Case Complexity Via Uniform Reductions. Computational Complexity 16(4), 331–364 (2007); Preliminary version in 17th CCC (2002)Google Scholar
  25. 25.
    Umans, C.: Pseudo-random Generators for all Hardness. JCSS 67(2), 419–440 (2002); Preliminary version in 34th STOC (2002)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Vadhan, S.: An Unconditional Study of Computational Zero Knowledge. SICOMP 36(4), 1160–1214 (2006); Preliminary version in 45th FOCS (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yao, A.C.: Theory and Application of Trapdoor Functions. In: 23rd FOCS, pp. 80–91 (1982)Google Scholar

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  • Oded Goldreich

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