computational complexity

, Volume 4, Issue 2, pp 158–174 | Cite as

The power of adaptiveness and additional queries in random-self-reductions

  • Joan Feigenbaum
  • Lance Fortnow
  • Carsten Lund
  • Daniel Spielman


We study random-self-reductions from a structural complexity-theoretic point of view. Specifically, we look at relationships between adaptive and nonadaptive random-self-reductions. We also look at what happens to random-self-reductions if we restrict the number of queries they are allowed to make. We show the following results:
  • ∘ There exist sets that are adaptively random-self-reducible but not nonadaptively random-self-reducible. Under plausible assumptions, there exist such sets inNP.

  • ∘ There exists a function that has a nonadaptive (k(n)+1)-random-self-reduction but does not have an adaptivek(n)-random-self-reduction.

  • ∘ Forany countable class of functionsC andany unbounded functionk(n), there exists a function that is nonadaptivelyk(n)-uniformly-random-self-reducible but is not inC/poly. This should be contrasted with Feigenbaum, Kannan, and Nisan's theorem that all nonadaptively 2-uniformly-random-self-reducible sets are inNP/poly.

Key words

Adaptiveness random-self-reducibility 

Subject classifications



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  1. [1]
    M. Abadi, J. Feigenbaum, andJ. Kilian, On Hiding Information from an Oracle.Journal of Computer and System Sciences 39 (1989), 21–50.Google Scholar
  2. [2]
    L. Babai, Random oracles separate PSPACE from the polynomial-time hierarchy.Information Processing Letters 26 (1987), 51–53.Google Scholar
  3. [3]
    L. Babai, L. Fortnow, andC. Lund. Non-deterministic exponential time has two-prover interactive protocols.Computational Complexity 1 (1991), 3–40.Google Scholar
  4. [4]
    D. Beaver andJ. Feigenbaum, Hiding instances in multioracle queries. InProceedings of the 7th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, volume 415, Springer-Verlag, 1990, 37–48.Google Scholar
  5. [5]
    D. Beaver, J. Feigenbaum, J. Kilian, andP. Rogaway, Security with low communication overhead. InAdvances in Cryptology—CRYPTO '90, Lecture Notes in Computer Science, volume 537, Springer-Verlag, 1991, 62–76.Google Scholar
  6. [6]
    R. Beigel, M. Bellare, J. Feigenbaum, andS. Goldwasser, Languages that are Easier than their Proofs. InProceedings of the 32nd Symposium on Foundations of Computer Science (1991), IEEE Computer Society, 19–28.Google Scholar
  7. [7]
    M. Blum andS. Kannan, Designing programs that check their work.Journal of the ACM, to appear. Extended abstract inProceedings of the 21st Symposium on the Theory of Computing (1989), ACM, 86–97.Google Scholar
  8. [8]
    M. Blum, M. Luby, andR. Rubinfeld, Self-testing/correcting with applications to numerical problems.Journal of Computer and System Sciences 47 (1993), 549–595.Google Scholar
  9. [9]
    M. Blum andS. Micali, How to generate cryptographically strong sequences of pseudo-random bits.SIAM Journal on Computing 13 (1984), 850–864.Google Scholar
  10. [10]
    J. Feigenbaum, Locally Random Reductions in Interactive Complexity Theory. InAdvances in Computational Complexity Theory, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 13, AMS, 1993, 73–98.Google Scholar
  11. [11]
    J. Feigenbaum andL. Fortnow, Random-self-reducibility of complete sets.SIAM Journal on Computing 22 (1993), 994–1005.Google Scholar
  12. [12]
    J. Feigenbaum, L. Fortnow, C. Lund, andD. Spielman,The Power of Adaptiveness and Additional Queries in Random-Self-Reductions. AT&T Bell Laboratories Technical Memorandum, January, 1992.Google Scholar
  13. [13]
    J. Feigenbaum, S. Kannan, andN. Nisan, Lower bounds on random-self-reducibility. InProceedings of the 5th Structure in Complexity Theory Conference (1990), IEEE Computer Society, 100–109.Google Scholar
  14. [14]
    S. Goldwasser andS. Micali, Probabilistic encryption.Journal of Computer and System Sciences 28 (1984), 270–299.Google Scholar
  15. [15]
    S. Goldwasser, S. Micali, andC. Rackoff, The knowledge complexity of interactive proof-systems.SIAM Journal on Computing 18 (1989), 186–208.Google Scholar
  16. [16]
    E. Hemaspaandra, A. Naik, M. Ogiwara, andA. Selman, P-Selective Sets, and Reducing Search to Decision vs. Self-Reducibility. Submitted for journal publication. Preliminary version, by A. Naik, M. Ogiwara, and A. Selman, inProceedings of the 8th Structure in Complexity Theory Conference (1993), IEEE Computer Society, 52–64.Google Scholar
  17. [17]
    R. Lipton, New directions in testing. InDistributed Computing and Cryptography, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 2, AMS, 1991, 191–202.Google Scholar
  18. [18]
    C. Lund, L. Fortnow, H. Karloff, andN. Nisan, Algebraic methods for interactive proof systems.Journal of the ACM 39 (1992), 859–868.Google Scholar
  19. [19]
    A. Shamir, IP=PSPACE.Journal of the ACM 39 (1992), 869–877.Google Scholar
  20. [20]
    M. Tompa andH. Woll, Random-self-reducibility and zero-knowledge interactive proofs of possession of information. InProceedings of the 28th Symposium on Foundations of Computer Science (1987), IEEE Computer Society, 472–482.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • Joan Feigenbaum
    • 1
  • Lance Fortnow
    • 2
  • Carsten Lund
    • 1
  • Daniel Spielman
    • 3
  1. 1.AT&T Bell LaboratoriesMurray HillUSA
  2. 2.Computer Science DepartmentUniversity of ChicagoChicagoUSA
  3. 3.Mathematics DepartmentMassachusetts Institute of TechnologyCambridgeUSA

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