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The complexity of computing hard core predicates

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1294)

Abstract

We prove that a general family of hard core predicates requires circuits of depth (l-0(1))log n/log log n or super-polynomial size to be realized. This lower bound is essentially tight. For constant depth circuits, an exponential lower bound on the size is obtained. Assuming the existence of one-way functions, we explicitly construct a one-way function f(x) such that for any circuit c from a family of circuits as above, c(x) is almost always predictable from f(x).

Keywords

pseudo-randomness small-depth circuit one-way function 

References

  1. 1.
    W. Alexi, B. Chor, O. Goldreich, and C. P. Schnorr: RSA and Rabin Functions: Certain Parts Are as Hard as the Whole. SIAM J. on Computing 17 (1988), no 2, pp. 194–209.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    P. Beame: A Switching Lemma Primer. Manuscript, 1994.Google Scholar
  3. 3.
    M. Blum and S. Micali: How to Generate Cryptographically Strong Sequences of Pseudorandom Bits. SIAM J. on Computing 13 (1986), no 4, pp. 850–864.CrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Furst, J. Saxe, and M. Sipser: Parity, Circuits, and the Polynomial Time Hierarchy. Proc. 22nd Symposium on Foundations of Computer Science, IEEE, 1981, pp. 260–270.Google Scholar
  5. 5.
    O. Goldreich and L. A. Levin: A Hard Core Predicate for all One Way Functions. Proc. 21st Symposium on Theory of Computing, ACM, 1989, pp. 25–32.Google Scholar
  6. 6.
    J. Håstad: Computational Limitations of Small-Depth Circuits. ACM doctoral dissertation award, 1986. MIT Press 1987.Google Scholar
  7. 7.
    J. Håstad, A. W. Schrift, and A. Shamir: The Discrete Logarithm Modulo a Composite Hides O(n) Bits. J. of Computer and System Sciences 47 (1993), pp. 376–403.zbMATHCrossRefGoogle Scholar
  8. 8.
    R. Impagliazzo and M. Naor: Efficient Cryptographic Schemes Provably as Secure as Subset Sum. J. of Cryptology 9 (1996), no 4, pp. 199–216.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    N. Linial, Y. Mansour, and N. Nisan: Constant Depth Circuits, Fourier Transform, and Learnability. J. of the ACM 40 (1993), no 3, pp. 607–620.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Y. Mansour, N. Nisan, and P. Tiwari: The Computational Complexity of Universal Hashing. Theoretical Computer Science 107 (1993), pp. 121–133.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    M. NÄslund: Universal Hash Functions & Hard Core Bits. Proc. Eurocrypt 1995, LNCS 921, Springer Verlag, pp. 356–366.Google Scholar
  12. 12.
    M. NÄslund: All Bits in ax+b mod p are Hard. Proc. Crypto 1996, LNCS 1109, Springer Verlag, pp. 114–128.Google Scholar
  13. 13.
    A. C. Yao: Theory and Applications of Trapdoor Functions. Proc. 23rd Symposium on Foundations of Computer Science, IEEE, 1982, pp. 80–91.Google Scholar
  14. 14.
    A. C. Yao: Separating the Polynomial-Time Hierarchy by Oracles. Proc. 26th Symposium on Foundations of Computer Science, IEEE, 1985, pp. 1–10.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  1. 1.Dept. of Numerical Analysis and Computing ScienceRoyal Institute of TechnologyStockholmSweden

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