If P ≠ NP then Some Strongly Noninvertible Functions Are Invertible

  • Lane A. Hemaspaandra
  • Kari Pasanen
  • Jörg RotheEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2138)


Rabi, Rivest, and Sherman alter the standard notion of noninvertibility to a new notion they call strong noninvertibility, and show—via explicit cryptographic protocols for secret-key agreement ([RS93,RS97] attribute this to Rivest and Sherman) and digital signatures [RS93,RS97]—that strongly noninvertible functions would be very useful components in protocol design. Their definition of strong noninvertibility has a small twist (“respecting the argument given”) that is needed to ensure cryptographic usefulness. In this paper, we show that this small twist has a large, unexpected consequence: Unless P = NP, some strongly noninvertible functions are invertible.


Computational and Structural Complexity 


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  1. [BC93]
    D. Bovet and P. Crescenzi. Introduction to the Theory of Complexity. Prentice Hall, 1993.Google Scholar
  2. [BDG95]
    J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. EATCS Texts in Theoretical Computer Science. Springer-Verlag, second edition, 1995.Google Scholar
  3. [Ber77]
    L. Berman. Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, Ithaca, NY, 1977.Google Scholar
  4. [BHHR99]
    A. Beygelzimer, L. Hemaspaandra, C. Homan, and J. Rothe. One-way functions in worst-case cryptography: Algebraic and security properties are on the house. SIGACT News, 30(4):25–40, December 1999.Google Scholar
  5. [GS88]
    J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309–335, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Hom00]
    C. Homan. Low ambiguity in strong, total, associative, one-way functions. Technical Report TR-734, University of Rochester, Department of Computer Science, Rochester, NY, August 2000.Google Scholar
  7. [HR99]
    L. Hemaspaandra and J. Rothe. Creating strong, total, commutative, associative one-way functions from any one-way function in complexity theory. Journal of Computer and System Sciences, 58(3):648–659, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Ko85]
    K. Ko. On some natural complete operators. Theoretical Computer Science, 37(1):1–30, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [KRS88]
    B. Kaliski Jr., R. Rivest, and A. Sherman. Is the data encryption standard a group? (Results of cycling experiments on DES). Journal of Cryptology, 1(1):3–36, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Pap94]
    C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.Google Scholar
  11. [RS93]
    M. Rabi and A. Sherman. Associative one-way functions: A new paradigm for secret-key agreement and digital signatures. Technical Report CS-TR-3183/UMIACS-TR-93-124, Department of Computer Science, University of Maryland, College Park, Maryland, 1993.Google Scholar
  12. [RS97]
    M. Rabi and A. Sherman. An observation on associative one-way functions in complexity theory. Information Processing Letters, 64(2):239–244, 1997.CrossRefMathSciNetGoogle Scholar
  13. [Sel92]
    A. Selman. A survey of one-way functions in complexity theory. Mathematical Systems Theory, 25(3):203–221, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [She86]
    A. Sherman. Cryptology and VLSI (a Two-Part Dissertation). PhD thesis, MIT, Cambridge, MA, 1986. Available as Technical Report MIT/LCS/TR-381.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Lane A. Hemaspaandra
    • 1
  • Kari Pasanen
    • 2
  • Jörg Rothe
    • 3
    Email author
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Nokia Networks and University of JyväskyläJyväskyläFinland
  3. 3.Abteilung für InformatikHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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