Journal of Mathematical Sciences

, Volume 188, Issue 1, pp 17–34 | Cite as

Feebly secure cryptographic primitives

  • E. A. Hirsch
  • O. Melanich
  • S. I. Nikolenko

In 1992, A. Hiltgen provided first construction of provably (slightly) secure cryptographic primitives, namely, feebly one-way functions. These functions are provably harder to invert than to compute, but the complexity (viewed as the circuit complexity over circuits with arbitrary binary gates) is amplified only by a constant factor (in Hiltgen’s works, the factor approaches 2).

In traditional cryptography, one-way functions are the basic primitive of private-key shemes, while public-key schemes are constructed using trapdoor functions. We continue Hiltgen’s work by providing examples of feebly secure trapdoor functions where the adversary is guaranteed to spend more time than honest participants (also by a constant factor). We give both a (simpler) linear and a (better) nonlinear construction. Bibliography: 25 titles.


Constant Factor Circuit Complexity Cryptographic Primitive Trapdoor Function Honest Participant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    E. Allender, “Circuit complexity before the dawn of the new millennium,” in Proceedings of the 16th Conference on Foundations of Software Technology and Theoretical Computer Science (1996), pp. 1–18.Google Scholar
  2. 2.
    N. Blum, “A boolean function requiring 3n network size,” Theoret. Comput. Sci., 28, 337–345 (1984).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A. Davydow and S. I. Nikolenko, “Gate elimination for linear functions and new feebly secure constructions,” Lect. Notes Comput. Sci., 6651, 148–161 (2001).CrossRefGoogle Scholar
  4. 4.
    W. Diffie and M. Hellman, “New directions in cryptography,” IEEE Trans. Inform. Theory, IT-22, 664–654 (1976).MathSciNetGoogle Scholar
  5. 5.
    O. Goldreich, Foundations of Cryptography. Basic Tools, Cambridge Univ. Press, Cambridge (2001).MATHCrossRefGoogle Scholar
  6. 6.
    D. Grigoriev. E. A. Hirsch, and K. Pervyshev, “ A complete pulic-key cryptosystem,” Groups Complex. Cryptol., 1, 1–12 (2009).Google Scholar
  7. 7.
    D. Harnik, J, Kilian, M. Naor, O. Reingold, and A. Rosen, “On robust combiners for oblivious transfers and other primitives,” Lect. Notes Comput. Sci., 3494, 96–113 (2005).MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Håstad, Computational Limitations for Small Depth Circuits, MIT Press, Cambridge, Massachusetts (1987).Google Scholar
  9. 9.
    A. P. Hiltgen, “Constructions of feebly-one-way families of permutations,” in: Proceedings of AsiaCrypt’ 92 (1992), pp. 422–434.Google Scholar
  10. 10.
    A. P. Hiltgen, “Cryptographically relevant contributions to combinatorial complexity theory,” ETH-Zürich Dissertation, Hartung–Gorre Verlag, Konstanz (1994).Google Scholar
  11. 11.
    A. P. Hiltgen, “Towards a better understanding of one-wayness: facing linear permutations,” Lect. Notes Comput. Sci., 1233, 319–333 (1998).MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. A. Hirsch and S. I. Nikolenko, “A feebly secure trapdoor function,” Lect. Notes Comput. Sci., 5675, 129–142 (2009).CrossRefGoogle Scholar
  13. 13.
    K. Iwama, O. Lachish, H. Morizumi, and R. Raz, “An explicit lower bound of 5no(n) for Boolean circuits,” in: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (2001), pp. 399–408.Google Scholar
  14. 14.
    E. A Lamagna and J. E. Savage, “On the logical complexity of symmetric switching functions in monotone and complete bases,” Technical Report, Brown University, Rhode Island (1973).Google Scholar
  15. 15.
    L. A. Levin, “One-way functions and pseudorandom generators,” Combinatorica, 7, No. 4, 357–363 (1987).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    J. Massey, “The difficulty with difficulty,” a guide to the transparencies from the EUROCRYPT’96 IACR distinguished lecture (1996).Google Scholar
  17. 17.
    O. Melanich, “Nonlinear feebly secure cryptographic primitives,” PDMI Preprint 12/2009 (2009).Google Scholar
  18. 18.
    W. J. Paul, “A 2.5n lower bound on the combinational complexity of boolean functions,” SIAM J. Comput., 6, 427–443 (1977).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    A. A. Razborov, “Bounded arithmetic and lower bounds in Boolean complexity.” in: P. Clote and J. Remmel (eds.), Feasible Mathematics II, Birkhäuser Boston, Boston (1995), pp. 344–386.CrossRefGoogle Scholar
  20. 20.
    R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystem,” Comm. ACM, 21, No. 2, 120–126 (1978).MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    J. E. Savage, The Complexity of Computing, Wiley, New York (1976).MATHGoogle Scholar
  22. 22.
    C.E. Shannon, “Communication theory of secrecy systems,” Bell System Tech. J., 28, No. 4, 656–717 (1949).MathSciNetMATHGoogle Scholar
  23. 23.
    L. Stockmeyer, “On the combinational complexity of certain symmetric Boolean functions,” Math. Systems Theory, 10, 323–326 (1977).MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    G. S. Vernam, “Cipher printing telegraph system for secret wire and radio telegraphic communications,” J. IEEE, 55, 109–115 (1926).Google Scholar
  25. 25.
    I. Wegener, The Complexity of Boolean Functions, B. G. Teubner, Stuttgart, and John Wiley & Sons, Chichester (1987).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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