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Feebly secure cryptographic primitives

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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.

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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.

  2. 2.

    N. Blum, “A boolean function requiring 3n network size,” Theoret. Comput. Sci., 28, 337–345 (1984).

  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).

  4. 4.

    W. Diffie and M. Hellman, “New directions in cryptography,” IEEE Trans. Inform. Theory, IT-22, 664–654 (1976).

  5. 5.

    O. Goldreich, Foundations of Cryptography. Basic Tools, Cambridge Univ. Press, Cambridge (2001).

  6. 6.

    D. Grigoriev. E. A. Hirsch, and K. Pervyshev, “ A complete pulic-key cryptosystem,” Groups Complex. Cryptol., 1, 1–12 (2009).

  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).

  8. 8.

    J. Håstad, Computational Limitations for Small Depth Circuits, MIT Press, Cambridge, Massachusetts (1987).

  9. 9.

    A. P. Hiltgen, “Constructions of feebly-one-way families of permutations,” in: Proceedings of AsiaCrypt’ 92 (1992), pp. 422–434.

  10. 10.

    A. P. Hiltgen, “Cryptographically relevant contributions to combinatorial complexity theory,” ETH-Zürich Dissertation, Hartung–Gorre Verlag, Konstanz (1994).

  11. 11.

    A. P. Hiltgen, “Towards a better understanding of one-wayness: facing linear permutations,” Lect. Notes Comput. Sci., 1233, 319–333 (1998).

  12. 12.

    E. A. Hirsch and S. I. Nikolenko, “A feebly secure trapdoor function,” Lect. Notes Comput. Sci., 5675, 129–142 (2009).

  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.

  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).

  15. 15.

    L. A. Levin, “One-way functions and pseudorandom generators,” Combinatorica, 7, No. 4, 357–363 (1987).

  16. 16.

    J. Massey, “The difficulty with difficulty,” a guide to the transparencies from the EUROCRYPT’96 IACR distinguished lecture (1996).

  17. 17.

    O. Melanich, “Nonlinear feebly secure cryptographic primitives,” PDMI Preprint 12/2009 (2009).

  18. 18.

    W. J. Paul, “A 2.5n lower bound on the combinational complexity of boolean functions,” SIAM J. Comput., 6, 427–443 (1977).

  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.

  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).

  21. 21.

    J. E. Savage, The Complexity of Computing, Wiley, New York (1976).

  22. 22.

    C.E. Shannon, “Communication theory of secrecy systems,” Bell System Tech. J., 28, No. 4, 656–717 (1949).

  23. 23.

    L. Stockmeyer, “On the combinational complexity of certain symmetric Boolean functions,” Math. Systems Theory, 10, 323–326 (1977).

  24. 24.

    G. S. Vernam, “Cipher printing telegraph system for secret wire and radio telegraphic communications,” J. IEEE, 55, 109–115 (1926).

  25. 25.

    I. Wegener, The Complexity of Boolean Functions, B. G. Teubner, Stuttgart, and John Wiley & Sons, Chichester (1987).

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Author information

Correspondence to E. A. Hirsch.

Additional information

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2012, pp. 32–64.

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Hirsch, E.A., Melanich, O. & Nikolenko, S.I. Feebly secure cryptographic primitives. J Math Sci 188, 17–34 (2013). https://doi.org/10.1007/s10958-012-1103-x

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  • Constant Factor
  • Circuit Complexity
  • Cryptographic Primitive
  • Trapdoor Function
  • Honest Participant