Towards a better understanding of one-wayness: Facing linear permutations

  • Alain P. Hiltgen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1403)


The one-wayness of linear permutations, i.e., invertible linear Boolean functions F: {0,1}n → {0, 1}n, is investigated. For linear permutations with a triangular matrix description (tlinear permutations), we prove that one-wayness, C(F−1)/C(F), is non-trivially upperbounded by 16√n, where C(.) denotes unrestricted circuit complexity. We also prove that this upper bound strengthens as the complexity of the inverse function increases, limiting the one-wayness of t-linear permutations with C(F−1) = n2/(c log2(n)) to a constant, i.e., a value that is independent of n. Direct implications for linear and also non-linear permutations are discussed. Moreover, and for the first time ever, a description is given about where, in the case of linear permutations, practical one-wayness would have to come from, if it exists.


  1. 1.
    V. L. Arlazarov, E. A. Dinic, M. A. Kronrod, and I. A. FaradŽev, “On economical construction of the transitive closure of an oriented graph,” Dokl. Akad. Nauk, vol. 194, pp. 487–488, 1970.MATHGoogle Scholar
  2. 1a.
    V. L. Arlazarov, E. A. Dinic, M. A. Kronrod, and I. A. FaradŽev, Engl. Transl.: Sov. Math. Dokl., vol. 11, pp. 1209–1210, 1970.MATHGoogle Scholar
  3. 2.
    R. B. Boppana and J. C. Lagarias, “One-way functions and circuit complexity,” Information and Computation, vol. 74, pp. 226–240, 1987. Conf. version: Proc. 1st Ann. IEEE Symp. Structure in Complexity Th., pp. 51–65, 1986.MATHMathSciNetCrossRefGoogle Scholar
  4. 3.
    S. W. Boyack, The Robustness of Combinatorial Measures of Boolean Matrix Complexity. Ph.D. thesis, Massachusetts Inst. of Techn., 1985.Google Scholar
  5. 4.
    D. Coppersmith and S. Winograd, “Matrix multiplication via arithmetic progressions,” J. Symbolic Comput., vol. 9, pp. 251–280, 1990. Conf. version: Proc. 19th Ann. ACM Symp. Theory of Comput., pp. 1–6, 1987.MATHMathSciNetCrossRefGoogle Scholar
  6. 5.
    W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. Inform. Theory, vol. IT-22, no. 6, pp. 644–655, Nov. 1976.MathSciNetCrossRefGoogle Scholar
  7. 6.
    S. Goldwasser, “The search for provably secure cryptosystems,” in Cryptology and Computational Number Theory: Proc. Symp. Appl. Math., vol. 42 (C. Pomerance, ed.), pp. 89–113, Providence, RI: Amer. Math. Soc., 1990.Google Scholar
  8. 7.
    A. P. Hiltgen, “Constructions of feebly-one-way families of permutations,” Advances in Cryptology: Proc. Auscrypt'92, Springer, pp. 422–434, 1993.Google Scholar
  9. 8.
    A. P. Hiltgen, Cryptographically Relevant Contributions to Combinational Complexity Theory, vol. 3 of ETH Series in Information Processing, ed. J. L. Massey. Konstanz: Hartung-Gorre, 1994. Reprint of: Ph.D. thesis no. 10382, Swiss Federal Institute of Technology, ETH-Zürich, 1993.Google Scholar
  10. 9.
    A. P. Hiltgen and J. Ganz, “On the existence of specific complexity-asymmetric permutations,” Technical Report, Signal and Inform. Proc. Lab, ETH-Zürich, 1992.Google Scholar
  11. 10.
    R. Impagliazzo and M. Luby, “One-way functions are essential for complexity based cryptography,” Proc. 30th Ann. IEEE Symp. Foundations of Computer Sci., pp. 230–235, 1989.Google Scholar
  12. 11.
    J. L. Massey, “The difficulty with difficulty,” A Guide to the Transparencies from the Eurocrypt '96 IACR Distinguished Lecture, Signal and Inform. Proc. Lab., ETH Zürich, 1996. Available from Scholar
  13. 12.
    A. Menezes, P. van Orschot, and S. Vanstone, Handbook of Applied Cryptography. CRC-Press Series on Discrete Mathematics and its Applications, CRC-Press: Boca Raton, 1997.Google Scholar
  14. 13.
    W. J. Paul, “Realizing Boolean functions on disjoint sets of variables,” Theoret. Comput. Sci., vol. 2, pp. 383–396, 1976.MATHMathSciNetCrossRefGoogle Scholar
  15. 14.
    V. Strassen, “Algebraic complexity theory,” in Handbook of Theoretical Computer Science, vol. A (J. van Leeuwen, ed.), ch. 11, Amsterdam: Elsevier, 1990.Google Scholar
  16. 15.
    C. Sturtivant and Z. Zhang, “Efficiently inverting bijections given by straight line programs,” Proc. 31th Ann. IEEE Symp. Foundations of Computer Sci., pp. 327–334, 1990.Google Scholar
  17. 16.
    D. Uhlig, “On the synthesis of self-correcting schemes from functional elements with a small number of reliable elements,” Matemat. Zametki, vol. 15, no. 6, pp. 937–944, June 1974.MATHMathSciNetGoogle Scholar
  18. 16a.
    D. Uhlig, Engl. Transl.: Math. Notes Acad. Sci. USSR, vol. 15, pp. 558–562, 1974.MATHMathSciNetCrossRefGoogle Scholar
  19. 17.
    I. Wegener, The Complexity of Boolean Functions. New York: Wiley (Stuttgart: Teubner), 1987.MATHGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Alain P. Hiltgen
    • 1
  1. 1.UBS - Corporate IT SecurityZurichSwitzerland

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