Abstract
In this work we give a non-trivial upper bound on the spectral norm of various Boolean predicates of the Diffie-Hellman function. For instance, we consider every individual bit and arbitrary unbiased intervals. Combining the bound with recent results from complexity theory we can rule out the possibility that such a Boolean function can be represented by simple functions like depth-2 threshold circuits with a small number of gates.
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References
D. Boneh. The decision Diffie-Hellman problem. Lecture Notes in Computer Science, 1423:48–63, 1998.
D. Coppersmith and I. Shparlinski. On polynomial approximation of the Discrete Logarithm and the Diffie-Hellman mapping. Journal of Cryptology, 13(3):339–360, March 2000.
Forster, Krause, Lokam, Mubarakzjanov, Schmitt, and Simon. Relations between communication complexity, linear arrangements, and computational complexity. FSTTCS: Foundations of Software Technology and Theoretical Computer Science, 21, 2001.
J. Forster. personal communication, 2002.
J. Forster and H. U. Simon. On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes. In Proceedings of the 13th International Conference on Algorithmic Learning Theory, pages 128–138, 2002.
Juergen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. In Proceedings of the Sithteenth Annual Conference on Computational Complexity, pages 100–106. IEEE Computer Society, 2001.
E. Kiltz. On the representation of boolean predicates of the diffie-hellman function. Manuscript, 2003.
N. M. Korobov. Exponential Sums and their Applications. Kluwer Academic Publishers, 1992.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, Cambridge, 1997.
T. Lange and A. Winterhof. Polynomial interpolation of the elliptic curve and XTR discrete logarithm. In Proceedings of the 8th Annual International Computing and Combinatorics Conference (COCOON’02), pages 137–143, 2002.
R. Paturi and J. Simon. Probabilistic communication complexity. Journal of Computer and System Sciences, 33(1):106–123, aug 1986.
K. Prachar. Primzahlverteilung. Springer-Verlag, Berlin, 1957.
R. Shaltiel. Towards proving strong direct product theorems. In Proceedings of the Sithteenth Annual Conference on Computational Complexity, pages 107–119. IEEE Computer Society, 2001.
I. Shparlinski. Communication complexity and fourier coefficients of the Diffie-Hellman key. Proc. the 4th Latin American Theoretical Informatics Conf., LNCS 1776, pages 259–268, 2000.
I. E. Shparlinski. Number Theoretic Methods in Cryptography. Birkhäuser Verlag, 1999.
I. E. Shparlinski. Cryptographic Application of Analytic Number Theory. Birkhäuser Verlag, 2002.
A. Winterhof. A note on the interpolation of the Diffie-Hellman mapping. In Bulletin of the Australian Mathematical Society, volume 64, pages 475–477, 2001.
A. Winterhof. Polynomial interpolation of the discrete logarithm. Designs, Codes and Cryptography, 25:63–72, 2002.
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Kiltz, E. (2003). On the Representation of Boolean Predicates of the Diffie-Hellman Function. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_21
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DOI: https://doi.org/10.1007/3-540-36494-3_21
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