Abstract
Diffie and Hellman proposed a key exchange scheme in 1976, which got their name in the literature afterwards. In the same epoch-making paper, they conjectured that breaking their scheme would be as hard as taking discrete logarithms. This problem has remained open for the multiplicative group modulo a prime P that they originally proposed. Here it is proven that both problems are (probabilisticly) polynomial-time equivalent if the totient of P-1 has only small prime factors with respect to a (fixed) polynomial in 2logP.
There is no algorithm known that solves the discrete log problem in probabilistic polynomial time for the this case, i.e., where the totient of P-1 is smooth. Consequently, either there exists a (probabilistic) polynomial algorithm to solve the discrete log problem when the totient of P-1 is smooth or there exist primes (satisfying this condition) for which Diffie-Hellman key exchange is secure.
Supported in pan by the Netherlands Organization for Scientific Research (NWO).
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References
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© 1990 Springer-Verlag Berlin Heidelberg
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den Boer, B. (1990). Diffie-Hellman is as Strong as Discrete Log for Certain Primes. In: Goldwasser, S. (eds) Advances in Cryptology — CRYPTO’ 88. CRYPTO 1988. Lecture Notes in Computer Science, vol 403. Springer, New York, NY. https://doi.org/10.1007/0-387-34799-2_38
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DOI: https://doi.org/10.1007/0-387-34799-2_38
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