Lower bounds on generic algorithms in groups (1998) [11 citations — 1 self]
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@INPROCEEDINGS{Maurer98lowerbounds,author = {Ueli Maurer and Stefan Wolf},
title = {Lower bounds on generic algorithms in groups},
booktitle = {72--84 in Advances in Cryptology - EUROCRYPT '98},
year = {1998},
pages = {72--84},
publisher = {Springer-Verlag}
}
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Abstract:
Abstract. In this paper we consider generic algorithms for computational problems in cyclic groups. The model of a generic algorithm was proposed by Shoup at Eurocrypt '97. A generic algorithm is a generalpurpose algorithm that does not make use of any particular property of the representation of the group elements. Shoup proved the hardness of the discrete logarithm problem and the Diffie-Hellman problem with respect to such algorithms for groups whose order contains a large prime factor. By extending Shoup's technique we prove lower bounds on the complexity of generic algorithms solving different problems in cyclic groups, and in particular of a generic reduction of the discrete logarithm problem to the Diffie-Hellman problem. It is shown that the two problems are not computationally equivalent in a generic sense for groups whose orders contain a multiple large prime factor. This complements earlier results which stated this equivalence for all other groups. Furthermore, it is shown that no generic algorithm exists that computes p-th roots efficiently in a group whose order is divisible by p 2 if p is a large prime. Keywords. Diffie-Hellman protocol, discrete logarithms, generic algorithms, roots in finite groups, complexity, lower bounds.
Citations
| 1752 | New directions in cryptography – Diffie, Hellman - 1976 |
| 58 | Algorithms for black-box fields and their application to cryptography – Boneh, Lipton - 1996 |
| 55 | Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms – Maurer - 1994 |
| 17 | Diffie-Hellman is as strong as discrete log for certain primes – Boer - 1989 |
