NDLI: The Relationship Between Breaking the Diffie--Hellman Protocol and Computing Discrete Logarithms

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Maurer, Ueli M. Wolf, Stefan
Copyright Year	1999
Abstract	Both uniform and nonuniform results concerning the security of the Diffie--Hellman key-exchange protocol are proved. First, it is shown that in a cyclic group G of order \|G\|=\prod{p_i^{e_i}}$, where all the multiple prime factors of \|G\| are polynomial in log\|G\|, there exists an algorithm that reduces the computation of discrete logarithms in G to breaking the Diffie--Hellman protocol in G and has complexity $\sqrt{\max\{\nu(p_i)\}}\cdot(\log\|G\|)^{O(1)}$, where $\nu(p)$ stands for the minimum of the set of largest prime factors of all the numbers d in the interval $[p-2\sqrt{p}+1,p+2\sqrt{p}+1]$. Under the unproven but plausible assumption that $\nu(p)$ is polynomial in log p, this reduction implies that the Diffie--Hellman problem and the discrete logarithm problem are polynomial-time equivalent in G. Second, it is proved that the Diffie--Hellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes: there exists a polynomial-time reduction algorithm that works for all those groups. Moreover, it is shown that breaking the Diffie--Hellman protocol for a small but nonnegligible fraction of the instances is equally difficult as breaking it for all instances. Finally, efficient constructions of groups are described for which the algorithm reducing the discrete logarithm problem to the Diffie--Hellman problem is efficiently constructible.
Starting Page	1689
Ending Page	1721
Page Count	33
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539796302749
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	28
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Subject Keyword	public-key cryptography finite fields discrete logarithms Algebraic coding theory; cryptography Diffie--Hellman protocol elliptic curves Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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