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The Relationship Between Breaking the DiffieHellman Protocol and Computing Discrete Logarithms
, 1998
"... Both uniform and nonuniform results concerning the security of the DiffieHellman keyexchange protocol are proved. First, it is shown that in a cyclic group G of order jGj = Q p e i i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that re ..."
Abstract

Cited by 38 (3 self)
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Both uniform and nonuniform results concerning the security of the DiffieHellman keyexchange protocol are proved. First, it is shown that in a cyclic group G of order jGj = Q p e i i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that reduces the computation of discrete logarithms in G to breaking the DiffieHellman protocol in G and has complexity p maxf(p i )g \Delta (log jGj) O(1) , where (p) stands for the minimum of the set of largest prime factors of all the numbers d in the interval [p \Gamma 2 p p+1; p+2 p p+ 1]. Under the unproven but plausible assumption that (p) is polynomial in log p, this reduction implies that the DiffieHellman problem and the discrete logarithm problem are polynomialtime equivalent in G. Second, it is proved that the DiffieHellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes: there exists a p...
The DiffieHellman Protocol
 DESIGNS, CODES, AND CRYPTOGRAPHY
, 1999
"... The 1976 seminal paper of Diffie and Hellman is a landmark in the history of cryptography. They introduced the fundamental concepts of a trapdoor oneway function, a publickey cryptosystem, and a digital signature scheme. Moreover, they presented a protocol, the socalled DiffieHellman protoco ..."
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Cited by 26 (0 self)
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The 1976 seminal paper of Diffie and Hellman is a landmark in the history of cryptography. They introduced the fundamental concepts of a trapdoor oneway function, a publickey cryptosystem, and a digital signature scheme. Moreover, they presented a protocol, the socalled DiffieHellman protocol, allowing two parties who share no secret information initially, to generate a mutual secret key. This paper summarizes the present knowledge on the security of this protocol.
Polynomial Representations of the DiffieHellman Mapping
"... We obtain lower bounds on the degrees of polynomials representing the DiffieHellman mapping (g x , g y ) # g xy , where g is a primitive root of a finite field IF q of q elements. These bounds are exponential in terms of log q. In particular, these results can be used to obtain lower bounds on the ..."
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Cited by 3 (0 self)
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We obtain lower bounds on the degrees of polynomials representing the DiffieHellman mapping (g x , g y ) # g xy , where g is a primitive root of a finite field IF q of q elements. These bounds are exponential in terms of log q. In particular, these results can be used to obtain lower bounds on the parallel arithmetic complexity of breaking the DiffieHellman cryptosystem. The method is based on bounds of numbers of solutions of some polynomial equations.