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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 ..."
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Cited by 42 (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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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.
DISCRETE LOGARITHMS, DIFFIEHELLMAN, AND REDUCTIONS
"... Abstract. We consider the OnePrimeNotp and AllPrimesButp variants of the Discrete Logarithm (DL) problem in a group of prime order p. We give reductions to the DiffieHellman (DH) problem that do not depend on any unproved conjectures about smooth or prime numbers in short intervals. We show t ..."
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Abstract. We consider the OnePrimeNotp and AllPrimesButp variants of the Discrete Logarithm (DL) problem in a group of prime order p. We give reductions to the DiffieHellman (DH) problem that do not depend on any unproved conjectures about smooth or prime numbers in short intervals. We show that the OnePrimeNotpDL problem reduces to DH in time roughly Lp(1/2); the AllPrimesButpDL problem reduces to DH in time roughly Lp(2/5); and the AllPrimesButpDL problem reduces to the DH plus Integer Factorization problems in polynomial time. We also prove that under the Riemann Hypothesis, with ε log p queries to a yesorno oracle one can reduce DL to DH in time roughly Lp(1/2); and under a conjecture about smooth numbers, with εlog p queries to a yesorno oracle one can reduce DL to DH in polynomial time. 1.