In addition, the primary exchange exponent is also one of the names for Diffie Hellman Key Exchange. This method produces decryption keys that are never sent directly by using the given numbers to a particular power and it is a digital encryption method. This Diffie Hellman Key Exchange was introduced by Whitfield Diffie and Martin Hellman in 1976 in “New Directions in Cryptography.” Diffie Hellman Key Exchange is also a way to generate secret between two people for the examples is Alice and Bob. The secrets produced cannot be seen only by observation. If to get the answer one must know the key to the secret decryption. This is because Alice and Bob do not share information during major exchange but they generate shared keys. Diffie Hellman is a public key or asymmetric cryptosystem but it not an encryption. Based on the two user examples above, they have agreed to use two prime numbers and, where p is a large number and is a primitive root modulo . Numbers and should not be kept secret from other users as it is a public key. Alice and Bob will choose a large random number as a private key. Alice will calculate that he has sent to Bob and Bob will also calculate that he has sent to Alice. After which they will produce . Now they’ve shared the K key to exchange information without worrying about other users getting the information they share with.
Elgamal, (1985) had studied a public key cryptosystem and a signature scheme based on discrete logarithms. In their study, he proposed a new signature scheme with an implementation if the Diffie Hellman key distribution scheme to achieve the public key cryptosystem. He has presented the system that very difficult to compute logarithms over finite fields. Therefore, breaking the system has not been proved to be the same as the discrete logarithm calculation. He also estimates the public file size of this scheme is large than RSA scheme but is at most a factor of two to the structure of both schemes. Hence, this scheme has a size of the ciphertext is double of the RSA system.
In the articles that present by Boneh, (1998) on the Decision Diffie Hellman (DDH) problem, he states the DDH assumption is the strong assumption. Besides that, the best to know the method for breaking DDH is by using discrete log. To make sure their assumption is a strong assumption, he gives the evidence for its security. The evidence that he states are no generic algorithm can break DDH, certain of Diffie Hellman secret are hard to compute as the entire secret, and statistically. DDH also use to construct a non-interactive oblivious transfer protocol. So, he briefly about some applications of DDH that show DDH is so interactive to cryptographers.
Raymond & Stiglic, (2000) on a study the security issue in the Diffie Hellman key agreement protocol. Diffie Hellman key agreement protocol implementation has been plagued by serious security flaws. Besides that, they present a mathematical background of the basics needed to understand the Diffie Hellman protocol. They also stated the types of attacks to know the system is vulnerable. Since they had stated the type of the attacks they give some example to attacks which are based on mathematical tricks. The attacks against the Diffie Hellman protocol come in a few flavours such as denial of service attacks, outsider attack and insider attacks. Apart from this attack, they also use the Man in the Middle attacks which is an active attacker, capable of removing and adding messages, and can easily break the core Diffie Hellman protocol presented. They also use attacks based on number theory such as simple exponents and simple substitution attacks. Lastly, they expose some subtleties that appear when using the Diffie Hellman shares secret to obtain a key which can be used in other cryptographic operation. They have pointed out to the designers of cryptographic protocols with the most important security issues related to Diffie Hellman protocol.
Based on Ayan Mahalanobis, (2005) were recognized the Diffie Hellman key Exchange Protocol and its generalization and Nilpotent groups. At the beginning their research, he presents the discrete logarithm problem which concentrates on the Diffie Hellman Key Exchange protocol. He also searches for the current state of security for the Diffie Hellman key exchange protocol. Diffie Hellman key exchange can be in terms of group automorphisms. After that, he studies about the similar key exchange protocol for the Diffie Hellman key exchange using an abelian subgroup of the automorphisms group of a nonabelian group. He also shows the attacks on the Discrete Logarithm problem to explore the public key cryptography. Since the keep of working with central automorphisms, so he generalizes to arbitrary nilpotent class.
Based on the research done by Cash, Kiltz, and Shoup, (2008) stated that the twin Diffie Hellman problem and its applications. They have proposed a new computational problem called the twin Diffie Hellman problem. Twin Diffie Hellman is most closely related to the Diffie Hellman problem and it also can be used in the cryptographic constructions that are based on the Diffie Hellman problem. Besides that, they also showed that twin Diffie Hellman problem still hard, even with the presence of a decision oracle that recognizes the solution to the problem – which is the feature not enjoyed by the usual Diffie-Hellman problem. Without knowing any of the corresponding discrete logarithms they have shown how to build the trapdoor test that allows the efficient answer. Lastly, they had present the variant ElGamal encryption with the short, simple and tight ciphertexts in the random oracle model, under the assumption that Diffie Hellman problem is hard.