IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. I (Jan. - Feb. 2017), PP 56-61 www.iosrjournals.org DOI: 10.9790/5728-1301015661 www.iosrjournals.org 56 | Page Elliptic curve Cryptography and Diffie- Hellman Key exchange Dr.S.Vasundhara Asst. Prof of Mathematics G.Narayanamma Institute of Technology & science(For Women) Shaikpet, Hyderabad. Telengana Abstract: In this paper an introduction of Elliptic curve cryptography explained Then the Diffie- Hellman algorithm was explained with clear examples. Keywords: Cryptography Elliptic curve cryptography, Diffie-Hellman Key exchange. I. Introduction The history of cryptography is long and interesting. It has a very considerable turning point when two researchers from Stanford, Whitfield Diffie and Martin Hellman, published the paper ―New Directions in Cryptography‖ in 1976. They preface the new idea of public key cryptography in the paper. Public-key cryptography and [4]symmetric-key cryptography are two main categories of cryptography. The Well-known public-key cryptography algorithms are RSA (Rivest, et al. 1978), El-Gamal and Elliptic Curve Cryptography. Presently, there are only three problems of public key cryptosystems that are considered to be both secure and effective (Certicom, 2001). Table 1.1 shows these mathematical problems and the cryptosystems that rely on such problems. Mathematical problem Detail Cryptosystem 1 Integer Factorization problem (IFP) Given an integer n find its prime factorization RSA 2 Discrete Logarithm problem(DLS) Given integer g and h find x‘ such that =g x mod n Diffie- Hellman(DH) 3 Elliptic curve discrete logarithmic problem(ECDLP) Given points P and Q on the curve find ‗x‘ such that Q=xP Diffie- Hellman(DH) Providing an equivalent level of security with smaller key size is an advantage of ECC compared to RSA. It is very efficient to[1] implement ECC.ECC obtains lower power consumption, and faster computation. It also gains small memory and bandwidth because of its key size length (Dormale, Bulens and Quisquater 2004), (Huang 2007). Such attributes are mainly fascinating in security applications in which calculative power and integrated circuit space are limited. Wireless devices and smart cards present a good example for the constrained devices with limited resources. Cryptography companies such as Certicom Corporation have already implemented ECC in their products for some commercial purposes which are RFID and Zigbee. This company has an agreement with NSA on a set of cryptographic algorithms called suite B. This suite uses Elliptic curves and works over the prime field. A modular arithmetic performs a main role in public key cryptographic systems (Dormale, et al. 2004). Some of these PKC are the Diffie-Hellman keys exchange algorithm (Diffie and Hellman 1976), the decipherment operation in the RSA algorithm (Quisquater and Couvreur 1982), the US Government Digital Signature Standard (FIPS 2000), and also elliptic curve cryptography (Koblitz 1987). Diffie–Hellman: Diffie–Hellman key exchange (D–H)[11] is a specific method of exchanging cryptographic keys. It is one of the earliest practical examples of key exchange implemented within the field of cryptography. The Diffie–Hellman key exchange method allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher. The scheme was first published by Whitfield Diffie and Martin Hellman in 1976, although it had been separately invented a few years earlier within GCHQ, the British signals intelligence agency, by Malcolm J. Williamson but was kept classified. In 2002, Hellman suggested the algorithm be called Diffie–Hellman–Merkle key exchange in recognition of Ralph Merkle's contribution to the invention of public-key cryptography (Hellman, 2002). Although Diffie–Hellman key agreement itself is an anonymous (non-authenticated) key-agreement protocol, it provides the basis for a variety of authenticated protocols, and is used to provide perfect forward