The Elliptic Curve Cryptosystem (ECC) is an emerging alternative for traditional Public-Key Cryptosystem like RSA, DSA and DH. It provides the highest strength-per-bit of any cryptosystem known today with smaller key sizes resulting in faster computations, lower power consumption and memory. It also provides a methodology for obtaining high-speed, efficient and scalable implementation of protocols for authentication and key agreement. This paper provides an introduction to Elliptic Curves and how they are used to create a secure and powerful cryptosystem. It provides an overview of the three hard mathematical problems that provide the basis for the security of public key cryptosystems used today: the Integer Factorization Problem (IFP), the Discrete Logarithm Problem (DLP), and the Elliptic Curve Discrete Logarithm Problem (ECDLP). It explains the proposed protocol which is improved to reduce the storage requirements for establishing a shared secret key between two parties, to sign and verify a document and to establish a mutual authentication between two parties. The result of implementation is also discussed.
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