Elliptic Curve Cryptography Fast Implementation

Elliptic curve cryptography is the developing trend of public key cryptography. Fast implementation problems of the elliptic curve cryptography arithmetic are the hotspot direction in the elliptic curve cryptography research. Elliptic curve cryptography based on the finite fields applies many cryptographic schemes such as data encryption, key exchange and digital signature. In this dissertation, with the emphasis of the algorithms for the fast implementation of the elliptic curve cryptography, the elliptic curve cryptosystems and the related algorithms are introduced. The application of elliptic curve cryptography in cryptographic schemes: Elliptic Curve Diffle-Hellman Key Exchange (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) are also implemented. It consists of four chapters. In chapter one, task background, some introduced materials including the motivations and recent research of the elliptic curve cryptosystems and main research content of this dissertation are briefly surveyed. In chapter two, some basic materials of the elliptic curve theory are discussed. In chapter three, the discrete logarithm problem of elliptic curve, the actual attack, and the structure of elliptic curve parameters are discussed. ECDH and ECDSA schemes are introduced in detail. In chapter four, first of all the fast implementation methods of the elliptic curve cryptography arithmetic are studied, including point addition, double-add basic operation implementation and scalar multiplication implementation. Then two applications of the cryptographic schemes based on elliptic curve: Elliptic Curve Diffle-Hellman Key Exchange (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) are implemented by using C++ language. The key problems in its implementation and the solutions are analysed. The main fuctions of the system are briefly described.