The highest safety strength of private key per bit in the Public-Key Cryptography systems is the Elliptic Curve Cryptography at present. Under similar secure conditions, the ECC has the advantages such as: less computation amounts, shorter length of private key, smaller storing and bandwidth. Moreover, it has been declared as standard documents adopted by many international standard institutions and regarded as the most universally used public key system.Firstly, we generalize and analyze the advantages and present research of Elliptic Curve Cryptography; secondly, we study the basic theory of the ECC; thirdly, we illustrate the safety of the ECC and discuss the Elliptic Curve Key Agreement Scheme, Elliptic Curve Encryption Scheme and Elliptic Curve Digital Signature Algorithm; fourthly, we study fast algorithms of the multiplication and inversion multiplication of the element of in the underlying finite field F2m whose characteristic is two represented by the two basis of optimal normal basis and polynomial basis. We make improvements to the fast algorithm of the polynomial basis multiplication by Hankerson and base on the experiments, we describe the properties and compare the advantages of the multiplication and inversion multiplication of the elements in F2m field under optimal normal bases and polynomial basis .Results concluding from the study car be used as references in the realization of the elliptic curve cryptosystem; fifthly, we overview the current fast algorithm of point multiplication, improve the fix base point comb algorithm, advance the speed of the whole system and remark the advantages and disadvantages of the popular algorithms based upon the experimental datas; sixthly we realize the algorithm library of elliptic curve cryptography based on the F2m. Only change slightly in our algorithm library can we realize the ECDH, ECES, ECDSA based onF2m of anysize; seventhly, we realize the ECC on two secure elliptic curves, including ECDH, ECES, ECDSA. One is the 191 bits secure curve of ANSI X9.62 , elements in F2m field represented by polynomial basis, the other is the 148bits secure curve in our secure curve library elements in F2m field represented by optimal normal basis; the conclusion summarizes the whole paper and preview the the further developments of the work we have done. |