The Research And Software Implementation Of The Elliptic Curve Over Binary Field

Elliptic curve public-key cryptosystem(ECC)'s security is based on the intractability of the elliptic curve discrete logarithm problem. How to ensure the security and improve the implementation efficiency of the elliptic curve cryptosystem has been a hot topic in the field of applied public-key cryptography in recent years. FIPS 186-2 adopted the ECC standard recommended by National Institute of standards and Technology(NIST) in 2000.We focus on the NIST ECC and its algorithms, including the optimistic algorithms, elliptic curve group operation and improvement and implementation of ECC over binary field.Firstly, some factors on the ECC running speed are analyzed. In order to improve the running speed of the implementation, we analyze and study the existing ECC over binary field group operation algorithm. We offer some efficient methods to improve the speed of the ECC. We also improve the fast module multiplication algorithms over the binary field and optimize the basic algorithms of the ECC.Secondly, we also analyze and study some typic digital signature algorithms, and design a new algorithm of non-inversion elliptic curve digital signature algorithm (ECDSA). Meanwhile, its security is analyzed and compared. The result shows that it has the same security level and less running time of the signing and verifying algorithm. Furthermore, we offer common non-inversion ECDSA, including the signing and verifying algorithm; design a non-inversion message recovery algorithm based on the ECC.Finally, a Libary of ECC over binary field is implemented from view point of the software engineering. We summarize some experiences in the coding and testing processs.