| Conic curve cryptology was a new orientation of cryptology in the recent research of the cryptology. Compared with ECC, the computing on conic curve is much easier, and the encoding/decoding procedure becomes simple. Moreover, a group modulo n on conic curve can be formed, on which we can construct cryptogram similar to RSA. As known to all, the discrete logarithms problem on conic curve is not easier than it is on elliptic curves if the steps are the same. Therefore, conic curve cryptology has become one of the most important directions in cryptography.The paper stresses the algorithms and protocols to implement conic curve cryptosystems. Although conic curves can be defined over binary fields, prime fields and ring, I only researched the conic curve cryptosystems defined over prime fields and ring. As such, the algorithms over these two type fields are thoroughly studied in the paper while the algorithms to implement conic curve point addition and point multiplication are presented as well. To go further, I reconsidered the addition on Conic curve of ring to reduce operating quantum. The paper studies the bilinear pairings based on conic curve, forming the bilinear pairings on conic curve.Based on these bilinear pairings, apply some signature and Signcryption algorithms of public-key cryptosystem,including Zhang-Safavi-Naini-Susilo signature algorithms, Hess signature algorithms based on user ID, Cheng-Liu-Wang algorithms based on user ID, Libert-Quisquater Signcryption algorithms, Chow-Yiu-Hui-Chow Signcryption algorithms based on user ID, in conic curve over ring. The security of all these algorithms mentioned above is based on several difficult problems of bilinear pairings. Based on the groundwork, I developed a conic curve cryptosystem over prime fields and ring, and then integrated it with symmetric encryption function and hash function. At the same time, I constructed a fundamental cryptographic library, named "CC-CSP", according to the framework of Windows CSP. |
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