The Conic Curves Over Z_n And Its Applications In Public-Key Cryptosystem

With the development of the computing technology of hardware and software, the classical public-key cryptosystem is faced up with growing security threats. The cryptosystem based on algebra curves, one branch of public-key cryptosystems, has aroused great interest from researchers, and corresponding standards have been issued for elliptic curve cryptosystem. Furthermore, we try to find more algebra curves except ECC, or at least more advantageous in some aspects to realize public-key cryptosystem.The dissertation makes systematic research into conic curve, generalized conic curve and conic curve cryptosystem over residue class ring, and then applies these to kinds of systems such as divisible E-cash, issuance of E-cash and etc. The research results of this dissertation are summarized as follows:1. The conic curve Cn(a,b) is described in two methods. Two additions are defined, proved same with each other and denoted by⊕. It is proved that the rational points of Cn(a,b) form an abelian group under the operation⊕, which is denoted by (Cn(a,b),⊕).2. The dissertation discusses basic properties of the group (Cn(a,b),⊕), including DLP, order computation, generator finding, and points out how to prove the properties of Cn(a,b) by those of Cp(a,b) and Cq(a,b), makes it possible to establish analogue over the conics for kinds of cryptographic protocols.