Design Of Public Key Cryptosystem Based On Special Conic Curve

Conic curve cryptography was firstly proposed by Cao Zhenfu in 1998.It is easier to conduct some calculations the conic curve group than the elliptic curve group,such as plaintext embedding and order of operations,especially on them to encode and decode,the cconic curve group is easier to perform.In this paper,we first introduce some basic definition and properties of the circle C_n(r),Pell equation C_n(D),parabola P_n(a)on the basis of the finite field Fq and ring Z_n,discuss properties of the circle,Pell equation and parabola on ring Z_n,testify the conclusion that the rational points on curve constitute a finite exchange group and give all the rational points on some simple curves.The RSA cryptosystem and ELGamal cryptosystem based on the ring Z_n of circle C_n(r)and Pell equations C_n(D)are deeply studied.Their security is depended on difficulty of large number factorization and the difficulty in calculating discrete logarithms on finite groups.Thus,the numerical simulation of the RSA cryptosystem and the ELGamal cryptosystem based on ring Z_n of circle C_n(r)and Pell equations C_n(D)is detailly discussed.Finally,the digital signature and multi-digital signcryption scheme based on a circle C_n(r)on ring Z_n is designed,and the security is analyzed.