基于特殊圆锥曲线的公钥密码算法设计

发布时间：2018-04-17 20:48

本文选题：剩余类环 + 有限域　；参考：《北方工业大学》2017年硕士论文

【摘要】：圆锥曲线密码学是曹珍富于1998年第一次提出的。圆锥曲线群上明文的嵌入、阶的运算等各项计算比椭圆曲线群上的更容易,特别是在它们上进行编码和解码,圆锥曲线群更容易执行。本文介绍了基于有限域F_p和基于环Z_n上的圆、Pell方程、抛物线等的定义和性质。重点对环Z_n上的圆C_n(r)、Pell方程C_n(D)、抛物线P_n(a)等曲线的性质进行了比较深入的讨论,证明了曲线上的有理点构成有限交换群,并给出了一些简单曲线上的全部有理点。深入研究了环Z_n上的圆C_n(r)、Pell方程C_n(D)的RSA密码体制和ELGamal密码体制,它们的安全性基于大数分解的困难性和有限域上计算离散对数的困难性,并详细地讨论了基于环Z_n上的圆C_n(r)、Pell方程C_n(D)的RSA密码体制和ELGamal密码体制的数值模拟。最后,设计了基于环Z_n上的圆曲线C_n(r)的数字签名及多重数字签密方案,并分析了安全性。
[Abstract]:Conic Cryptography was first proposed by Cao Zhenfu in 1998.The computation of plaintext embedding and order operation on conic curve group is easier than that on elliptic curve group, especially when coding and decoding on them, conic curve group can be executed more easily.In this paper, the definitions and properties of the circular Pell equation and parabola based on the finite field FSP and the ring Zn are introduced.In this paper, we focus on the discussion of the properties of the C _ S _ n / Pell's equation C _ S _ n _ D _ (?) and the parabola _ P _ n _ (a) curves, and prove that the rational points on the curves form a finite commutative group, and give all the rational points on some simple curves.In this paper, the RSA cryptosystem and the ELGamal cryptosystem of the C _ S _ n / P _ ell equation are studied. Their security is based on the difficulty of the large number decomposition and the difficulty of computing the discrete logarithm on the finite field.The numerical simulation of RSA cryptosystem and ELGamal cryptosystem based on the circle C _ S _ n / Pell equation C _ S _ n over Z _ S _ n is also discussed in detail.Finally, a digital signature and multiple digital signcryption scheme based on the circle curve C _ s _ n over Zs _ n are designed, and the security is analyzed.
【学位授予单位】：北方工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TN918.4

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