基于Eisenstein环上圆锥曲线的数字签名

Digital Signature Based on Conic Curve over Eisenstein Ring

为了使曲线上的密码体制更加安全有效,引进Eisenstein环Z[ω],介绍剩余类环Z[ω]/(r)上的圆锥曲线Cr(a,b),其中,r为Z[ω]上满足()()Nπ1≠Nπ2的2个不同的不可分数π1,π2的乘积。给出基于RSA的盲签名方案在圆锥曲线Cr(a,b)上的模拟,并以电子支付系统中的可分电子现金为例讨论Cr(a,b)上数字签名的应用,其安全性是基于大数分解和有限Abel群Cr(a,b)上计算离散对数的困难性。圆锥曲线Cr(a,b)上的数字签名方案体现了圆锥曲线所具有的明文嵌入方便、运算速度快、更易于实现等优点。

In order to make the cryptosystem over curves safe and efficient, this paper introduces Eisenstein ring Z[w] and introduces the conic curve Cr （a,b） over the residue class ring Z[w]/（r）, where r is the product of two different impartibility numbers, which satisfies N（π1）≠ N （π2） over Z[w] . Conic analog of blind signature based on RSA over conic curve Cr （a,b） is presented and E-cash in the E-payment system is taken as an example to discuss the application of digital signature over Cr （a,b）. The security is based on the difficulty in factorizing large integer and computing discrete logarithm on Abel group Cr （a,b）. The digital signature based on the conic curve Cr （a,b） shows the merits on conic curve that it is easy to embed plaintext, and its computing speed is rapid and is easy to implement.