基于双难题的环Z_n上圆锥曲线的数字签名

Signature on conic curve over Z_n based on two hard problems

通过对一个剩余类环Zn上圆锥曲线Cn(a,b)数字签名方案(X iao 06方案)的安全性分析,发现该方案的公开参数选取和算法设计存在问题,导致利用韦达定理可以分解模数n,说明X iao06方案的安全性不是基于整数分解难题的.针对此缺陷,采取保密部分参数和修改验证算法的方法,提出了一个改进的环Zn上圆锥曲线的数字签名方案,并且给出了改进方案的数值模拟.分析表明,改进的方案是一个同时基于离散对数和整数分解双难题的环Zn上圆锥曲线的数字签名方案,不仅保留了原X iao 06方案的优点(明文嵌入方便,求逆元速度快,元素阶的计算及曲线上点的运算容易),还具有很强的抗破解能力.

The security of the digital signature scheme （Xiao06 scheme） on conic curve C （a,b） over the residue class ring Z, was analyzed. The analysis result indicates that the published parameters can make the modulus n be factorized using the Weda＇s theorem, and shows that the Xiao 06 scheme is not a scheme whose security based on the integer factorization problem. To address this issue, an improved digital signature scheme on conic curve over Z was proposed. Some parameters were kept secretly, and the verification algorithm was modified in the improved scheme. Furthermore, the numerical simulation of the improved scheme was given. The analysis shows that the improved scheme is a digital signature scheme based on two hard problems in computing discrete logarithm and factorizing integer simultaneously, and that the improved scheme has not only the merits （convenience for plaintext embedding, quickness for the inverse operation, and easiness for element order and points computing in curve） of the Xiao06 scheme, but also the advantage of strong anticracking ability.