基于椭圆曲线的数字签名系统的设计与实现

The Design and Implementation of Digital Signature System Based on Elliptic Curve

基于椭圆曲线的数字签名系统是目前主流的数字签名系统之一,并且被认为是经典的RSA系统的最佳替代者。基于椭圆曲线离散对数问题的数字签名系统使用的签名协议主要来自于签名等式的不同变形,通过对协议进行面向实现的优化可以使整个系统更加高效。在协议的实现过程中底层算法对系统的效率有着至关重要的影响。基于椭圆曲线的数字签名系统主要包括两个层次的底层运算:椭圆曲线上点的运算;有限域上元素的运算。对曲线上点的运算的优化主要是通过对标量乘算法和曲线上点的坐标系统的优化(减少求元素逆的操作)实现的,对有限域上元素运算的优化主要是通过使用类Mersenne素数模数优化求模操作,从而加快模乘和模平方操作。经过以上优化设计与实现的系统比以往实现的系统更加高效。

The digital signature system based on elliptic curve is one of the main stream digital signature systems and it has been regarded as the best replacer of RSA—— — a classical cryptosystem.The digital signature protocols based on elliptic curve discrete logarithm problem come mainly from the transmutation of signature equation and when the proto-col has been optimized for implementing,the whole system will get better performance.The low level algorithms have great influence on the performance of digital signature system.There are two kinds of main low level algorithm,one is the computation of points on elliptic curve,the other is the computation of finite field element.The better performance of computation of points on elliptic curve can be obtained through optimizing the algorithm of scale multiplication of points and using the mixed coordinates(reduce the operations of computing finite field element inverse).Using the Mersenne-like prime number as the modulus can optimize the modular arithmetics greatly and make the computation of modular multiplication and modular squaring more easily than before.The digital signature system that has been de-signed and implemented using all above optimize methods has better performance than the systems that has been imple-mented in the past.