Research on public-key cryptosystems from hyperelliptic-curves

我们从安全性和效率两个方面设计了超椭圆曲线密码体制，并在软件和硬件上实现了所设计的超椭圆曲线密码体制，研究了GF（p）上的超椭圆曲线密码体制与GF（2 ^n）上的超椭圆曲线密码体制的性能，分析了Jacobian群律的复杂性，并比较了GF（p）上的超椭圆曲线密码体制与GF（2 ^n）上的超椭圆曲线密码体制的性能。相对于GF（2^n）。考虑到单词大小的有效性，（32位或64位）的应用CPU（Alpha和Pentium）在定义域上的算法，我们还发展了有限域GF（p^n）上由方程y ^2 = x^p-x+1$定义的超椭圆曲线的雅可比矩阵的有效算法我们首先确定zeta函数的曲线产生的顺序雅可比。我们还研究了Frobenius算子，并利用它证明了对于n次与p互素的域扩张GF（p^n），雅可比矩阵具有循环群结构。此外，我们提出了一种方法，更快的标量乘法的雅可比矩阵，使用有效的运营商比Frobenius具有较小的特征值。

We have designed hyperelliptic curve cryptosystems with considering security and efficiency.We have implemented our designed hyperelliptic curve cryptosystems both over software and over hardware, and confirm their practical performance.We consider the performance of hyperelliptic curve cryptosystems over GF(p) vs. over GF(2^n).We analyze the complexity of the group law of Jacobians and make comparison of their performance between over over GF(p) vs. over GF(2^n). with considering the effectiveness of the word size (32-bit or 64-bit) of the applied CPU (Alpha and Pentium) on the arithmetic on the definition field.We also develop efficient algorithms for the jacobian of the hyperelliptic curve defined by the equation $y^2 = x^p-x+1$ over a finite field GF(p^n) of odd characteristic p. We first determine the zeta function of the curve which yields the order of the jacobian. And we investigate the Frobenius operator and use it to show that, for field extensions GF(p^n) of degree n prime to p, the jacobian has a cyclic group structure. We furthermore propose a method for faster scalar multiplication in the jacobian by using efficient operators other than the Frobenius that have smaller eigenvalues.