Research on public-key cryptosystems from hyperelliptic-curves

我们设计了兼顾安全性和效率的超椭圆曲线密码系统。我们已经在软件和硬件上实现了我们设计的超椭圆曲线密码系统，并验证了它们的实际性能。研究了GF(p)和GF（2^n）上超椭圆曲线密码系统的性能。我们分析了雅可比矩阵群律的复杂性，并比较了它们在over / GF(p)和over GF（2^n）之间的性能。考虑到应用CPU （Alpha和Pentium）的字长（32位或64位）对定义字段的算术的有效性。我们还开发了求解超椭圆曲线雅可比矩阵的有效算法，该雅可比矩阵由方程$y^2 = x^p-x+1$在奇数特征p的有限域GF（p^n）上定义。我们首先确定曲线的zeta函数，从而得到雅可比矩阵的阶数。我们研究了Frobenius算子，并利用它证明了对于n '到p次的域扩展GF(p^n)，雅可比矩阵具有循环群结构。我们进一步提出了一种在雅可比矩阵中使用比具有较小特征值的Frobenius算子更有效的标量乘法的方法。

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.