Research on public-key cryptosystems from hyperelliptic-curves

本文综合考虑安全性和效率，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，验证了它们的实用性能。我们考虑了环GF(P)上的超椭圆曲线密码体制与环GF(2^n)上的超椭圆曲线密码体制的性能，分析了雅可比群律的复杂性，并比较了环GF(P)上和环GF(2^n)上的性能。考虑到应用处理器(Alpha和Pentium)的字长(32位或位)在定义域上的算术有效性，我们还给出了由方程$y^2=x^p-x+1$定义的超椭圆曲线在具有奇特征p的有限域GF(p^n)上雅可比的有效算法。我们首先确定了产生雅可比阶数的曲线的Zeta函数。我们研究了Frobenius算子，并利用它证明了对于n次素数到p的域扩张GF(p^n)，雅可比具有循环群结构。此外，我们还提出了一种通过使用除Frobenius以外的具有较小本征值的有效算子来加快雅可比中标量乘法的方法。

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.