Fast implementation and security analysis of hyperelliptic curve cryptosystems

1.在基于代数曲线的密码体制中，只有基因ra小于等于3的超椭圆曲线是安全的。在本研究中，我们首先发展出亏格为三的超椭圆曲线的快速演算法。基于这些曲线的密码系统在64位的廉价处理器上实现，具有单一判决，因此比椭圆曲线密码系统更有效的密码系统是可能的。特别地，获得了具有最小计算代价的快速加法算法。这些算法的实现实现了173微秒快速标量乘法的新记录。2.在安全性分析方面，我们第一次证明了大量被认为是安全的但可以被GHS攻击破坏的椭圆曲线的存在性。特别是，我们明确表明类的椭圆和超椭圆曲线的低属定义的扩展领域，有弱覆盖，即他们的井限制可以攻击的指数演算攻击超椭圆曲线或迪姆最近的攻击非超椭圆曲线。得到了此类弱曲线的一个完整分类。此外，我们展示了如何从这些曲线构造这样的覆盖，并分析了这些弱曲线的密度。

1. It is known that among the algebraic curve based cryptosystems, only hyperelliptic curves of gene ra less or equal to three are secure. In this research, we first developed fast algorithms for hyper elliptic curves of genus three. Cryptosystems based on these curves are implemented on cheap processors of 64 bits with single decision, thus more efficient cryptosystems than elliptic curve crypt osystems are possible. In particular, fast addition algorithms with the least computational cost are obtained. These algorithms are implemented to achieve a new record of fast scalar multiplication with173 microseconds.2. As to security analysis, we show for the first time the existence of a huge number of elliptic curves which are believed to be secure but can be broken by GHS attack. In particular, we show explicitly classes of elliptic and hyperelliptic curves of low genera defined over extension fields, which have weak coverings, i.e. their Well restrictions can be attacked by either index calculus attacks to hyperelliptic curves or Diem's recent attack to non-hyperelliptic curves. A complete classification of such weak curves is obtained. Besides, we show how to construct such coverings from these curves and analyze density of these weak curves.