Algebraic function fields and cryptography

拟议的研究追求三个目标。前两个目标与代数函数域的算法有关，第三个目标是代数函数域算法在密码学中的实际应用。第一个研究是发展雅可比和理想类群具有良好结构的代数函数域的构造方法，并研究函数域类群的分布。第二个目标是确定与Drinfeld模曲线相关的Galois表示的图像。最后一项研究是为基于配对的密码学发展计算超椭圆曲线除数类群上的Tate配对的可行而有效的方法。由于信息安全是当今非常重要的问题之一，密码学受到了极大的关注。数论在密码学中产生了很好的应用。这项研究计划的资金将有助于确保加拿大在这些领域的研究继续发展和扩大，无论是在与重要应用相关的基础科学方面，还是在为学术界和工业界培养高素质人才方面。

The proposed research pursues three objectives. The first two objectives are related to the arithmetic of algebraic function fields, and the third one is a practical application of the arithmetic of algebraic function fields to Cryptography. The first proposed research is the development of constructive methods of algebraic function fields whose Jacobians and ideal class groups have nice structures, and investigation of the distribution of class groups of function fields. And the second objective is to determine the image of the Galois representation associated to the Drinfeld modular curves. The last proposed research is developing computationally feasible and efficient methods for computing the Tate-pairing over divisor class groups of hyperelliptic curves for pairing-based cryptography. Cryptography has been getting much attention since information security is one of the very significant issues nowadays. Number Theory has produced nice applications to Cryptography. Funding for this research program will help to ensure that Canada's research in these areas continues to develop and expand, both in terms of basic science related to important applications and the development of highly qualified personnel for academia and industry.