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。首先提出的研究是发展代数函数域的构造方法，其Jacobian和理想类群具有良好的结构，并调查函数域的类群分布。第二个目标是确定与Drinfeld模曲线相关联的Galois表示的图像。最后提出的研究是开发计算上可行的和有效的方法来计算基于配对的密码学的超椭圆曲线的除数类组上的Tate配对。信息安全是当今社会的重要问题之一，密码学一直受到人们的广泛关注。数论在密码学中有很好的应用。对这一研究方案的资助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有关的基础科学方面，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.