代数几何和组合方法在Hash函数族构造中的应用

Hash函数族的构造是编码理论与密码学中的主要研究课题之一.应用有限域上的代数曲线和组合方法是构造Hash函数族的重要方法.明确构造具有优良渐近行为的Hash函数族具有很重要的理论价值,许多问题有待研究解决.本项目的重点是应用有限域上的代数曲线与组合方法相结合,研究Hash函数族的构造问题,特别是研究具有优良渐近行为的分离Hash函数族的构造问题.D.R.Stinson和G.Zaverucha提出公开问题：当t≥3时，是否可以确定性构造出具有优良渐近行为的{w1,w2,...,wt}-分离Hash函数族？本项目将探索解答此公开问题。同时,本项目还研究应用组合方法和技巧给出的Hash函数族的新的递归构造方法；研究t≥3且w1+w2+...+wt≥4时Hash函数族存在性并设计构造Hash函数族的组合算法，分析算法的有效性；研究Hash函数族在编码理论与密码学中的应用。

Hash families are of great importance in coding theory and cryptography. Algebraic curves over finite fields accompanying combinatorial methods, have many significant applications on constructions of Hash families. Explicit constructions of Hash families asymptotically good are of great value in practice. There are many efforts on the topic, moreover, there are still many open problems.The project is devoted to the constructions of Hash families applying the algebraic curves over finite fields. Especially, we plan to present an infinite class of explicitly constructed Hash families with good asymptotical behavior.We are going to study on the open problem given by D.R.Stinson and G.Zaverucha, which says that when t≥3, whether there exist explicit constructions of {w1,w2,...,wt}-separating Hash families asymptotically well；meanwhile, we study recursive construction of Hash families derived from combinatorial methods and techniques，find lower and upper bounds on separating Hash families when t≥3 and w1+w2+...+wt≥4. The combinatorial algorithms for the construction of Hash families also would be designed and analyzed. Furthermore, we contribute ourselves to applying Hash families to the coding theory and cryptography in order to design more useful codes and schemes.