Computations in finite fields and probabilistic analysis of algorithms

我的研究领域是算法的设计和分析。我最近的研究集中在两条线：有限域中的算法及其在现代通信领域中的应用的研究，以及考虑随机性时的算法研究。 我的主要研究领域涉及有限域的计算及其应用。有限域算法在通信、密码学、编码理论和信息论等领域的无数实际应用中起着至关重要的作用。我在这方面的目标是设计有效的算法在有限领域，并使用元素的理论有限领域的工程应用。例子是使用多项式和正规元素的算术的有效实现，以及有限域上某些类型的多项式（不可约，原始等）的构造。这些研究的主要应用是密码学，其他领域包括代码和组合结构。 我的第二个研究方向是算法的概率分析，包括使用随机性时的算法研究。特别是，算法的平均情况分析侧重于理解算法在随机输入上的行为。该领域的一个基本目标是促进对随机性的理解。近期目标是更好地理解特定的算法，并在此基础上开发新的有效算法。科学方法是基于数学证明，涉及生成函数，用于计算算法分析和渐近分析的感兴趣的属性。我在这个领域的目标是双重的：开发算法的基本问题，如搜索比以前已知的算法更好的性能，并有助于在有限域上的随机多项式的数学理解。

The overall area of my research is the design and analysis of algorithms. My recent research has centered around two lines: the investigation of algorithms in finite fields and their applications in modern communication areas, and the study of algorithms when randomness is considered. A main area of my research involves computations in finite fields and their applications. Algorithms that work with finite fields play a crucial role in innumerous practical applications in communications, cryptography, coding theory and information theory. My goal in this area is the design of efficient algorithms in finite fields, and the usage of elements from the theory of finite fields in engineering applications. Examples are efficient implementation of arithmetic using polynomials and normal elements, and constructions of certain types of polynomials over finite fields (irreducible, primitive, among others). The main applications of these studies are in cryptography, with other areas including codes and combinatorial constructions. My second line of research is the probabilistic analysis of algorithms that comprises the study of algorithms when randomness is used. In particular, average-case analysis of algorithms focuses on understanding the behaviour of algorithms on random inputs. An essential goal of the area is to advance in the understanding of randomness. Immediate goals are better understanding of particular algorithms, and the development of new efficient algorithms based on this knowledge. The scientific approach is based on mathematical proofs involving generating functions for counting the properties of interest for the analysis of the algorithms, and asymptotic analysis. My goal in this area is two-fold: to develop algorithms for fundamental problems like searching with better performance than previously known algorithms, and to contribute to the mathematical understanding of random polynomials over finite fields.