Computations in finite fields and probabilistic analysis of algorithms

我的整个研究领域是算法的设计和分析。我最近的研究集中在两个方面：有限域中的算法及其在现代通信领域中的应用的研究，以及考虑随机性的算法的研究。我研究的一个主要领域涉及有限域中的计算及其应用。处理有限域的算法在通信、密码学、编码理论和信息论的无数实际应用中起着至关重要的作用。我在这个领域的目标是设计有限域上的有效算法，并将有限域理论中的元素用于工程应用。例如，使用多项式和正规元素有效地实现算术，以及在有限域上构造某些类型的多项式(不可约、本原等)。这些研究的主要应用是密码学，其他领域包括编码和组合构造。我的第二条研究是算法的概率分析，这包括使用随机性时的算法研究。特别是，算法的平均案例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推进对随机性的理解。眼前的目标是更好地理解特定的算法，并在此基础上开发新的高效算法。科学的方法是基于数学证明，涉及用于计算算法分析的感兴趣属性的生成函数和渐近分析。我在这个领域的目标有两个：为搜索等基本问题开发比以前已知的算法更好的算法，以及对有限域上随机多项式的数学理解做出贡献。

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.