Computations in finite fields and probabilistic analysis of algorithms

我研究的总体领域是算法的设计和分析。我最近的研究集中在两条线上：有限域的算法研究及其在现代通信领域的应用，以及考虑随机性时的算法研究。我研究的一个主要领域涉及有限域的计算及其应用。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在有限域中设计有效的算法，并在工程应用中使用有限域理论中的元素。例如使用多项式和正规元素的有效算术实现，以及有限域（不可约的，原始的，等等）上某些类型多项式的构造。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提高对随机性的理解。当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。科学的方法是建立在数学证明的基础上的，包括生成函数来计算算法分析中感兴趣的性质，以及渐近分析。我在这个领域的目标是双重的：开发算法来解决基本问题，比如搜索，比以前已知的算法性能更好，并有助于对有限域上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.