finite fields and their applications

本论文的研究方向是有限域及其应用。我最近的研究集中在对象/结构及其在有限域上的属性的理论研究，以及它们在数学，计算机科学和信息论的其他分支中的应用。这些对象包括有限域上的多项式/函数/序列，它们在编码理论、组合学、通信和密码学中有大量的应用。多项式的置换行为和算术性质，如因式分解、可除性、不可约性和不可约性，以及序列的伪随机性，是基础研究的中心课题。实际上，由于诸如置换多项式（PP）、不可约多项式、本原多项式、特殊函数和反馈移位寄存器（FSR）序列在分组密码和流密码以及无线通信中的信号集合中的应用，对它们的进一步研究的需求一直在增加。例如，可靠的流密码的设计需要良好的伪随机序列;抵抗线性/差分密码分析的良好S盒（排列）的设计需要有用的特殊函数，例如有限域上的几乎完美非线性（APN）排列。因此，我的长期目标是双重的：1）更好地

The proposed research is in finite fields and their applications. My recent research has centered on the theoretical study of objects/structures and their properties over finite fields, as well as on their applications to other branches of mathematics, computer science, and information theory. These objects include polynomials/functions/sequences over finite fields, which have a large number of applications in coding theory, combinatorics, communications and cryptography. The permutation behavior and arithmetic properties of polynomials such as factorization, divisibility, irreducibility and primitivity, as well as the pseudo-randomness of sequences, are central topics of fundamental research. Indeed, there has been an increasing demand for further studies of objects such as permutation polynomials (PPs), irreducible polynomials, primitive polynomials, special functions, and feedback shift register (FSR) sequences due to their applications in block ciphers and stream ciphers, as well as signal sets in wireless communications. For example, the design of reliable stream ciphers requires good pseudo-random sequences; the design of good S-boxes (permutations) which are resistant against linear/differential cryptanalysis requires useful special functions such as almost perfect nonlinear (APN) permutations over finite fields. My long term goal is thus two-fold: 1) to better