Finite fields and applications in coding theory and cryptography

提出的研究领域是有限域及其在编码理论和密码学中的应用。我最近的研究集中在有限域上离散对象/结构及其性质的理论研究，以及它们在数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和序列，它们在编码理论、通信和密码学中有大量的应用。在离散数学、数论、理论计算机科学和信息论的交叉中，这是一个迷人而充满活力的研究领域。有限域上的许多公开问题和猜想都源于信息论中的有用问题。我的长期愿景是为这一研究领域做出重大而持久的贡献。多项式的组合性质(如排列和值集大小)、多项式的算术性质(如不可约性、原始性、可除性和因式分解)以及序列的伪随机性是基础研究的中心课题。例如，由于其在分组密码和流密码以及无线通信中的信号集中的应用，对诸如置换多项式、不可约多项式、本原多项式和反馈移位寄存器序列等对象的进一步研究的需求越来越大。事实上，抗线性/差分密码分析的好的S盒(置换)的设计需要有用的特殊函数，如几乎完美非线性置换；提高Reed-Solomon码列表译码算法的复杂性需要进一步研究具有给定范围的多项式；可靠的流密码的设计需要良好的伪随机序列；线性反馈移位寄存器(LFSR)序列的实现需要了解二进制域上具有一定低权(即3或5个非零系数)的本原多项式的存在性。*因此，我的长期目标有两个：1)更好地了解有限域上这些基本对象的组合和算术性质，以及它们的构造、分布和计数；2)更好地了解有限域上不同对象和性质之间的相互作用，并找到真正的应用，如构造好的码和S盒。我的科学方法不仅需要广泛的理论努力，还需要大量的计算实验。这一探索涉及到组合学、数论、代数、计算机科学和信息论的综合知识。对其中一些问题的积极解决方案不仅会对研究界产生重大影响，而且还会带来直接的技术进步。

The proposed research area is finite fields and applications in coding theory and cryptography. My recent research has centered on the theoretical study of discrete objects/structures and their properties over finite fields, as well as on their applications to other branches of mathematics and information theory. These objects include polynomials and sequences over finite fields, which have a large number of applications in coding theory, communications and cryptography. This is a fascinating and vibrant area of research in the intersection of discrete math, number theory, theoretical computer science and information theory. Many open problems and conjectures over finite fields arise from useful problems in information theory. It is my long term vision to play a significant and lasting contribution to this area of research. *********The combinatorial properties of polynomials such as permutations and value set sizes, arithmetic properties of polynomials such as irreducibility, primitivity, divisibility and factorization, as well as the pseudo-randomness of sequences, are central topics of fundamental research. For example, there has been an increasing demand for further studies of objects such as permutation polynomials, irreducible polynomials, primitive polynomials, and feedback shift register sequences due to their applications in block ciphers and stream ciphers, as well as signal sets in wireless communications. Indeed, 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; improving the complexity of list decoding algorithm for Reed-Solomon codes requires the further study of polynomials with prescribed ranges; the design of reliable stream ciphers requires good pseudo-random sequences; the implementation of linear feedback shift register (LFSR) sequences requires the understanding of existence of primitive polynomials with certain low weight (i.e., 3 or 5 nonzero coefficients) over the binary field. ******My long term goal is thus two-fold: 1) to better understand the combinatorial and arithmetic properties of these fundamental objects over finite fields, and their construction, distribution and enumeration; 2) to better understand the interplay among different objects and properties over finite fields and find genuine applications such as constructing good codes and S-boxes. My scientific approach requires not only extensively theoretical efforts, but also massive computational experiments. This quest involves a combination of knowledge from combinatorics, number theory, algebra, computer science, and information theory. Positive solutions to some of these problems would not only have significant impact on the research community but also have direct technology advance.