finite fields and their applications

有限域及其应用

• 批准号: 312588-2012
• 负责人: Wang, Qiang(Steven)
• 金额: $ 0.87万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571780
• 关键词: finite; fields; their; applications

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 understand the permutation behavior and arithmetic properties of these polynomials/functions 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 in constructing good codes, combinatorial designs, signal sets for wireless communications 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.

拟议的研究是在有限领域及其应用中。我最近的研究集中在对象/结构及其对有限领域的特性的理论研究，以及它们在数学，计算机科学和信息理论的其他分支上的应用。这些对象包括有限字段上的多项式/函数/序列，在编码理论，组合学，通信和密码学中具有大量应用。多项式的置换行为和算术特性，例如分解，划分性，不可减至性和原始性以及序列的伪随机性，是基本研究的核心主题。 的确，人们对诸如置换多项式（PPS），不可删除的多项式，原始多项式，特殊功能和反馈移位寄存器（FSR）序列等物体的进一步研究的需求越来越不断增加。例如，可靠的流密码的设计需要良好的伪随机序列。对线性/差分密码分析具有抗性的良好S盒（排列）的设计需要有用的特殊功能，例如几乎完美的非线性（APN）排列有限场上。因此，我的长期目标是两个方面：1） 了解这些多项式/功能在有限场上的置换行为和算术特性及其结构，分布和枚举； 2）更好地了解有限字段上不同对象和属性之间的相互作用，并在构建良好的代码，组合设计，用于无线通信的信号集和S框时找到真正的应用程序。我的科学方法不仅需要广泛的理论努力，还需要大规模的计算实验。这项追求涉及组合学，数字理论，代数，计算机科学和信息理论的知识结合。