CIF: Small: RUI: Low Correlation and Highly Nonlinear Structures for Communications and Sensing

CIF：小型：RUI：用于通信和传感的低相关性和高度非线性结构

• 批准号: 1815487
• 负责人: Daniel Katz
• 金额: $ 33.09万
• 依托单位: The University Corporation, Northridge
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-10-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1815487&HistoricalAwards=false
• 关键词: CIF; Small; RUI; Low; Correlation

Many communications and remote sensing systems require modulation protocols that are described by digital sequences, which may be regarded as words composed of symbols from a prescribed alphabet, such as the binary alphabet with symbols 0 and 1. The efficiency of the system will often depend on producing sequences that are as uncorrelated as possible: they should not resemble shifted (time-delayed) versions of each other, nor even of themselves. Lack of resemblance between a sequence and shifted versions of itself aids in synchronization and timing, which is useful in radar and sonar. Lack of resemblance between two different sequences (no matter how they are shifted) prevents confusion between different users in communications networks. Random sequences are not ideal for these applications, as even random sequences are expected to have occasional repetitions. It is more advantageous to use pseudorandom sequences that avoid repetition to a greater degree than random sequences do. These pseudorandom sequences and related mathematical structures, such as Boolean functions, are also significant in other information-theoretic problems, such as in cryptography, where one seeks to design permutations that have a simple underlying mathematical form (to ease encryption and decryption) but avoid resembling easily detectable patterns (to resist cryptanalysis). Pseudorandom sequences find further applications in error-correcting codes, antenna arrays, scientific instrumentation, and acoustic design, and thus science and technology benefit both from the analysis of known digital sequences and the discovery of new ones.The goal of this project is to create and investigate digital sequences and related mathematical structures with good correlation properties. This project considers both periodic and aperiodic forms of correlation, as both are important in applications. In periodic correlation, the shifting of the sequences is cyclic, and the maximum length linear feedback shift register sequences (m-sequences) are a common building block in the design of digital sequences with low periodic correlation. Finding pairs of m-sequences with low mutual correlation is equivalent to finding highly nonlinear permutations of finite fields, which can be used to make cryptosystems resilient to linear cryptanalysis. This project will investigate m-sequence pairs with exceptional correlation properties, which translate into exceptional nonlinearity properties of the corresponding permutations. Extremal properties, such as exceptionally high nonlinearity or exceptionally few correlation values, are sought out, and this project will investigate bounds and limitations on these extremes using tools from abstract algebra, combinatorics, and number theory, as well as empirical computational explorations. In aperiodic correlation, the shifting of sequences is a non-cyclic translation, and various families of sequences whose correlation properties make them superior to random sequences are known, but their analysis has been difficult and many open questions remain. This project will analyze the performance of these sequences both empirically and theoretically, and will seek new families of sequences with good correlation propertiesThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

许多通信和遥感系统都需要通过数字序列描述的调制协议，这些协议可以被视为由处方字母的符号组成的词，例如具有符号0和1的二进制字母。系统的效率通常会取决于产生不正确的序列：它们不应同时换句话说：它们本身不相处（时间段）。序列和移动版本之间缺乏相似之处，有助于同步和时机，这在雷达和声纳中很有用。两个不同的序列之间缺乏相似之处（无论它们如何转移）都阻止了通信网络中不同用户之间的混淆。随机序列对于这些应用而言并不理想，因为甚至随机序列也有望偶尔重复。使用伪随机序列比随机序列避免重复更大的序列更有利。这些伪随机序列和相关的数学结构（例如布尔函数）在其他信息理论问题（例如在加密术中）也很重要，在该问题中，人们试图设计具有简单基础数学形式的排列（以避免加密和解密），但避免避免易于检测的模式（以抗抗药性）。伪随机序列在错误校正的代码，天线阵列，科学仪器和声学设计中找到了进一步的应用，因此，科学和技术也从对已知数字序列的分析和发现新的序列的分析中受益。该项目的目标是创建和研究具有良好率属性的数字序列以及相关的数学序列以及相关的数学结构。该项目考虑了周期性和周期性形式的相关性，因为两者在应用中都很重要。在周期性相关性中，序列的转移是循环的，最大长度线性反馈移位寄存器序列（M序列）是具有低周期相关性的数字序列设计中的常见构建块。找到与低相关性低相关的M序列对等同于找到有限场的高度非线性排列，这些磁场可用于使加密系统具有弹性对线性隐式分析的弹性。该项目将研究具有特殊相关属性的M序列对，这些属性转化为相应排列的非线性特性。寻找极端属性，例如异常高的非线性或异常相关值，并且该项目将使用抽象代数，组合学和数量理论以及经验计算探索的工具研究这些极端的界限和局限性。在基质相关性中，序列的转移是一种非环状翻译，其相关性能使它们优于随机序列的各种序列家族是已知的，但是它们的分析很困难，并且仍然存在许多空旷的问题。该项目将从经验和理论上分析这些序列的性能，并将寻求具有良好相关性Propertiesthis奖的新序列家庭反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响来通过评估来获得支持的。

项目成果

Sequences with Low Correlation

• DOI: 10.1007/978-3-030-05153-2_8
• 发表时间: 2018-06
• 期刊: ArXiv
• 作者: D. Katz

Peak Sidelobe Level and Peak Crosscorrelation of Golay–Rudin–Shapiro Sequences

Golay-Rudin-Shapiro 序列的峰值旁瓣电平和峰值互相关

• DOI: 10.1109/tit.2021.3135564
• 发表时间: 2021
• 期刊: IEEE Transactions on Information Theory
• 影响因子: 2.5
• 作者: Katz, Daniel J.;Van der Linden, Courtney M.
• 通讯作者: Van der Linden, Courtney M.

An improved uncertainty principle for functions with symmetry

对称函数的改进不确定性原理

• DOI: 10.1016/j.jalgebra.2021.07.017
• 发表时间: 2021
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Garcia, Stephan Ramon;Karaali, Gizem;Katz, Daniel J.
• 通讯作者: Katz, Daniel J.

Rudin-Shapiro-Like Sequences With Maximum Asymptotic Merit Factor

具有最大渐进优值因子的类 Rudin-Shapiro 序列

• DOI: 10.1109/tit.2020.3011853
• 发表时间: 2020
• 期刊: IEEE Transactions on Information Theory
• 影响因子: 2.5
• 作者: Katz, Daniel J.;Lee, Sangman;Trunov, Stanislav A.
• 通讯作者: Trunov, Stanislav A.

Sequence Pairs with Lowest Combined Autocorrelation and Crosscorrelation

具有最低组合自相关和互相关的序列对

• DOI: 10.1109/tit.2022.3187923
• 发表时间: 2022
• 期刊: IEEE Transactions on Information Theory
• 影响因子: 2.5
• 作者: Katz, Daniel J.;Moore, Eli
• 通讯作者: Moore, Eli