CAREER: Finite-Field Wavelets for Cryptography and Error Control Coding

有限域小波用于密码学和差错控制编码Faramarz Fekri今天的成功越来越依赖于能够随时随地根据需要访问、共享和使用信息。这种信息的广泛可获得性和传输要求采用新的密码学和差错控制编码方法。复杂性、可伸缩性和适应性的概念正在成为差错控制编码和数据安全的关键挑战。本研究探讨了有限域上的小波变换及其在卷积编码和数据安全中的应用。它研究了一套丰富的信号处理技术，可用于构建新的编码和安全方案。这项研究不是循序渐进的。它在三个非常大的研究领域的边界上定义了一个新的研究领域，即数字信号处理、通信和计算机科学。它影响着1.基础科学，2.技术和产品，以及3.教育和学习。本研究探讨了有限域小波变换、差错控制编码和数据加密的交集问题。本研究的主要重点是推进有限域上的小波和滤波器组的研究及其在差错控制编码和数据安全中的应用。它发展了有限域上的小波变换理论，提供了定义在有限域上的序列的一般小波分解。对于实值信号的表示，这是一种在信号处理方面有着丰富历史的方法，但在有限域情况下一直缺乏这种方法。特别是，研究了有限域上的多分辨小波和超完备滤波器组。与有限域上的小波理论一起，本文首次研究了有限域上的小波理论在新型时变卷积码中的应用，这些卷积码具有特殊的格状结构，降低了复杂度。利用小波的多分辨率分解，研究人员打算为手持设备构造速率兼容的小波卷积码，以支持灵活的数据速率(语音、传真、视频等)。为了解决手持设备的安全问题，本研究利用有限域小波作为有效联合设计数据加密和差错控制编码的统一框架。在该框架中，用户的公钥和私钥决定了小波系统，安全性与小波基函数的长度有关。这项研究的目标是围绕手持设备的创新编码/安全协议的开发。

Finite-Field Wavelets for Cryptography and Error-Control CodingFaramarz Fekri Today's success is increasingly dependent on being able to access, share and use information whenever and wherever needed. The widespread availability and transmission of such information demands new approaches to cryptography and error-control coding. Notions of complexity, scalability, and adaptability are becoming critical challenges for error-control coding and data security. This research explores wavelet transform over finite fields and their applications to convolutional coding and data security. It investigates a rich set of signal processing techniques that can be exploited for the construction of new coding and security schemes. The research is not evolutionary. It defines a new research area on the boundary of three very large research fields; namely digital signal processing, communications, and computer science. It impacts 1. basic science, 2. technology and products, and 3. education and learning. This research explores the intersection of finite-field wavelet transforms, error-control coding and data encryption. The primary focus of this research is to advance the study of wavelets and filter banks over finite fields and their applications to error -control coding and data security. It develops the theory of wavelet transforms over finite fields which provides a general wavelet decomposition of sequences defined over finite fields. This is an approach that has a rich history in signal processing for the representation of real-valued signals, but it has been lacking in the finite-field case. In particular, the research explores multiresolution wavelets and overcomplete filter banks over finite fields. Along with wavelet theory on finite fields, this work investigates the first application of the finite-field wavelet theory to new types of time varying convolutional codes that have unusual trellis structures with reduced complexity. Using multiresolution decomposition of wavelets, the researchers intend to construct rate-compatible wavelet convolutional codes for handheld devices to support flexible data rates (voice, fax, video,...). To address security issues with handheld devices, this research exploits the finite-field wavelet as a unifying framework for effective joint design of data encryption and error control coding. In this framework, the public and secret keys of the user determine the wavelet system and the security is tied to the length of the wavelet basis function. The goal of the research is centered around the development of innovative coding/security protocols for handheld devices.