几类数论函数的密码学应用研究

项目名称： 几类数论函数的密码学应用研究

项目编号： No.61462077

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 计算机科学学科

项目作者： 杜小妮

作者单位： 西北师范大学

项目金额： 46万元

中文摘要： 费马商的工程应用是2011年新兴的研究方向。本课题拟针对费马商及其扩展、Legendre符号等数论函数的性质及其在密码学和通信领域的应用开展研究。主要包括：对已有的费马商及其扩展函数序列的迹函数表示、线性复杂度和稳定性等重要的密码学指标进行进一步研究，并将现有的二元序列研究成果推广到多元情形；将Legendre符号等数论函数与费马商及其扩展函数相融合构造新商式及新序列，研究其线性复杂度、相关性等伪随机性指标，并对部分结果进行编程论证；基于费马商及其扩展函数构造具有优的汉明相关值的跳频序列，结合新的序列构造技术探讨其他构造跳频序列的一些有效的工具和方法，设计达到或者接近相关值理论界的序列集。通过以上研究将为费马商及其他数论函数在密码学以及通信领域的应用提供理论基础，进一步拓广和深化伪随机序列基本理论。

中文关键词： 伪随机序列；流密码；线性复杂度；跳频序列

英文摘要： The applications of Fermat quotient to engineering is a brand new research direction proposed in 2011. In this project, we will focus on the study of some number theory functions such as Fermat quotient and its extensions, Legendre symbol, etc., and their applications in the field of cryptography and communication. Our research includes: (1) Study the trace function representation, linear complexity and stability of some existing sequences constructed by Fermat quotient and its extensions, extend the research of binary case to the k-ary case. (2) Combine Legendre symbol and other number theory functions with Fermat quotient and its extensions to construct new extension function sequence， then study their pseudo-random measures such as linear complexity, stability and correlations etc.. Moreover, partial results will be demonstrated by programming. (3) Based on Fermat quotient and their extensions, we design some class frequency hopping sequences set with optimal Hamming correlation values. With the comprehensive utilization of sequences design techniques, we will explore some effective tools and methods to construct frequency hopping sequences set meet or close to Hamming correlation value theoretical bounds. Our study will provide a theoretical basis for the applications of Fermat quotient and their extensions in cryptography and communications, and further improve the basic theory of pseudo-random sequence.

英文关键词： psuedorandom sequences;stream cipher;linear complexity;frequency-hopping sequences