流密码系统中若干问题的进位模拟研究

在流密码体制中，非线性序列生成器的设计及输出序列的密码性质是当前序列密码体制研究的重点。本课题综合利用代数学、数论、有限域、格理论、N-adic数理论等数学工具，对序列密码系统中基于多项式现象的进位模拟的若干问题进行研究。具体包括：确定二元l序列紧的错误复杂度的下界，研究最长多元进位反馈移位寄存器(FCSR)序列的线性复杂度、稳定性和相关函数等性质；探究研究二元d-FCSR序列的伪随机性质的新工具；确定包括legendre序列在内的多种二元序列的2-adic复杂度，研究非二元序列的算数相关性的统计特性。探索研究布尔函数算数模拟性质的工具，并确定具有特定制约关系的布尔函数的算数Walsh-Hadamard变换以及算术相关性之间的关系，进一步完善算数布尔函数的进位模拟的这一全新的理论体系，深化和拓展流密码体系中相关问题的研究。

The design and analysis of nonlinear sequences generators and the cryptograophic properties of its output sequences are the key point in the research work of stream cipher system. We combine the theory of algebraic, number theory, finite fields, N-adic number theory and lattice theory together to study the "with carry"analogs to certain problems in the stream cipher system. More precisely, we determine a tight lower bound for the binary sequence with maximum-period generated by feedback with carry shift register (FCSR); study the linear complexity, stability and correlation properties of the N-ary l sequences; explore the new tools to analysis the pseudo-random properties of binary d-FCSR sequences. We estimate the the 2-adic complexity of the sequences such as Legendre sequences and the expected arithmetic cross-correlation of fixed sequences and the expected arithmetic autocorrelation of a fixed sequence in the non-binary cases. Moreover, we will study the relationship of the arithmetic Walsh-hadamard transform and the arithmetic correlation of the boolean functions with certain relationship. Our research work will improve the new theory of arithmetic analog of Boolean function, deepen and expand the research on the related problem in the stream cipher system.