环Z/(2^e-1)上二元导出非线性序列的分析

Recently, primitive sequences over Z/(2^e-1) are proposed to be used in stream cipher designs since they are shown to have many desirable cryptographical properties. In particular, primitive sequences over Z/(2^31-1) have been used in the ZUC algorithm, a new stream cipher accepted by 3GPP SA3 as a new inclusion in the LTE standards. However, on the other hand, theoretical results on primitive sequences over Z/(2^e-1) are incomplete and there still exist several important problems needed to be further explored. In this project, we study the cryptographic properties of nonlinear binary sequences derived from primitive sequences over Z/(2^e-1). The main contents of the project consist of three problems: the modulo 2 distinctness of primitive sequences generated by a primitive polynomial over Z/(2^e-1); the modulo 2 distinctness of primitive sequences generated by two distinct primitive polynomials over Z/(2^e-1); and the reconstruction of primitive sequences over Z/(2^e-1) from their modulo 2 reductions. Our aims are to provide more theoretical foundations and technologies for their further applications.

近年来，环Z/(2^e-1)上本原序列的提出及优良密码性质的发现，使得它们非常适合用作序列密码算法设计的驱动序列。特别地，环Z/(2^31-1)上的本原序列已经应用于3GPP LTE加密标准ZUC算法的设计中。但另一方面，当前环Z/(2^e-1)上本原序列密码性质的分析还很不完善，还存在多个重要的问题有待进一步研究。本申请项目研究环Z/(2^e-1)上本原序列导出的二元非线性序列的密码性质，主要内容有：环Z/(2^e-1)上本原序列的模2保熵性，环Z/(2^e-1)上不同本原多项式生成的本原序列的模2互异性，环Z/(2^e-1)上本原序列的还原问题。通过本项目的研究，可提高对环Z/(2^e-1)上本原序列密码性质的认识，丰富相关基础理论，并为它们的密码应用进一步提供理论支撑和应用指导。

整数剩余类环Z/(2^e-1)上的本原序列是一类具有重要密码研究价值和密码应用潜力的伪随机序列。特别地，环Z/(2^31-1)上的本原序列已经应用于3GPP LTE空中接口加密标准ZUC算法的设计中。本项目围绕环Z/(2^e-1)上本原序列导出的二元非线性序列的密码性质展开研究，取得的成果如下：（1）给出了环 Z/(p^e*q) 上本原序列是模2保熵的充分条件；（2）给出了形如 g(a_(e-1) )+η(a_0,a_1,… ,a_(e-2))的权位压缩导出序列是局部保熵的充分必要条件；（3）证明了环 Z/(m) 上一类本原序列是模H保熵的；（4）对e属于{4,8,16,32}，给出了环Z/(2^e-1)上由两个不同的本原多项式生成的本原序列是模2互异的充分条件；（5）利用格基约化算法给出了环Z/(m)上截位序列的还原算法；（6）给出了模m加法最低两个比特线性逼近概率的精确计算公式。这些成果提高了对环Z/(2^e-1)上本原序列密码性质的认识，丰富相关基础理论，并将为它们的密码应用进一步提供理论支撑和应用指导。此外，在完成本项目研究内容的同时，我们还对非线性反馈移位寄存器的子簇问题进行了研究，取得一些有价值的研究成果。

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