On the distinctness of modular reductions of primitive sequences over Z/(232?)

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• 作者：Qun-Xiong Zheng (1)
  Wen-Feng Qi (1)
  Tian Tian (1)
  
• 关键词：Stream ciphers ; Integer residue rings ; Linear recurring sequences ; Primitive sequences ; Modular reductions ; 11B50 ; 94A55 ; 94A60
• 刊名：Designs, Codes and Cryptography
• 出版年：2014
• 出版时间：March 2014
• 年：2014
• 卷：70
• 期：3
• 页码：359-368
• 全文大小：169 KB
• 作者单位：Qun-Xiong Zheng (1)
  Wen-Feng Qi (1)
  Tian Tian (1)
  
  1. Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, People’s Republic of China
  
• ISSN：1573-7586

文摘

This paper studies the distinctness of modular reductions of primitive sequences over ${\mathbf{Z}/(2^{32}-1)}$ . Let f(x) be a primitive polynomial of degree n over ${\mathbf{Z}/(2^{32}-1)}$ and H a positive integer with a prime factor coprime with 232?. Under the assumption that every element in ${\mathbf{Z}/(2^{32}-1)}$ occurs in a primitive sequence of order n over ${\mathbf{Z}/(2^{32}-1)}$ , it is proved that for two primitive sequences ${\underline{a}=(a(t))_{t\geq 0}}$ and ${\underline{b}=(b(t))_{t\geq 0}}$ generated by f(x) over ${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$ if and only if ${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$ for all t??0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.