Further results on the distinctness of modulo 2 reductions of primitive sequences over \(\mathbf{Z}/(2^{32}-1)\)

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• 作者：Dong Yang (1)
  Wen-Feng Qi (1)
  Qun-Xiong Zheng (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：2015
• 出版时间：February 2015
• 年：2015
• 卷：74
• 期：2
• 页码：467-480
• 全文大小：197 KB
• 参考文献：1. ETSI/SAGE Specification: Specification of the 3GPP confidentiality and integrity algorithms 128-EEA3 & 128-EIA3. Document 4: Design and Evaluation Report, Version: 2.0. Technicl Report, ETSI 2011. http://www.gsmworld.com/our-work/programmes-and-initiatives/fraud-and-security/gsm_security_algorithms.htm (2011). Accessed 9 Sept 2011.
  2. Zhu X.Y., Qi W.F.: On the distinctness of modular reduction of maximal length modulo odd prime numbers. Math. Comput. 77(7), 1623鈥?637 (2008).
  3. Zheng Q.X., Qi W.F., Tian T.: On the distinctness of modular reduction of primitive sequences over \({ Z}/(2^{32}-1)\) . Des. Codes Cryptogr. 112(22), 872鈥?75 (2012). doi:10.1007/s10623-012-9698-y .
  4. Lidl R., Niedereiter H.: Finite Field. Addison-Wesley, Don Mills (1983).
  5. Hall M.: An isomorphism between linear recurring sequences and algebraic rings. Trans. Am. Math. Soc. 44, 196鈥?18 (1938).
  6. Rueppel R.A.: Analysis and Design of Stream Ciphers. Springer, New York (1986).
  7. Tian T., Qi W.F.: Typical primitive polynomials over integer residue rings. Finite Fields Appl. 15, 796鈥?07 (2009).
  8. Wang Y.K.: Residue-to-binary converters based on Chinese remainder theorems. IEEE Trans. Circut. Syst. II 40(3), 197鈥?05 (2000).
  9. Bugeaud Y., Corvaja P., Zannier U.: An upper bound for the G.C.D. of \(a^{n}-1\) and \(b^{n}-1\) . Math. Z. 243, 79鈥?4 (2003).
  
• 作者单位：Dong Yang (1)
  Wen-Feng Qi (1)
  Qun-Xiong Zheng (1)
  
  1. State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhengzhou Information Science and Technology Institute, Zhengzhou, People鈥檚 Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

Recently, primitive sequences over \(\mathbf{Z}/(2^{32}-1)\) are shown to have many desirable properties, which makes them of potential interest for cryptographic applications. To further support the applications of this kind of sequences, in this paper, we consider the problem whether primitive sequences generated by two distinct primitive polynomials over \(\mathbf{Z} /(2^{32}-1)\) are pairwise distinct modulo 2. A sufficient condition is given for ensuring that the answer to this problem is positive.