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Injective maps on primitive sequences over Z/(p e)

  • Sun Zhonghua 
  • Qi Wenfeng 
Article

Abstract

Let Z/(p e) be the integer residue ring modulo p e with p an odd prime and integer e ≥ 3. For a sequence \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \) over Z/(p e), there is a unique p-adic decomposition \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _0 + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _1 \cdot p + \cdots + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _{e - 1} \cdot p^{e - 1} \), where each \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _i \) can be regarded as a sequence over Z/(p), 0 ≤ ie − 1. Let f(x) be a primitive polynomial over Z/(p e) and G′(f(x), p e) the set of all primitive sequences generated by f(x) over Z/(p e). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and ged(1 + deg(μ(x)), p − 1) = 1, set
$$\varphi _{e - 1} (x_0 ,x_1 , \cdots ,x_{e - 1} ) = x_{e - 1} \cdot \left[ {\mu (x_{e - 2} ) + \eta _{e - 3} (x_0 ,x_1 , \cdots ,x_{e - 3} )} \right] + \eta _{e - 2} (x_0 ,x_1 , \cdots ,x_{e - 2} ),$$
which is a function of e variables over Z/(p). Then the compressing map
$$\varphi _{e - 1} :G'(f(x),p^e ) \to (Z/(p))^\infty ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \mapsto \varphi _{e - 1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _0 ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _1 , \cdots \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _{e - 1} )$$
is injective. That is, for \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} \in G'(f(x),p^e ),\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} \) and only if \(\varphi _{e - 1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _0 ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _1 , \cdots \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _{e - 1} ) = \varphi _{e - 1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} _0 ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} _1 , \cdots ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} _{e - 1} )\). As for the case of e = 2, similar result is also given. Furthermore, if functions ε e − 1 and ψ e − 1 over Z/(p) are both of the above form and satisfy
$$\varphi _{e - 1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _0 ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _1 , \cdots \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} _{e - 1} ) = \psi _{e - 1} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} _0 ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} _1 , \cdots ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} _{e - 1} )$$
for \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} \in G'(f(x),p^e )\), the relations between \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \) and \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} \), ε e − 1 and ψ e − 1 are discussed.

Keywords

integer residue ring linear recurring sequence primitive sequence injective map 

MR Subject Classification

11B50 94A55 

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References

  1. 1.
    Dai Z D. Binary sequences derived from ML-sequences over rings I: periods and minimal polynomials, J Cryptology, 1992, 5(4): 193–207.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Huang M Q, Dai Z D. Projective maps of linear recurring sequences with maximal p-adic periods, Fibonacci Quart, 1992, 30(2): 139–143.zbMATHMathSciNetGoogle Scholar
  3. 3.
    Kuzmin A S, Nechaev A A. Linear recurring sequences over Galois ring, Russian Mathematical Surveys, 1993, 48(1): 171–172.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Huang M Q. Analysis and cryptological evaluation of primitive sequences over an integer residue ring, Ph.D. dissertation, Beijing: Graduate School of USTC, 1988.Google Scholar
  5. 5.
    Qi W F, Zhou J J. Classes of injective maps on primitive sequences over Z/(2d), Prog Nat Sci, 1999, 9(3): 209–215.Google Scholar
  6. 6.
    Qi W F. Compressing maps of primitive sequences over Z/(2e) and analysis of their derived sequences, Ph.D. dissertation, Zhengzhou: Zhengzhou Inform Eng Univ, China, 1997.Google Scholar
  7. 7.
    Qi W F, Yang J H, Zhou J J. ML-sequences over rings Z/(2e), In: Advances in Cryptology-ASIACRYPT’98, Lecture Notes in Computer Science 1514. Berlin, Heidelberg: Springer-Verlag, 1998, 315–325.CrossRefGoogle Scholar
  8. 8.
    Zhu X Y, Qi W F. Compressing mappings on primitive sequences over Z/(2e) and its Galois extension, Finite Fields and Their Applications, 2002, 8(4): 570–588.zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dai Z D, Beth T, Gollman D. Lower bounds for the linear complexity of sequences over residue ring, In: Advances in Cryptology-EUROCRYPT’90, Lecture Notes in Computer Science 473, Berlin, Heidelberg: Springer, 1991, 189–195.Google Scholar
  10. 10.
    Kuzmin A S. Lower estimates for the ranks of coordinate sequences of linear recurrent sequences over primary residue rings of integers, Russian Mathematical Surveys, 1993, 48(3): 203–204.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Qi W F, Zhou J J. The distribution of 0 and 1 in the highest level of primitive sequences over Z/(2e), Science in China Ser A, 1997, 27(4): 311–316.Google Scholar
  12. 12.
    Qi W F, Zhou J J. The distribution of 0 and 1 in the highest level of primitive sequences over Z/(2e) (II), Chinese Science Bulletin, 1997, 42(18): 1938–1940.MathSciNetGoogle Scholar
  13. 13.
    Fan S Q, Han W B. The distribution of 0 and 1 in the highest level of primitive sequences over Z/(2e), Science in China Ser A, 2002, 32(11): 983–990.Google Scholar
  14. 14.
    Zhu X Y, Qi W F. Uniqueness of the distribution of zeros of primitive level sequences over Z/(p e), Finite Fields and Their Applications, 2005, 11(1): 30–44.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zhu X Y, Qi W F. Compressing mappings on primitive sequences over Z/(p e), IEEE Transactions on Information Theory, 2004, 50(10): 2442–2448.CrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics 2007

Authors and Affiliations

  • Sun Zhonghua 
    • 1
  • Qi Wenfeng 
    • 1
  1. 1.Dept. of Appl. Math.Inform. Eng. Univ.ZhengzhouChina

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