环Z/(p~e)上本原序列压缩映射的新结果
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摘要
令Z/(pe)表示整数剩余类环,其中p为素数且e 2为正整数.令f(x)表示Z/(pe)上的n次本原多项式,G′(f(x),pe)表示Z/(pe)上所有由f(x)生成的本原序列构成的集合.设序列a∈G′(f(x),pe),它有唯一的p进制展开a=a0+a1p+···+ae-1pe-1.令φ(x0,x1,...,xe-1)=g(xe-1)+μ(x0,x1,...,xe-2)表示由Fe p到Fp的一个e变元多项式.那么,φ可以诱导出一个从G′(f(x),pe)到F∞p的压缩映射.在p为奇素数且f(x)为强本原多项式的条件下,人们已经证明该压缩映射是保熵的.而本文证明该压缩映射在f(x)为本原多项式的条件下仍然是保熵的.当deg(g(x))2时,我们还要求deg(g(x))为奇数,或者g(x)=xk+∑k-2i=0cixi.
Let Z/(pe) be the integer residue ring with p prime and e 2. Let f(x) be a primitive polynomial over Z/(pe) with degree n and G′(f(x), pe) the set of all primitive sequences over Z/(pe) generated by f(x).For any sequence a ∈ G′(f(x), pe), it has a unique p-adic expansion a = ae0 + a1p + · · · + ae-1p-1. Letφ(x0, x1,..., xe-1) = g(xe-1) + μ(x0, x1,..., xe-2) be a function from Fe p to Fp. Then φ can induce a compression mapping from G′(f(x), pe) to F∞p. In recent years, Zhu, Tian and Qi have proved that the compression mapping is injective under the condition that p is an odd prime and f(x) is a strongly primitive polynomial. In this paper,we improve their results. More exactly, we prove that the compression mapping is also injective only under the condition that p is an odd prim∑e and f(x) is a primitive polynomial. Of course we should ask that deg(g(x)) is an odd number or g(x) = xk+k-2ii=0ci x.
Let Z/(pe) be the integer residue ring with p prime and e 2. Let f(x) be a primitive polynomial over Z/(pe) with degree n and G′(f(x), pe) the set of all primitive sequences over Z/(pe) generated by f(x).For any sequence a ∈ G′(f(x), pe), it has a unique p-adic expansion a = ae0 + a1p + · · · + ae-1p-1. Letφ(x0, x1,..., xe-1) = g(xe-1) + μ(x0, x1,..., xe-2) be a function from Fe p to Fp. Then φ can induce a compression mapping from G′(f(x), pe) to F∞p. In recent years, Zhu, Tian and Qi have proved that the compression mapping is injective under the condition that p is an odd prime and f(x) is a strongly primitive polynomial. In this paper,we improve their results. More exactly, we prove that the compression mapping is also injective only under the condition that p is an odd prim∑e and f(x) is a primitive polynomial. Of course we should ask that deg(g(x)) is an odd number or g(x) = xk+k-2ii=0ci x.
引文
1 Qi W F,Yang J H,Zhou J J.ML-sequences over rings Z/(2e).In:Lecture Notes in Computer Science,vol.1514.Berlin:Springer,1998,315–326
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5 田甜.带进位的反馈移位寄存器序列分析.郑州:信息工程大学,2010
6 Huang M Q,Dai Z D.Projective maps of linear recurring sequences with maximal p-adic periods.Fibonacci Quart,1992,30:139–143
7 Kuzmin A S,Nechaev A A.Linear recurring sequences over Galois ring.Russian Math Surveys,1993,48:171–172
8 黄民强.环上本元序列分析及其密码学评价.北京:中国科学技术大学研究生院,1988
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10 Zhu X Y,Qi W F.Compression mappings on primitive sequences over Z/(pe).IEEE Trans Inform Theory,2004,50:2442–2448
11 Zhu X Y,Qi W F.Further result of compressing maps on primitive sequences modulo odd prime powers.IEEE Trans Inform Theory,2007,53:2985–2990
12 Tian T,Qi W F.Injectivity of compressing maps on primitive sequences over Z/(pe).IEEE Trans Inform Theory,2007:53:2960–2966
13 Dai Z D.Binary sequences derived from ML-sequences over rings I:Periods and minimal polynomials.J Cryptology,1992,5:193–207
14 Kurankin V L,Kuzmin A S,Mikhalev A V,et al.Linear recurring sequences over rings and modules.J Math Sci,1995,76:2793–2915
15 朱宣勇.环上本原序列保熵压缩映射的研究.郑州:信息工程大学,2004
2 Dai Z D,Huang M Q.A criterion for primitiveness of polynomials over Z/(2d).Chin Sci Bull,1991,36:892–895
3 戚文峰,周锦君.环Z/(2e)上本原序列最高权位的0,1分布.中国科学A辑,1997,27:311–316
4 Qi W F,Zhou J J.Distribution of 0 and 1 in highest level of primitive sequences over Z/(2e)(II).Chin Sci Bull,1998,43 :633–635
5 田甜.带进位的反馈移位寄存器序列分析.郑州:信息工程大学,2010
6 Huang M Q,Dai Z D.Projective maps of linear recurring sequences with maximal p-adic periods.Fibonacci Quart,1992,30:139–143
7 Kuzmin A S,Nechaev A A.Linear recurring sequences over Galois ring.Russian Math Surveys,1993,48:171–172
8 黄民强.环上本元序列分析及其密码学评价.北京:中国科学技术大学研究生院,1988
9 Qi W F,Zhu X Y.Compressing mappings on primitive sequences over and its Galois extension.Finite Fields Appl,2002,8:570–588
10 Zhu X Y,Qi W F.Compression mappings on primitive sequences over Z/(pe).IEEE Trans Inform Theory,2004,50:2442–2448
11 Zhu X Y,Qi W F.Further result of compressing maps on primitive sequences modulo odd prime powers.IEEE Trans Inform Theory,2007,53:2985–2990
12 Tian T,Qi W F.Injectivity of compressing maps on primitive sequences over Z/(pe).IEEE Trans Inform Theory,2007:53:2960–2966
13 Dai Z D.Binary sequences derived from ML-sequences over rings I:Periods and minimal polynomials.J Cryptology,1992,5:193–207
14 Kurankin V L,Kuzmin A S,Mikhalev A V,et al.Linear recurring sequences over rings and modules.J Math Sci,1995,76:2793–2915
15 朱宣勇.环上本原序列保熵压缩映射的研究.郑州:信息工程大学,2004