环Z/(p~2q)上本原序列的模2保熵性
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摘要
设整数N>1,Z/(N)表示整数模N的剩余类环。大量的实验数据表明,Z/(N)上的n>1次本原多项式生成的本原序列应该是模2保熵的。然而,除N是素数方幂时已被完全解决以外,其它情形没有一个完整的理论证明。目前的研究成果主要集中在N是无平方因子奇合数上,给出了若干个模2保熵的充分条件。文章首次研究了环Z/(p2q)上本原序列的模2保熵性,其中,p,q是两个不同的奇素数,给出了Z/(p2q)上n>1次本原多项式生成的本原序列是模2保熵的一个充分条件。
Let N be an integer greater than 1 and Z /( N) the integer residue ring modulo N. Extensive experiments seem to imply that primitive sequences of order n > 1 over Z /( N) are pairwise distinct modulo 2. However,the proof has been quite resistant to complete except for the case when N is an odd prime power. Recent research mainly focuses on square-free odd integers and several sufficient conditions have been given. This paper,for the first time,studies the distinctness of primitive sequences over Z /( p2q) modulo 2,where p and q are two distinct odd primes. A sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial over Z /( p2q)are pairwise distinct modulo 2.
Let N be an integer greater than 1 and Z /( N) the integer residue ring modulo N. Extensive experiments seem to imply that primitive sequences of order n > 1 over Z /( N) are pairwise distinct modulo 2. However,the proof has been quite resistant to complete except for the case when N is an odd prime power. Recent research mainly focuses on square-free odd integers and several sufficient conditions have been given. This paper,for the first time,studies the distinctness of primitive sequences over Z /( p2q) modulo 2,where p and q are two distinct odd primes. A sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial over Z /( p2q)are pairwise distinct modulo 2.
引文
[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[EB/OL].[2011-09-01].http://zuc.dacas.cn/thread.aspx ID=2304,2011.
[2]Zheng Q X,Qi W F,Tian T.On the distinctness of modular reduction of primitive sequences over Z/(232-1)[J].Des.Codes Cryptogr.,2014,70(3):359-368.
[3]Zhu X Y,Qi W F.On the distinctness of modular reductions of maximal length sequences modulo odd prime powers[J].Mathematics of Computation,2008,77(263):1623-1637.
[4]Chen H J,Qi W F.On the distinctness of maximal length sequences over Z/(pq)modulo 2[J].Finite Fields and Their Applications,2009,15(2):23-39.
[5]Zheng Q X,Qi W F.A new result on the distinctness of primitive sequences over Z/(pq)modulo 2[J].Finite Fields Appl.,2011,17:254-274.
[6]Zheng Q X,Qi W F.On the distinctness of binary sequences derived from primitive sequences modulo square-free odd integers[J].IEEE Transactions on Information Theory,2013,59(1):680-690.
[7]Zheng Q X,Qi W F.Further results on the distinctness of binary sequences derived from primitive sequences modulosquarefree odd integers[J].IEEE Trans.Inf.Theory,2013,59:4013-4019.
[8]Yang D,Qi W F,Zheng Q X.Further results on the distinctness of modulo 2 reductions of primitive sequences over Z/(232-1)[J].Des.Codes Cryptogr,2015,74(2):467-480.
[9]Ward M.The arithmetical theory of linear recurringseries[J].Trans.Amer.Math.Soc.,1933,35:600-628.
[10]黄民强.环上本原序列的分析及其密码学评价[D].合肥:中国科技大学,1988.
[11]Kamlovskii O V.Frequency characteristics of coordinate sequences of linear recurrences over Galois rings[J].Izv.RAN.Ser.Mat.,2013,77:71-96.
[12]Kurakin V L,Kuzmin A S,Mikhalev A V,et al.Linear recurring sequences over rings and modules[J].J.Math.Sci.,1995,76:2793-2915.
[13]Hall M.An isomorphism between linear recurring sequences and algebraic rings[J].Trans.Amer.Math.Soc.,1938,44:196-218.
[14]Bylkov D N,Kamlovskii O V.Occurrence indices of elements in linear recurrence sequences over primary residue rings[J].Problems of Information Transmission,2008,44:161-168.
[15]Lidl R,Niedereiter H.Finite Field[M].Canada:Addison-Wesley,1983.
[16]Bugeaud Y,Corvaja P,Zannier U.An upper bound for the G.C.D.of an-1 and bn-1[J].Mathematische Zeitschrift,2003,243(1):79-84.
[17]Mc Donald B R.Finite Rings with Identity[M].New York:Marcel Dekker,1974.
[18]Wan Z X.Finite fields and Galois Rings[M].Singapore:World Scientific Publisher,2003.
[19]戚文峰,周锦君.Z/(pe)上多项式分裂环及线性递归序列根表示[J].中国科学(A辑),1994,24(7):692-696.
[20]Rueppel R A.Analysis and Design of Stream Cipher[M].New York:Springer,1986.
[2]Zheng Q X,Qi W F,Tian T.On the distinctness of modular reduction of primitive sequences over Z/(232-1)[J].Des.Codes Cryptogr.,2014,70(3):359-368.
[3]Zhu X Y,Qi W F.On the distinctness of modular reductions of maximal length sequences modulo odd prime powers[J].Mathematics of Computation,2008,77(263):1623-1637.
[4]Chen H J,Qi W F.On the distinctness of maximal length sequences over Z/(pq)modulo 2[J].Finite Fields and Their Applications,2009,15(2):23-39.
[5]Zheng Q X,Qi W F.A new result on the distinctness of primitive sequences over Z/(pq)modulo 2[J].Finite Fields Appl.,2011,17:254-274.
[6]Zheng Q X,Qi W F.On the distinctness of binary sequences derived from primitive sequences modulo square-free odd integers[J].IEEE Transactions on Information Theory,2013,59(1):680-690.
[7]Zheng Q X,Qi W F.Further results on the distinctness of binary sequences derived from primitive sequences modulosquarefree odd integers[J].IEEE Trans.Inf.Theory,2013,59:4013-4019.
[8]Yang D,Qi W F,Zheng Q X.Further results on the distinctness of modulo 2 reductions of primitive sequences over Z/(232-1)[J].Des.Codes Cryptogr,2015,74(2):467-480.
[9]Ward M.The arithmetical theory of linear recurringseries[J].Trans.Amer.Math.Soc.,1933,35:600-628.
[10]黄民强.环上本原序列的分析及其密码学评价[D].合肥:中国科技大学,1988.
[11]Kamlovskii O V.Frequency characteristics of coordinate sequences of linear recurrences over Galois rings[J].Izv.RAN.Ser.Mat.,2013,77:71-96.
[12]Kurakin V L,Kuzmin A S,Mikhalev A V,et al.Linear recurring sequences over rings and modules[J].J.Math.Sci.,1995,76:2793-2915.
[13]Hall M.An isomorphism between linear recurring sequences and algebraic rings[J].Trans.Amer.Math.Soc.,1938,44:196-218.
[14]Bylkov D N,Kamlovskii O V.Occurrence indices of elements in linear recurrence sequences over primary residue rings[J].Problems of Information Transmission,2008,44:161-168.
[15]Lidl R,Niedereiter H.Finite Field[M].Canada:Addison-Wesley,1983.
[16]Bugeaud Y,Corvaja P,Zannier U.An upper bound for the G.C.D.of an-1 and bn-1[J].Mathematische Zeitschrift,2003,243(1):79-84.
[17]Mc Donald B R.Finite Rings with Identity[M].New York:Marcel Dekker,1974.
[18]Wan Z X.Finite fields and Galois Rings[M].Singapore:World Scientific Publisher,2003.
[19]戚文峰,周锦君.Z/(pe)上多项式分裂环及线性递归序列根表示[J].中国科学(A辑),1994,24(7):692-696.
[20]Rueppel R A.Analysis and Design of Stream Cipher[M].New York:Springer,1986.