基于semi-bent函数正交序列集的构造

基于semi-bent函数正交序列集的构造 Constructions of Orthogonal Sets Using Semi-bent Functions first-author 季霄鹏，男，硕士生，现从事密码学方面的研究. E-mail:JiXiaopeng@cumt.edu.cn 季霄鹏，男，硕士生，现从事密码学方面的研究. E-mail: JiXiaopeng@cumt.edu.cn JiXiaopeng@cumt.edu.cn corresp 夏士雄，E-mail:xiasx@cumt.edu.cn 夏士雄，E-mail:xiasx@cumt.edu.cn xiasx@cumt.edu.cn 中国矿业大学 计算机科学与技术学院, 江苏 徐州 221116 中国矿业大学 计算机科学与技术学院, 江苏 徐州 221116 School of Computer Science and Technology, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China School of Computer Science and Technology, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China 蜂窝技术在码分多址(code division multiple access, CDMA)系统中起着非常重要的作用.本文给出了用于构造CDMA规则蜂窝系统正交序列集的不同函数.首先，基于多输出semi-bent函数的性质，给出了一个构造编码长度为2 6 的蜂窝系统.研究发现，通过选取不同的参数 d ( d ∈{2, 3})，所构造蜂窝系统的蜂窝排列难易程度和重用距离可以相应的调整.其次，利用“直和构造”将semi-bent函数变元增加，进而得到编码长度为2 n ( n ≥6且为偶数)，重用距离 D ≥4的规则蜂窝系统.研究表明，所构造的蜂窝系统内，每个蜂窝所包含的正交序列个数达到了2 n -2 (即与目前已知的最好结果相同). The cellular technology plays an important role in the code division multiple access (CDMA) system. In this paper, we present a different function for constructing orthogonal sequence set of CDMA system. Firstly, based on the properties of multi-dimension semi-bent function, we presented a cellular system with the code length of 2 6 . By choosing different parameter d ( d ∈{2, 3}), we can assign the sets of orthogonal sequences to a regular tessellation of hexagonal cells more easily and adjust the re-use distance accordingly. Secondly, the number of variables of this semi-bent function increases by using direct sum construction and then a regular tessellation of hexagonal cells with the length of 2 n ( n ≥6, even) is obtained. It is showed that the number of users per cell reaches 2 n -2 (which achieves the same as the best known results). 布尔函数 码分多址 正交序列 semi-bent函数 Boolean function CDMA(code division multiple access) orthogonal sequence semi-bent function 引言 1 蜂窝系统在不同分区中可以使用相同频率完成不同的数据传输，与单一基站相比具有更高的容量，在卫星传输、移动通信等方面有重要的应用 [ 1 1 , 2 2 ] .码分多址(code division multiple access，CDMA)系统是第三代蜂窝通信系统的主流接入技术 [ 3 3 ] .在CDMA系统的同一个小区内，给每个用户一个惟一的编码来惟一标识该用户，为了给小区内更多用户提供服务，需要在同一个小区内提供更多的编码序列.又知，CDMA系统内的编码序列需要满足一定的基本条件，如在同一个蜂窝内的编码不会产生串扰，且彼此相邻的蜂窝之间也不会产生串扰.这就要求在同一个蜂窝内的编码序列满足两两正交，相邻的蜂窝之间的编码序列也应两两正交，而彼此不相邻的蜂窝间的相关性要尽可能的小.一种常用构造展频码的基本方法是使用Hadamard矩阵的相关约束集 [ 4 4 ~ 6 6 4~6 ] ，该方法可以保证不同Hadamard矩阵的行之间的相关性在[2 n /2 , 2 ⌊( n +2)/2⌋ ] 内.另一种更为有效的方法是基于Hadamard矩阵以及semi-bent函数.这种方法使用一阶Reed-Muller码的子码的4个陪集构造出12个正交序列集，在 n 为3或9，重用距离为4这种特殊情况下，单个蜂窝内的正交序列个数可达2 n －2 . Zhang等人 [ 7 7 ] 给出了一个更完善的适用于规则蜂窝结构的正交序列集的方法.该方法先借助线性变换，使用Maiorana-McFarland(M-M)类函数构造技术构造出一个多输出semi-bent函数，然后将该函数和任意的多输出bent函数级联.使用这种方法构造的蜂窝系统能够在 n ≠4, 5时，每个蜂窝内的序列集的个数达到2 n －2 .与之前方法相比，单个蜂窝内的序列集个数扩充了1倍.在重用距离方面，该方法在 n 为奇数的情况下，可以始终保证重用距离 D =4，在 n 为偶数情况下，重用距离 D ≥4.虽然文献[ 7 7 ]的方法在构造蜂窝系统上增加了单个蜂窝内的正交序列数量，也解决了蜂窝的排列问题，但由于该方法使用M-M技术构造semi-bent函数时使用的是线性变换，这使得在级联bent函数产生正交序列集时semi-bent函数的选择面较窄.另外，由于线性函数在密码学上是不安全的，难以抵抗各类攻击 [ 8 8 ] ，因此在产生编码序列时可能存才潜在的不安全. 基于文献[ 7 7 ]的构造方法，本文给出了新的正交序列集.在重用距离方面，直接使用该函数构造的蜂窝结构满足 D ≥4. 基本概念与引理 1 一个 n 元布尔函数是从 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258708&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258708&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258708&type=middle 到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258723&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258723&type=middle 的映射，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258730&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258730&type=middle 表示二元有限域， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258734&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258734&type=middle 表示 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258740&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258740&type=middle 上的 n 维向量空间.所有 n 元布尔函数所组成的集合用 B n 来表示. 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258744&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258744&type=middle ， a 与 b 的内积定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258265&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258265&type=middle 其中符号“⊕”表示模2加，在不引起歧义情况下，该符号可以简单记为“+”. 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258280&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258280&type=middle 形如 f ( x 1 , …, x n )= a · x + c = a 1 x 1 +…+ a n x n + c 的函数称为仿射函数.特别地，当常数项 c =0时，该函数称为线性函数.所有 n 个变元的线性函数所组成的集合可以用 L n 表示，即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258748&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258748&type=middle . 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258943&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258943&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258943&type=middle ，则一个 n 元布尔函数 f ( x )在点 ω 处的Walsh谱可以表示为 [ 9 9 ] ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258291&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258291&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258291&type=middle 特别地，函数 f ( x )是平衡的，当且仅当 W f (0 n )=0.任意布尔函数的Walsh谱都满足Parseval’s等式 [ 10 10 ] ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258298&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258298&type=middle 定义1 [ 11 11 ]   设 f i ∈ B n ( i =1, 2, …, n ), π ( x )=( f 1 ( x ), f 2 ( x ), …, f n ( x )).如果对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258946&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258946&type=middle 都有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258949&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258949&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258949&type=middle ，则称 π ( x )是一个 n 元布尔置换. 定义2 [ 12 12 ]   设 n 是一个偶数， f ( x )是一个 n 元布尔函数，如果 W f ( ω )=±2 n /2 对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258955&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258955&type=middle 都成立，则称 f ( x )是一个bent函数. 定义3 [ 13 13 ]   设 f ∈ B n ，如果存在一个偶数 r , 0≤ r ≤ n ，使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258958&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258958&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258958&type=middle ，并且对任意 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258961&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258961&type=middle ， W f ( ω )的值只能是±2 (2 n － r )/2 或0，那么 f ( x )称为 n 元 r 阶plateaued函数.特别地，当 n 为奇数时，对任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258966&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258966&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258966&type=middle , W f ( ω )的值是±2 ( n +1)/2 或0，则称 f 为 n 元semi-bent函数.即 n 元 n －1阶plateaued函数.当 n 为偶数时，对任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258969&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258969&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258969&type=middle ， W f ( ω )的值是±2 ( n +2)/2 或0，则称 f 为 n 元semi-bent函数.即 n 元 n －2阶plateaued函数. 设 f ∈ B n ，则该函数的(1, －1)序列(简称为序列)定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258304&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258304&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258304&type=middle 如果函数 f 是一个bent函数，则该函数产生的序列 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258976&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258976&type=middle 称为bent序列.相对应地，semi-bent函数(plateaued函数)产生的序列称为semi-bent序列(plateaued序列). 定义4   设任意的正整数 s 和 t 满足 n = s + t ，一个 n 元M-M类布尔函数定义为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258311&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258311&type=middle 其中 ϕ 是一个从 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258980&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258980&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258980&type=middle 到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258984&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258984&type=middle 的映射， τ ∈ B s .当 s ≤ t 且 ϕ 是一个单射时，此时的函数是plateaued函数.特别地，当 s = t 且 ϕ 是双射时，该方法构造的函数是bent函数. 设 f 1 , f 2 是两个 n 元布尔函数，它们对应的序列分别为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258986&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258986&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258990&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258990&type=middle ，二者的内积可以定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258320&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258320&type=middle 因此，一个函数 f 在 ω 处的Walsh谱也看作是序列 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258994&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258994&type=middle 和一个线性函数 l 产生的序列 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258998&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258998&type=middle 的乘积，其中 l = ω · x .函数 f 1 , f 2 的分量积(componentwise pro-duct)定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258327&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258327&type=middle 显然， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258335&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258335&type=middle 定义5 [ 7 7 ]   设 f 1 , f 2 是两个 n 元布尔函数，它们对应的序列分别为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258340&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258340&type=middle 可以用符号⊥表示二者的正交关系，当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258347&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258347&type=middle 设任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259001&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259001&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259004&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259004&type=middle ，如果 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259006&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259006&type=middle ，则 S 是一个基数为 k 的正交序列集，记为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258357&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258357&type=middle 设 S 1 和 S 2 是两个正交序列集，如果对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259008&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259008&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259011&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259011&type=middle 都有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259014&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259014&type=middle 就说集合 S 1 和 S 2 是正交的，记为 S 1 ⊥ S 2 . 引理1 [ 7 7 ]   设 f 1 , f 2 是两个 n 元布尔函数，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258364&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258364&type=middle 一个2 n 阶的Sylvester-Hadamard矩阵(记为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259017&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259017&type=middle )是一个递归定义的方阵： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258370&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258370&type=middle 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259027&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259027&type=middle ， i 是 ω 在整数域的映射.如果用 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259032&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259032&type=middle 表示 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259035&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259035&type=middle 的第 i 行(0≤ i ≤2 n －1)，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259042&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259042&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259042&type=middle 可以看作是线性函数 l = ω · x 所产生的序列.由Hadamard矩阵的每一行序列所构成的序列集 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258377&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258377&type=middle 称为Hadamard序列集.显然， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258382&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258382&type=middle 对于任意两个线性函数 l , l ′都有 W l + l ′ ( 0 n )=0，因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259046&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259046&type=middle , 也就是说 H 是一个正交序列集. 定义6 [ 14 14 ]   设 f i ( x )∈ B n , i =1, 2, …, t ，一个 n 维输入， t 维输出的函数可以表示为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258400&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258400&type=middle 其中每一个函数 f i ( x )称作函数 F ( x )的分量函数.如果函数 F ( x )的分量函数 f i ( x )的非零线性组合是一个plateaued函数，则函数 F ( x )被称作多维plateaued函数.相应的，如果对于任意的非零 c =( c 1 , …, c t )∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259050&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259050&type=middle ，函数 c 1 f 1 +…+ c t f t 是一个semi-bent函数(bent函数)，则 F ( x )是一个多维semi-bent函数(多维bent函数). 文献[ 7 7 ]中给出了一个 n ≥6且为偶数情况下构造规则蜂窝系统的具体方法. 引理2 [ 7 7 ]   设正整数 n , k 满足 n =2 k +2.设 γ 是有限域 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259052&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259052&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259052&type=middle 的本原元，且{1, γ , …, γ k －1 }是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259054&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259054&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259054&type=middle 在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259055&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259055&type=middle 上的多项式基， π 是一个 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259059&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259059&type=middle 的映射，定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258409&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258409&type=middle 设 ϕ i , i =1, …, k 是一个双射，定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258420&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258420&type=middle 其中[ x ]是 x 的十进制表示，即对于一个固定的 x ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259065&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259065&type=middle . 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259071&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259071&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259073&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259073&type=middle ，函数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258426&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258426&type=middle 记为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258429&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258429&type=middle 可以构造一个 n 维输入， k 维输出的函数 F ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258434&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258434&type=middle 设 d ∈{2, 3}，对于任意固定的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259074&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259074&type=middle ，定义一个线性函数的集合： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258437&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258437&type=middle 对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259075&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259075&type=middle ，令 f c ( x , y )= c · F ( x , y )，可以构造出2 k + d 个如下的正交序列集 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258443&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258443&type=middle 每个正交序列集内包含元素个数可以达到2 n － d . 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259076&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259076&type=middle , u ≥2 k ， H ( z )=( h 1 , …, h k )是一个任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259079&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259079&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259079&type=middle 多维bent函数，可以构造如下一个 n + u 维输入， k 维输出的函数 G ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258449&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258449&type=middle 设 d ∈{2, 3}， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259081&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259081&type=middle ，定义 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258456&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258456&type=middle 对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259083&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259083&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259083&type=middle ，令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258467&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258467&type=middle 其中 h c = c · H , f c = c · F .可以构造如下2 k + d 个正交序列集 S ′ c , β ，每一个正交序列集内包含2 u + n － d 个正交序列 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258477&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258477&type=middle 构造出的正交序列集可以排列在重用距离 D ≥4的规则蜂窝系统内，单个蜂窝内的容量 N 可达2 n －2 . 正交序列集的构造及蜂窝排列 1 基于引理2，给出一种新的直接构造正交序列集的方法.借助任意的布尔置换，利用M-M类函数构造技术给出一个多输出semi-bent函数，并直接构造出 n =6的蜂窝系统.进一步利用该函数与引理2的方法相结合构造出 n > 6且为偶数时的正交序列集. 正交序列集的直接构造 2 引理3 [ 15 15 ]   设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259084&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259084&type=middle ， A 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259085&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259085&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259085&type=middle 上的周期为2 n －1 的某一 m 序列的状态转移矩阵， I 是 n 阶单位阵，则 I , A , A 2 , …, A n －1 的任意非零线性组合 A c = c 1 I + c 2 A +…+ c n A n －1 都可逆，因而它们都是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259087&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259087&type=middle 上的可逆线性变换，记为 A c -1 . 借助引理2和引理3，可以给出下面的构造： 构造1   设 n , k 是两个正整数且满足 n =2 k +2.设 ϕ i , i =1, …, k 是一个 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259089&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259089&type=middle 的双射，记为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258488&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258488&type=middle 其中 φ ( x )是一个任意的布尔置换， A 是引理3中定义的 k 阶状态转移矩阵. 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259091&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259091&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259092&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259092&type=middle ，函数 f i ( x , y ), i =1, …, k 记为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258496&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258496&type=middle 可以构造一个 n 维输入， k 维输出的函数 F ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258505&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258505&type=middle 设 d ∈{2, 3}，对于任意固定的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259093&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259093&type=middle ，定义一个线性函数的集合： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258514&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258514&type=middle 对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259094&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259094&type=middle ，令 f c ( x , y )= c · F ( x , y )，可以构造出2 k + d 个如下的正交序列集 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258521&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258521&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258521&type=middle 每个正交序列集内包含元素个数可以达到2 n － d . 命题1   构造1中的函数 F 是一个semi-bent函数. 证   设非零向量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259096&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259096&type=middle ，对于 x ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259099&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259099&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259099&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259101&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259101&type=middle ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258531&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258531&type=middle 因此，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258536&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258536&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258539&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258539&type=middle 由定义3知函数 F 是一个多维semi-bent函数. 证毕 定理1   设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259104&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259104&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259105&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259105&type=middle ， S c , β 是构造1中(2)式所定义的正交序列集，则有： 1) S c , β 中的所有序列都是semi-bent序列； 2) 对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259106&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259106&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259107&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259107&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259107&type=middle ， S c , β 是一个包含2 n －d 个元素的正交序列集； 3) 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259108&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259108&type=middle ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259109&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259109&type=middle ，对于任意给定的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259110&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259110&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259111&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259111&type=middle ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258544&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258544&type=middle 特别地， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259113&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259113&type=middle ； 4) 在 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259118&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259118&type=middle 中存在3·2 k + d －2 +2 d －2 －1个正交序列集与 S e , δ 正交. 证  1)由命题1知 S c , β 中的所有序列都是semi-bent序列. 2) 对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259123&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259123&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259127&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259127&type=middle 且 β ≠ β ′，设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259129&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259129&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259130&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259130&type=middle .由于 l β + l β ′ 是非零线性函数，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258549&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258549&type=middle 由引理1，知 S c , β 是一个正交序列集. 对于任意给定的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259131&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259131&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259131&type=middle l ∈ L β }=# L β =2 n － d ，其中 L β 如(1)式所定义. 3) 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259134&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259134&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259137&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259137&type=middle ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258558&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258558&type=middle 其中， L β 如(1)式所定义.设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259140&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259140&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259140&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259143&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259143&type=middle , Γ e , δ = f e + L δ ={ f e + l | l ∈ L δ }, 其中 L δ = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259147&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259147&type=middle .对于任意的 f c , β ∈ Γ c , β , f e , δ ∈ Γ e , δ ，存在两种情况 ⅰ) c ≠ e ，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258565&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258565&type=middle 因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258572&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258572&type=middle 由引理2和(3)式，有 S c , β ⊥ S e , δ ，当且仅当 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258579&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258579&type=middle 又因为 φ ( x )· A c + e 是由 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259149&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259149&type=middle 到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259151&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259151&type=middle 的双射， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258587&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258587&type=middle 是由 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259153&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259153&type=middle 到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259155&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259155&type=middle 的单射，所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258617&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258617&type=middle 所以 S c , β ⊥ S e , δ ⇔( β d －1 , β d )≠( δ d －1 , δ d ). ⅱ) c = e ，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258624&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258624&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258624&type=middle 因此 β ≠ δ ⇔ l α + l θ ≠0⇔ S e , β ⊥ S e , δ . 4) ⅰ)当 c ≠ e 时 对于一个固定的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259157&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259157&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259159&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259159&type=middle ，#{( β d －1 , β d )≠( δ d －1 , δ d )}=(2 d －2 d －2 )=3·2 d －2 ，而 c 的可取值个数共有2 k －1个.此种情况共有3·2 d －2 (2 k －1)=3·2 k + d －2 －3·2 d －2 个正交序列集与 S e , δ 正交. ⅱ)当 c = e 时 对于任意给定的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259161&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259161&type=middle ， δ 共有2 d －1种选择，因此有2 d －1个正交序列集与 S e , δ 正交. 综上两种情况， Ω 中共有3·2 k + d －2 －3·2 d －2 +2 d －1=3·2 k + d －2 +2 d －2 －1个集合与 S e , δ 正交. 证毕 例1   设 n =6， A 是2阶状态转移矩阵.设 x = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259162&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259162&type=middle ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259163&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259163&type=middle ， F ( x , y )是构造1中定义的一个6维输入，2维输出的多输出semi-bent函数，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258631&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258631&type=middle 设 f c ( x , y )= c · F ( x , y ), http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259165&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259165&type=middle ，取 d =2，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259168&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259168&type=middle ，每个 S c , β 包含2 n － d =16个正交序列.设任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259170&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259170&type=middle ，令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259172&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259172&type=middle l α ∈ L β }= S 00, β .显然 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259173&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259173&type=middle 把Hadamard矩阵分成4部分.即， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258635&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258635&type=middle 如果，对于任意的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259174&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259174&type=middle 都有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259175&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259175&type=middle ，就称 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259176&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259176&type=middle 与 H β 是正交的，记为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259177&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259177&type=middle .对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259178&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259178&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259180&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259180&type=middle ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259181&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259181&type=middle .因此， S c , β ⊥ S c ′, β ′ 当且仅当 β ≠ β ′. 表 1 表 1 给出了 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259182&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259182&type=middle 和 H β 的正交关系.从 表 1 表 1 所列出的正交关系中，易给出在重用距离 D =4的条件下的规则蜂窝的排列结构(如 图 1 图 1 ).从图中可以看出，相邻的蜂窝之间是正交的关系. H 00 H 01 H 10 H 11 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259185&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259185&type=middle ⊥ ⊥ ⊥ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259188&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259188&type=middle ⊥ ⊥ ⊥ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259189&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259189&type=middle ⊥ ⊥ ⊥ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259190&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259190&type=middle ⊥ ⊥ ⊥ 注1   这里由于函数的输入变量个数限制，在使用M-M构造技术构造semi-bent函数时，选择的是线性置换.但在 n =6时使用该函数构造的正交序列集和文献[ 7 7 ]相比，只有 S 00, 00 , S 00, 01 , S 00, 10 , S 00, 11 这4个的正交序列集相同.下面给出的例子，将借助非线性置换构造8维输入，3维输出的semi-bent函数，并构造出相应的正交序列集. 例2   设 n =8， A 是3阶状态转移矩阵.设 x = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259194&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259194&type=middle ， F ( x , y )是构造1中定义的一个8维输入，3维输出的多输出semi-bent函数，其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258664&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258664&type=middle 设 f c ( x , y )= c · F ( x , y ), http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259197&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259197&type=middle ，取 d =2，由定理1可以构造序列集 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259199&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259199&type=middle ，且有#{ S c , β | http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259200&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259200&type=middle ，每个 S c , β 包含2 n － d =64个正交序列. D =4的蜂窝系统的正交序列集构造 2 在实际应用中，重用距离 D =4即可满足通信需求.使用构造1中的semi-bent函数构造的正交序列集的个数随着semi-bent函数输入维数的增加而增加，因此当 n ≥8时无法直接使用该semi-bent函数构造重用距离 D =4的蜂窝系统.但可以将此多维semi-bent函数与多维bent函数相互级联来构造出满足 D =4的蜂窝系统. 在文献[ 7 7 ]中，作者通过级联带有线性变换的semi-bent函数和任意bent函数给出 n ≥8且为偶数时一个具体的构造.现结合构造1中的semi-bent函数与引理1的方法给出适用于 n ≥8且为偶数时的规则蜂窝系统构造. 构造2   设 F ( x , y )=( f 1 , …, f k )是构造1中的多维semi-bent函数，其中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259201&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259201&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259202&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259202&type=middle .设 u 是一个偶数， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259203&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259203&type=middle ，且 u ≥2 k .设 H ( z )=( h 1 , …, h k )是一个从 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259205&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259205&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259205&type=middle 到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259208&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259208&type=middle 的多维bent函数.可以构造出一个 n + u 维输入， k 维输出的多维plateaued函数 G ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258671&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258671&type=middle 对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259212&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259212&type=middle ，令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258679&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258679&type=middle 对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259215&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259215&type=middle ，令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258689&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258689&type=middle 其中 h c = c · H , f c = c · F .可以如下构造2 k + d 个正交序列集 S ′ c , β ，每一个正交序列集内包含2 u + n － d 个正交序列： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258695&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258695&type=middle 基于文献[ 7 7 ]的证明方法，可以直接得出下面的结论. 定理2   对于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259216&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259216&type=middle ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259217&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259217&type=middle ，设 S c , β 、 S ′ c , β 分别是构造1和构造2所构造的正交序列集，那么 1) 对于固定的 e , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259218&type=middle ， S ′ c , β ⊥ S ′ e , δ 当且仅当 S c , β ⊥ S e , δ ； 2) 所有 S ′ c , β 中的序列都是semi-bent正交序列，且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259219&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259219&type=middle . 注2   这里使用了与文献[ 7 7 ]相同的方法来构造适用于蜂窝系统的正交序列集.通过对参数 d 取不同的值来调节重用距离和每个蜂窝内的序列个数之间的关系，保证每个蜂窝内的序列个数尽可能多的同时使重用距离尽可能小. 例3   设 n =6， B 是2阶状态转移矩阵， x = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259221&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259221&type=middle ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259223&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259223&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259223&type=middle 是例1中的6维输入，2维输出的多输出semi-bent函数.设偶数 u =2 k =4, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259225&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259225&type=middle ， H ( z )=( h 1 ( z ), h 2 ( z ))是一个从 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259226&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259226&type=middle 到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259227&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259227&type=middle 的多维bent函数.其中： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258701&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258701&type=middle 可以构造出一个10维输入， k =2维输出的多维plateaued函数 G ： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258705&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258705&type=middle 设 g c ( x , y )= c · G ( x , y ), http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259229&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259229&type=middle ，取 d =2，由定理1和定理2知每一个序列集 S ′ c , β 都是正交序列集，且有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259230&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259230&type=middle ，因此重用距离 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259231&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4259231&type=middle ，每个 S ′ c , β 包含2 u + n － d =2 8 个正交序列.这里使用构造2构造的 n =6的正交序列集与使用构造1构造的 n ′= n + u =10的序列集相比，在序列总数上个数相同，但使用构造2可以保证重用距离始终为4，单个蜂窝内的序列个数都为2 n ′－2 ，因此对于任意 n ′≥10的偶数都可以使用该semi-bent函数级联bent函数构造出重用距离 D =4，单个蜂窝序列个数为2 n ′－2 的正交序列集. 与已知方法比较 2 在构造出的蜂窝系统主要参数方面，当 n ≥6且为偶数情况下，借助非线性置换的多输出semi-bent函数所构造的蜂窝系统重用距离 D ≥4，单个蜂窝容量(用户数) N 可达2 n －2 、构造的序列集个数可达2 n /2+1 ，与文献[ 7 7 ]中多输出semi-bent函数构造的蜂窝系统的主要参数相比较具有相同结果，即目前已知最好结果. 在构造蜂窝系统时semi-bent函数的选择范围方面，借助非线性置换的多输出semi-bent函数与使用带有线性置换的semi-bent函数相比有更大的数量优势.在 n ≥3情况下，借助非线性置换的多输出semi-bent函数的数量大于使用带有线性置换的semi-bent函数的数量，且前者的数量优势随着 n 的增大而增大.因此，在选择多输出semi-bent函数构造正交序列集及蜂窝系统时选择范围更广. 在通信安全方面，借助非线性置换的M-M类semi-bent函数具有更高的代数次数，相比文献[ 7 7 ]中所使用的semi-bent函数具有更高的代数免疫性.同时，由于使用非线性置换，在面对线性攻击时，该函数比文献[ 7 7 ]中的semi-bent函数具有更高的免疫性.因此在面对潜在的通信安全时这里所使用的函数可以提供更安全的通信环境. 结论 1 本文给出了不同于已知用来构造CDMA蜂窝系统的函数.首先借助非线性置换，利用M-M类函数构造方法，给出了一个多输出semi-bent函数.在编码长度相同情况下，使用该函数构造的蜂窝系统的系统容量、重用距离和使用已知函数所能达到的最好结果相同.其次，由于给出了一个通用的一般函数构造方法，因此在构造蜂窝系统时semi-bent函数的选择面得到了扩展.最后，当变元数量增多时，由于可以使用没有非零线性结构的映射，因此与使用已知函数相比较在面对可能的潜在危险时，具有更高的安全性. 参考文献 啜钢, 王文博, 常永宇, 等.移动通信原理与系统[M].北京:北京邮电大学出版社, 2015:161-211. 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On plateaued functions[J]. IEEE Transactions on Information Theory, 2001, 47 (3): 1215-1223. DOI:10.1109/18.915690. 张凤荣, 谢敏, 马华.多维Plateaued函数的构造[J].计算机工程, 2009, 35 (20): 170-172. DOI:10.3969/j.issn.1000-3428.2009.20.060. ZHANG F R, XIE M, MA H. Construction of multi-dimension Plateaued functions[J]. Computer Engineering, 2009, 35 (20): 170-172.DOI:10.3969/j.issn.1000-3428.2009.20.060(Ch). 丁石孙.线性移位寄存器序列[M].上海:上海科学技术出版社, 1982:19-23. DING S S. Linear Shift Register Sequence [M]. Shanghai: Shanghai Scientific & Technical Publishers, 1982: 19-23(Ch). 10.14188/j.1671-8836.2018.02.011 TN918.1 whdxxblxb-64-2-175 国家重点研发计划项目(2016YFC0600908), 国家自然科学基金资助项目(61303263) 2017-04-19 2018-04-24 2020-11-30 版权所有© 2018 《武汉大学学报(理学版)》编辑部 Copyright © 2018 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2018 武汉大学学报(理学版) 2 64