首页
社区
课程
招聘
问答 CTF 排行榜 知识库 工具下载 峰会 看雪商城 证书查询
  1. 社区
  2. 密码应用
    3. 发新帖
[转帖]布尔函数的WALSH谱
发表于: 2011-12-6 18:36 5434

[转帖]布尔函数的WALSH谱

lilianjie 活跃值
1
2011-12-6 18:36
5434
http://www.docin.com/p-69599801.html

http://www.cnki.com.cn/Article/CJFDTotal-GDJX401.007.htm

[注意]传递专业知识、拓宽行业人脉——看雪讲师团队等你加入！

上传的附件：
收藏
免费 ・0
支持
分享
最新回复 (3)
没有姓名
雪    币： 67
活跃值： (30)
能力值： ( LV2，RANK：10 )
在线值：
发帖
回帖
粉丝
私信
2 楼
哈哈，这本书我有，讲很多布尔函数的。
2011-12-6 21:49
0
lilianjie
雪    币： 433
活跃值： (45)
能力值： ( LV4，RANK：50 )
在线值：
发帖
回帖
粉丝
私信
3 楼
这书难，还是杨晓元的好懂点

Z:=Matrix(GF(2), 1 ,[1]);
Z;
X := RandomMatrix(GF(2), 3, 3);
> X;
Y := RandomMatrix(GF(2), 4, 4);
> Y;

KZX:=KroneckerProduct(Z, X);
KZX;
KZX1:=KroneckerProduct(Z, KZX);
KZX1;
KZX2:=KroneckerProduct(Z, KZX1);
KZX2;

KZX3:=KroneckerProduct(Z, KZX2);
KZX3;

KZX4:=KroneckerProduct(Z, KZX3);
KZX4;

[1]
[0 1 1]
[0 1 1]
[0 1 0]
[1 0 1 0]
[1 1 0 1]
[1 1 0 0]
[0 0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]

Z:=Matrix(GF(2), 1 ,[1]);
Z;
X := RandomMatrix(GF(2), 2, 2);
> X;
Y := RandomMatrix(GF(2), 4, 4);
> Y;

沃尔什转换矩阵：

KXY:=KroneckerProduct(X, Y);
KXY;
KXY1:=KroneckerProduct(X, KXY);
KXY1;
KXY2:=KroneckerProduct(X, KXY1);
KXY2;

[1]
[1 1]
[0 1]
[0 0 1 1]
[1 0 0 1]
[0 0 0 0]
[1 1 0 0]
[0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0][0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]
[0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]

逆针

[1 0 0 0 1 0 0 0]
[0 1 1 0 0 1 1 0]
[1 0 1 1 1 0 1 1]
[1 1 1 1 1 1 1 1]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 1 1][1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0]
[0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0]
[1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0]
[0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0]
[1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]

转换式与反转换式只差了一个常数，这是由于沃尔什转换矩阵的反矩阵就是自己的转置矩阵乘上一个常数

Transpose(KXY);

[1 1 1 1 0 0 0 0]
[0 1 0 0 0 0 0 0]
[1 0 1 1 0 0 0 0]
[1 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]

序数顺序(沃尔什顺序) 双积顺序(培力顺序) 自然顺序(哈德码得顺序) W[m,n]
Sequency Ordering(Walsh Ordering) Dyadic Ordering(Paley Ordering) Natural Ordering (Hadamard Ordering)

双积顺序的二进制编号是序数顺序的格雷码编码
G：格雷码 B：二进制码
G(N) = B(n+1) XOR B(n)

自然顺序的二进制编号是双积顺序的位元反转。

沃尔什转换矩阵的每个列是互相正交的
SwapColumns(KXY, 1, 2) ;

SwapColumns(KXY, 2, 3) ;
SwapColumns(KXY, 3, 4) ;

[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
2011-12-28 14:17
0
lilianjie
雪    币： 433
活跃值： (45)
能力值： ( LV4，RANK：50 )
在线值：
发帖
回帖
粉丝
私信
4 楼
零交(zero-crossing)性质

奇偶性质

沃尔什转换矩阵(沃尔什顺序)中，编号为偶数的列是偶对称，编号为奇数的列是奇对称。(有第0列)

[0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0

================
chrestenson谱：

扩沃尔什从2^n到k^n,3进制以上计算机？
上传的附件：
2011-12-28 15:51
0
游客
登录 | 注册 方可回帖
回帖 表情 雪币赚取及消费
高级回复
返回
//