在Matlab中如何在GF(2)中执行逆运算并在GF(256)中进行乘法运算? [英] How to perform inverse in GF(2) and multiply in GF(256) in Matlab?

问题描述

我在Galois字段(256)中有一个二进制矩阵A(仅1和0)和向量D.向量C的计算公式为:

I have a binary matrix A (only 1 and 0), and a vector D in Galois field (256). The vector C is calculated as:

  C = (A^^-1)*D

其中，A^^-1表示GF(2)中矩阵A的逆矩阵，*是乘法运算.结果向量C必须在GF(256)中.我试图在Matlab中做到这一点.

where A^^-1 denotes the inverse matrix of matrix A in GF(2), * is multiply operation. The result vector C must be in GF(256). I tried to do it in Matlab.

A= [ 1     0     0     1     1     0     0     0     0     0     0     0     0     0;
     1     1     0     0     0     1     0     0     0     0     0     0     0     0;
     1     1     1     0     0     0     1     0     0     0     0     0     0     0;
     0     1     1     1     0     0     0     1     0     0     0     0     0     0;
     0     0     1     1     0     0     0     0     1     0     0     0     0     0;
     1     1     0     1     1     0     0     1     0     1     0     0     0     0;
     1     0     1     1     0     1     0     0     1     0     1     0     0     0;
     1     1     1     0     0     0     1     1     1     0     0     1     0     0;
     0     1     1     1     1     1     1     0     0     0     0     0     1     0;
     0     0     0     0     1     1     1     1     1     0     0     0     0     1;
     0     1     1     1     1     0     1     1     1     0     1     1     1     0;
     0     0     0     1     0     0     0     1     0     0     0     0     0     0;
     0     0     1     0     0     0     0     1     0     0     0     0     0     0;
     1     1     1     1     0     0     0     0     0     0     0     0     0     0]

D=[0;0;0;0;0;0;0;0;0;0;103;198;105;115]
A=gf(A,1);
D=gf(D,8); %%2^8=256
C=inv(A)*D
%% The corrected result must be
%%C=[103;187;125;210;181;220;161;20;175;175;187;187;220;115]

但是，对于上述代码，我无法达到预期的效果

However, for above code, I cannot achieved as my expected result

C=[103;187;125;210;181;220;161;20;175;175;187;187;220;115]

它产生错误为

Error using  *  (line 14)
Orders must match.

您能帮助我达到预期的结果吗?

Could you help me achieve my expected result?

推荐答案

错误

使用*时出错(第14行)
订单必须匹配.

Error using * (line 14)
Orders must match.

会发生变化，因为Matlab不支持将标准数学运算(+，*，.*，.^，\等)应用于不同顺序的GF(2)和.反向运算inv(A)是有效的运算，如果您以 Ander Biguri 将成功并在GF(2)中生成A的逆.当您尝试将inv(A)和D相乘时，由于这些Galois字段的顺序不匹配，计算会中断.

arises because Matlab does not support applying standard mathematical operations (+, *, .*, .^, \, etc.) to Galois Fields of different order, GF(2) and GF(256). The inverse operation, inv(A) is a valid operation and if you perform it on its own as shown by @Ander Biguri it will succeed and generate the inverse of A in GF(2). Your calculation breaks down when you attempt to multiply inv(A) and D as the orders of these Galois Fields do not match.

在此示例中，没有必要将A显式定义为2的Galois字段，因为GF(256)中的乘法发生在GF(2)中.那是1 + 1 = 0.

In this example it is unnecessary to explicitly define A as a Galois Field of order 2, GF(2) as multiplication in GF(256) takes place in GF(2). That is 1 + 1 = 0.

如果我们因此将为A创建Galois字段的代码修改为

If we thus modify the code that creates the Galois Field for A to

A = gf(A, 8);

然后我们可以执行

C = inv(A)*D

产生

C = GF(2^8) array. Primitive polynomial = D^8+D^4+D^3+D^2+1 (285 decimal)

Array elements = 

         103
         187
         125
         210
         181
         220
         161
          20
         175
         175
         187
         187
         220
         115

因此，

C在GF(256)中，并产生预期的结果.

C is thus in GF(256) and produces the expected result.

这篇关于在Matlab中如何在GF(2)中执行逆运算并在GF(256)中进行乘法运算?的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持IT屋！