在伽罗瓦域 256 中执行乘法的有效方法答案

【问题标题】：An efficient way to perform multiplication in Galois field 256在伽罗瓦域 256 中执行乘法的有效方法
【发布时间】：2016-09-02 14:42:07
【问题描述】：

我正在使用 MATLAB 2015b 在两个矩阵 A、B 之间执行乘法运算。在MATLAB中，我们可以使用代码

A=randi(255,[128 128]);
B=randi(255,[128 128]);
f2=@()(gf(A,8)*gf(B,8))
t2=timeit(f2)

在我的电脑上大约需要 0.03 秒。我尝试使用第二种方式来加速代码：

function A = gfMatrixMult( H, G )
% Matrix multiplication where both H, G matrices have elements in GF(256)
%
% Input: 
%  H,G - Input matrices
% Output: 
%    A - Output matrix - A = H*G
%
[HROWS HCOLS] = size(H);
[GROWS GCOLS] = size(G);
if ( HCOLS ~= GROWS )
    error('Matrix sizes do not fit!')
end

A = zeros(HROWS,GCOLS);
for ii = 1:HROWS
    for jj = 1:GCOLS
        for kk = 1:HCOLS
            if ((H(ii,kk) == 0)||(G(kk,jj) == 0))
                coeff = 0;
            elseif (H(ii,kk) == 1)
                coeff = G(kk,jj);
            elseif (G(kk,jj) == 1)
                coeff = H(ii,kk);
            else
                coeff = gfmult(H(ii,kk),G(kk,jj));
            end
            A(ii,jj) = bitxor(A(ii,jj),coeff);
        end
    end
end

其中gfmult 函数用于从表中查找值。这是

function val = gfmult( u, v )

OCT_EXP = [ 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38,...
   76, 152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192, 157,...
   39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159, 35,...
   70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111, 222,...
   161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30, 60,...
   120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223, 163,...
   91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26, 52,...
   104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147, 59,...
   118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,...
   169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,...
   170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,...
   145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,...
   75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,...
   50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,...
   162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,...
   18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,...
   22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,...
   142, 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38,...
   76, 152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192,...
   157, 39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159,...
   35, 70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111,...
   222, 161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30,...
   60, 120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223,...
   163, 91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26,...
   52, 104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147,...
   59, 118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,...
   169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,...
   170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,...
   145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,...
   75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,...
   50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,...
   162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,...
   18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,...
   22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,...
   142 ];

OCT_LOG = [ 0, 1, 25, 2, 50, 26, 198, 3, 223, 51, 238, 27, 104, 199, 75, 4,...
   100, 224, 14, 52, 141, 239, 129, 28, 193, 105, 248, 200, 8, 76, 113, 5,...
   138, 101, 47, 225, 36, 15, 33, 53, 147, 142, 218, 240, 18, 130, 69,...
   29, 181, 194, 125, 106, 39, 249, 185, 201, 154, 9, 120, 77, 228, 114,... 
   166, 6, 191, 139, 98, 102, 221, 48, 253, 226, 152, 37, 179, 16, 145,...
   34, 136, 54, 208, 148, 206, 143, 150, 219, 189, 241, 210, 19, 92,...
   131, 56, 70, 64, 30, 66, 182, 163, 195, 72, 126, 110, 107, 58, 40,...
   84, 250, 133, 186, 61, 202, 94, 155, 159, 10, 21, 121, 43, 78, 212,...
   229, 172, 115, 243, 167, 87, 7, 112, 192, 247, 140, 128, 99, 13, 103,...
   74, 222, 237, 49, 197, 254, 24, 227, 165, 153, 119, 38, 184, 180,...
   124, 17, 68, 146, 217, 35, 32, 137, 46, 55, 63, 209, 91, 149, 188,...
   207, 205, 144, 135, 151, 178, 220, 252, 190, 97, 242, 86, 211, 171,...
   20, 42, 93, 158, 132, 60, 57, 83, 71, 109, 65, 162, 31, 45, 67, 216,...
   183, 123, 164, 118, 196, 23, 73, 236, 127, 12, 111, 246, 108, 161,...
   59, 82, 41, 157, 85, 170, 251, 96, 134, 177, 187, 204, 62, 90, 203,...
   89, 95, 176, 156, 169, 160, 81, 11, 245, 22, 235, 122, 117, 44, 215,...
   79, 174, 213, 233, 230, 231, 173, 232, 116, 214, 244, 234, 168, 80,...
   88, 175 ];

if (( u == 0 )||( v == 0 ))
    val = 0;
else
    val = OCT_EXP( OCT_LOG(u) + OCT_LOG(v) + 1);
      
end

这样，大约需要 11 秒。我认为通过 MATLAB 社区的支持，以下方式可以更快。你能帮我加快速度，而不是使用第一种方式吗？我推荐第二种方式，因为它是标准Sec. 5.7推荐的。

【问题讨论】：

标签： performance matlab matrix sparse-matrix mex

【解决方案1】：

这是一个矢量化版本：

function A = gfMatrixMult( H, G )
% Matrix multiplication where both H, G matrices have elements in GF(256)
%
% Input: 
%  H,G - Input matrices
% Output: 
%    A - Output matrix - A = H*G
%
    [HROWS HCOLS] = size(H);
    [GROWS GCOLS] = size(G);
    if ( HCOLS ~= GROWS )
    error('Matrix sizes do not fit!')
    end
    [ii,jj,kk] = ndgrid(1:HROWS,1:GCOLS,1:HCOLS);
    H_ = H(sub2ind([HROWS HCOLS],ii,kk));
    G_ = G(sub2ind([GROWS GCOLS],kk,jj));
    A = zeros(HROWS,GCOLS);
    coeff = zeros(size(G_));
    H_1 = (H_== 1);
    coeff(H_1) = G_(H_1);
    G_1 = (G_== 1);
    coeff(G_1) = H_(G_1);
    G_1H_1 = ~G_1 & ~H_1;
    coeff(G_1H_1) = gfmult(H_(G_1H_1), G_(G_1H_1));
    for k = 1:HCOLS
    A = bitxor(A,coeff(:,:,k));
    end
end


function val = gfmult( u, v )

persistent OCT_EXP = [ 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38,...
   76, 152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192, 157,...
   39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159, 35,...
   70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111, 222,...
   161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30, 60,...
   120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223, 163,...
   91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26, 52,...
   104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147, 59,...
   118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,...
   169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,...
   170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,...
   145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,...
   75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,...
   50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,...
   162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,...
   18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,...
   22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,...
   142, 1, 2, 4, 8, 16, 32, 64, 128, 29, 58, 116, 232, 205, 135, 19, 38,...
   76, 152, 45, 90, 180, 117, 234, 201, 143, 3, 6, 12, 24, 48, 96, 192,...
   157, 39, 78, 156, 37, 74, 148, 53, 106, 212, 181, 119, 238, 193, 159,...
   35, 70, 140, 5, 10, 20, 40, 80, 160, 93, 186, 105, 210, 185, 111,...
   222, 161, 95, 190, 97, 194, 153, 47, 94, 188, 101, 202, 137, 15, 30,...
   60, 120, 240, 253, 231, 211, 187, 107, 214, 177, 127, 254, 225, 223,...
   163, 91, 182, 113, 226, 217, 175, 67, 134, 17, 34, 68, 136, 13, 26,...
   52, 104, 208, 189, 103, 206, 129, 31, 62, 124, 248, 237, 199, 147,...
   59, 118, 236, 197, 151, 51, 102, 204, 133, 23, 46, 92, 184, 109, 218,...
   169, 79, 158, 33, 66, 132, 21, 42, 84, 168, 77, 154, 41, 82, 164, 85,...
   170, 73, 146, 57, 114, 228, 213, 183, 115, 230, 209, 191, 99, 198,...
   145, 63, 126, 252, 229, 215, 179, 123, 246, 241, 255, 227, 219, 171,...
   75, 150, 49, 98, 196, 149, 55, 110, 220, 165, 87, 174, 65, 130, 25,...
   50, 100, 200, 141, 7, 14, 28, 56, 112, 224, 221, 167, 83, 166, 81,...
   162, 89, 178, 121, 242, 249, 239, 195, 155, 43, 86, 172, 69, 138, 9,...
   18, 36, 72, 144, 61, 122, 244, 245, 247, 243, 251, 235, 203, 139, 11,...
   22, 44, 88, 176, 125, 250, 233, 207, 131, 27, 54, 108, 216, 173, 71,...
   142 ];

persistent OCT_LOG = [ 0, 1, 25, 2, 50, 26, 198, 3, 223, 51, 238, 27, 104, 199, 75, 4,...
   100, 224, 14, 52, 141, 239, 129, 28, 193, 105, 248, 200, 8, 76, 113, 5,...
   138, 101, 47, 225, 36, 15, 33, 53, 147, 142, 218, 240, 18, 130, 69,...
   29, 181, 194, 125, 106, 39, 249, 185, 201, 154, 9, 120, 77, 228, 114,... 
   166, 6, 191, 139, 98, 102, 221, 48, 253, 226, 152, 37, 179, 16, 145,...
   34, 136, 54, 208, 148, 206, 143, 150, 219, 189, 241, 210, 19, 92,...
   131, 56, 70, 64, 30, 66, 182, 163, 195, 72, 126, 110, 107, 58, 40,...
   84, 250, 133, 186, 61, 202, 94, 155, 159, 10, 21, 121, 43, 78, 212,...
   229, 172, 115, 243, 167, 87, 7, 112, 192, 247, 140, 128, 99, 13, 103,...
   74, 222, 237, 49, 197, 254, 24, 227, 165, 153, 119, 38, 184, 180,...
   124, 17, 68, 146, 217, 35, 32, 137, 46, 55, 63, 209, 91, 149, 188,...
   207, 205, 144, 135, 151, 178, 220, 252, 190, 97, 242, 86, 211, 171,...
   20, 42, 93, 158, 132, 60, 57, 83, 71, 109, 65, 162, 31, 45, 67, 216,...
   183, 123, 164, 118, 196, 23, 73, 236, 127, 12, 111, 246, 108, 161,...
   59, 82, 41, 157, 85, 170, 251, 96, 134, 177, 187, 204, 62, 90, 203,...
   89, 95, 176, 156, 169, 160, 81, 11, 245, 22, 235, 122, 117, 44, 215,...
   79, 174, 213, 233, 230, 231, 173, 232, 116, 214, 244, 234, 168, 80,...
   88, 175 ];

    uv0 =  (~(( u == 0 )|( v == 0 )));
    val = zeros(size(u));
    val(uv0) = OCT_EXP( OCT_LOG(u(uv0)) + OCT_LOG(v(uv0)) + 1);

end

【讨论】：

不错的选择。但是，您的代码看起来像 Octave 风格。我更改了一些东西以在 MATLAB 中使用。最后一行看起来是val = (~(( u == 0 )|( v == 0 )))' .* OCT_EXP( OCT_LOG(u) + OCT_LOG(v) + 1);，而不是val = (~(( u == 0 )|( v == 0 ))).* OCT_EXP( OCT_LOG(u) + OCT_LOG(v) + 1); 你同意吗？
@user3051460 是的，您是正确的，答案已更新。 && 和 ||改为 |和＆。应该应用转置
谢谢。在我原来的gfmult 函数中（输入是两个比例数，而不是向量）。我试图将该函数转换为 mex 代码以加速它。但是，它并没有太大的优势。你认为我们可以用两个刻度数字u,v 输入加速我的gfmul 函数吗，例如u=200,v=200
我不明白，你能解释一下吗？现在你是两个比例数字。应该改变什么？
抱歉我的英语不好。只是忘记了您的解决方案并在我的问题中返回我的代码。在我的代码中，函数gfmult 用于查找输入为u,v 的比例值。因为函数被多次调用，所以函数的耗时会影响到整个项目。我也想加快这个功能。在您的代码中，您通过使用矢量化方式加速了它，但它需要更改两个函数，在我的项目中，函数 gfmult 被称为多个位置，因此，很难将所有更改为矢量化。因此，我试图保持函数的输入为刻度值