Question

zlib＆＃39; crc32_combine（）接受crcA，crcB和lengthB来计算crcAB。

# returns crcAB
crc32_combine(crcA, crcB, lenB)

使用Mark Adler的精彩帖子here和here中的概念，我能够生成crc32_trim_trailing.pl，它需要crcAB，crcB和lengthB来计算crcA（我用这个剥掉已知长度和值的填充物。）

# prints crcA
perl crc32_trim_trailing.pl $crcAB $crcB $lenB

不幸的是，这使用了所描述的慢速方法的原理，其中每个空字节必须一次剥离一个。它很慢，但却是一个很好的概念证明。

我一直致力于制作crc32_trim_trailing的快速版本，该版本利用了Mark的帖子中描述的矩阵技巧，并在zlib的crc32_combine（）中实现了组合用例。< / p>

这是我对crc32_trim_trailing.c。

的尝试

/* crc32_trim_trailing.c
  This code is borrows heavily from crc32.c from zlib version 1.2.8, but has
  been altered.
*/

#include <stdio.h>

#define GF2_DIM 32      /* dimension of GF(2) vectors (length of CRC) */

/* ========================================================================= */
unsigned long gf2_matrix_times(mat, vec)
    unsigned long *mat;
    unsigned long vec;
{
    unsigned long sum;

    sum = 0;
    while (vec) {
        if (vec & 1)
            sum ^= *mat;
        vec >>= 1;
        mat++;
    }
    return sum;
}

/* ========================================================================= */
void gf2_matrix_square(square, mat)
    unsigned long *square;
    unsigned long *mat;
{
    int n;

    for (n = 0; n < GF2_DIM; n++)
        square[n] = gf2_matrix_times(mat, mat[n]);
}

/* ========================================================================= */
int main(int argc, char *argv[])
{
    unsigned long crc1;
    unsigned long crc2;
    int len2;

    sscanf(argv[1], "%lx", &crc1);
    sscanf(argv[2], "%lx", &crc2);
    sscanf(argv[3],  "%d", &len2);

    int n;
    unsigned long row;
    unsigned long even[GF2_DIM];    /* even-power-of-two zeros operator */
    unsigned long odd[GF2_DIM];     /* odd-power-of-two zeros operator */

    /* degenerate case (also disallow negative lengths) */
    if (len2 <= 0)
        return crc1;

    /* get crcA0 */
    crc1 ^= crc2;

    /* put operator for one zero bit in odd */
    odd[0] = 0x82608edbUL;          /* used sage math to get inverse matrix polynomial */
    row = 1;
    for (n = 1; n < GF2_DIM; n++) {
        odd[n] = row;
        row <<= 1;
    }

    /* put operator for two zero bits in even */
    gf2_matrix_square(even, odd);

    /* put operator for four zero bits in odd */
    gf2_matrix_square(odd, even);

    /* apply len2 zeros to crc1 (first square will put the operator for one
       zero byte, eight zero bits, in even) */
    do {
        /* apply zeros operator for this bit of len2 */
        gf2_matrix_square(even, odd);
        if (len2 & 1)
            crc1 = gf2_matrix_times(even, crc1);
        len2 >>= 1;

        /* if no more bits set, then done */
        if (len2 == 0)
            break;

        /* another iteration of the loop with odd and even swapped */
        gf2_matrix_square(odd, even);
        if (len2 & 1)
            crc1 = gf2_matrix_times(odd, crc1);
        len2 >>= 1;

        /* if no more bits set, then done */
    } while (len2 != 0);

    printf("\nCRC: %lx\n", crc1);

    return 0;
}

我将xor移动到矩阵乘法之前。这似乎没有问题，并通过xoring crcAB和crcB给我们crcA0。

接下来，使用sage数学我能够找到crc32_combine（）中使用的初始矩阵的逆矩阵。

通过3个方格运行这些基质中的每一个导致矩阵crc32_combine（）用于添加1个空字节（matrixA）并且它是反向的（matrixB）。

使用sage数学我确认了以下内容。

matrixA * matrixB = identity
crc * identity = crc
crc * matrixA * matrixB = crc

码

M = MatrixSpace(GF(2),32,32)
A = M([0,1,1,1,0,1,1,1,0,0,0,0,0,1,1,1,0,0,1,1,0,0,0,0,1,0,0,1,0,1,1,0,
1,1,1,0,1,1,1,0,0,0,0,0,1,1,1,0,0,1,1,0,0,0,0,1,0,0,1,0,1,1,0,0,
0,0,0,0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,1,
0,0,0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,
0,0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,
0,0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,
0,1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,0,
1,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0])

B = A^-1

I = A*B

print "matrixA"
print A.str()
print "matrixB"
print B.str()
print "identity"
print I.str()

N = MatrixSpace(GF(2),1,32)
THIS=N([1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,1,1,1])

print "'this' crc * identity"
print THIS * I
print "'this' crc * maxtrixA"
print THIS * A
print "'this' crc * maxtrixA * matrixB"
print THIS * A * B

输出：

matrixA
[0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0]
[1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0]
[0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1]
[0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0]
[0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0]
[0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0]
[0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0]
[1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
matrixB
[1 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 0 0 1 1 0]
[0 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 1 1]
[1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 1 1 0]
[0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1]
[1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 1 0]
[0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1]
[1 1 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0]
[1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
identity
[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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0]
[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 0 0 0 0 0 0]
[0 0 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 0 0 0 0]
[0 0 0 0 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 0 0]
[0 0 0 0 0 0 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]
'this' crc * identity
[1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1]
'this' crc * maxtrixA
[1 1 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0]
'this' crc * maxtrixA * matrixB
[1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1]

我使用crc和单位矩阵测试了gf2_matrix_times（），正如预期的那样导致crc没有变化。

由于gf2_matrix_times（crc，matrixA）可用于向crc添加1个空字节，我曾希望gf2_matrix_times（crc，matrixB）可用于删除1个空字节。但是，这似乎没有开箱即用。

此外，当lengthB为1时，sage数学中的crc * matrixA与crc32_combine（）中的crcA0（0xa5f45be9）产生不同的结果（0xc05e2dda）。

为什么sage math和gf2_matrix_times（）之间的GF（2）矩阵乘法存在差异？当matrixA和matrixB是反向时，为什么gf2_matrix_times（crc，matrixB）不会反转gf2_matrix_times（crc，matrixA）？

Answer 1

我们将首先看一下标准CRC-32的简单逐位实现（作为CRC的自包含定义，此例程返回初始CRC，即空字符串的CRC，当{ {1}}是data）：

NULL

我们可以简化将#include <stddef.h> #include <stdint.h> #define POLY 0xedb88320 uint32_t crc32(uint32_t crc, void const *data, size_t len) { if (data == NULL) return 0; crc = ~crc; while (len--) { crc ^= *(unsigned char const *)data++; for (int k = 0; k < 8; k++) crc = crc & 1 ? (crc >> 1) ^ POLY : crc >> 1; } crc = ~crc; return crc; }零应用于CRC：

现在让我们仔细研究单个零位到CRC的应用：

uint32_t crc32_zeros(uint32_t crc, size_t n) {
    crc = ~crc;
    while (n--)
        for (int k = 0; k < 8; k++)
            crc = crc & 1 ? (crc >> 1) ^ POLY : crc >> 1;
    crc = ~crc;
    return crc;
}

应用该位时可以采用两种路径。在最后一个操作中，多项式要么与CRC排除，要么不是。如果我们想要扭转这种局面，我们想知道它走了哪条路。

我们可以通过查看结果的高位来判断。我们可以看到，如果多项式不是排他性的，那么高位必须为0.但如果它被排他性呢？在这种情况下，结果的高位是crc = crc & 1 ? (crc >> 1) ^ POLY : crc >> 1;的高位。我们可以看到高位是1.所以我们可以通过查看结果的高位来判断。事实上，对于任何有效的CRC多项式都必须如此，因为对于x ⁰项，它们都具有系数1。（该项在该反射多项式的高位中。）

通过检查，我们可以轻松地反转该操作，其中POLY进入后是应用0位后的最终CRC，而crc出现的是CRC在应用0位之前的情况：

crc

这将采用最终CRC并反转在单个0位上计算CRC的操作。请注意，我们必须插入导致异或的低1位。

我们可以将crc = crc & 0x80000000 ? ((crc ^ POLY) << 1) + 1 : crc << 1;分解为：

POLY

这与使用多项式crc = crc & 0x80000000 ? (crc << 1) ^ ((POLY << 1) + 1) : crc << 1;将追加 0位到非 - 反射的CRC的操作完全相同，这只是{{向左旋转一位。

然后我们可以编写一个函数来从标准CRC-32中删除(POLY << 1) + 1个零字节：

POLY

现在我们可以使用zlib中使用的相同方法，但是使用非反射CRC，编写一个函数来从O中的CRC-32中删除 n 零（log（ n ））时间。我们不需要反转任何矩阵，因为我们已经颠倒了原始操作。

其余部分留给读者练习。

crc32_combine（）的矩阵技巧的反转是什么？

1 个答案: