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crc32_combine（）的矩陣技巧的反轉是什么？

[英]What is the inverse of crc32_combine()'s matrix trick?

 原文  2016-07-30 04:58:16  2   1    matrix/ zlib/ checksum/ crc

問題描述

zlib的crc32_combine（）接受crcA，crcB和lengthB來計算crcAB。 

# returns crcAB
crc32_combine(crcA, crcB, lenB)

使用Mark Adler 在這里和這里的精彩帖子中的概念，我能夠生成crc32_trim_trailing.pl ，它使用crcAB，crcB和lengthB來計算crcA（我用它來剝離已知長度和值的填充）。 

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

不幸的是，這使用了所描述的慢速方法的原理，其中每個空字節必須一次剝離一個。 它很慢，但卻是一個很好的概念證明。

我一直在努力制作crc32_trim_trailing的快速版本，它利用了Mark的帖子中描述的矩陣技巧，並在zlib的crc32_combine（）中實現了組合用例。

這是我對crc32_trim_trailing.c的嘗試。 

/* 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;
}

我把xor移到了矩陣乘法之前。 這似乎沒有問題，並通過xoring crcAB和crcB給我們crcA0。

接下來，使用sage數學我能夠找到crc32_combine（）中使用的初始矩陣的逆矩陣。

通過3個方塊運行這些基質中的每一個導致矩陣crc32_combine（）用於添加1個空字節（matrixA）並且它是反向的（matrixB）。

使用sage數學我確認了以下內容。

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

碼： 

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

輸出： 

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和單位矩陣測試了gf2_matrix_times（），如預期的那樣導致crc沒有變化。

由於gf2_matrix_times（crc，matrixA）可用於向crc添加1個空字節，我曾希望gf2_matrix_times（crc，matrixB）可用於刪除1個空字節。 但是，這似乎不是開箱即用的。

此外，當lengthB為1時，sc數學中的crc * matrixA與crc32_combine（）中的crcA0（0xa5f45be9）產生不同的結果（0xc05e2dda）。

為什么sage數學和gf2_matrix_times（）之間的GF（2）矩陣乘法存在差異？ 當matrixA和matrixB是反向時，為什么gf2_matrix_times（crc，matrixB）不會反轉gf2_matrix_times（crc，matrixA）？

1 個解決方案

解決方案1
 4 已采納  2016-07-30 16:48:26

我們將首先看一下標准CRC-32的簡單逐位實現（作為CRC的自包含定義，此例程返回初始CRC，即空字符串的CRC，當data為NULL ） ： 

#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;
}

我們可以簡化將n零應用於CRC的方法： 

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;
}

現在讓我們仔細研究單個零位到CRC的應用： 

crc = crc & 1 ? (crc >> 1) ^ POLY : crc >> 1;

應用該位時可以采用兩種路徑。 在最后一個操作中，多項式要么與CRC排除，要么不是。 如果我們想要扭轉這種局面，我們想知道它走向何方。

我們可以通過查看結果的高位來判斷。 我們可以看到，如果多項式不是排他性的，那么高位必須為0.但如果它被排他性呢？ 在這種情況下，結果的高位是POLY的高位。 我們可以看到高位是1.所以我們可以通過查看結果的高位來判斷。 事實上，對於任何有效的CRC多項式都必須如此，因為x 0項都具有系數1。 （該項對於此反射多項式處於高位。）

通過檢查，我們可以輕松地反轉該操作，其中crc進入是應用0位后的最終CRC，並且crc出來是應用0位之前的CRC： 

crc = crc & 0x80000000 ? ((crc ^ POLY) << 1) + 1 : crc << 1;

這將采用最終的CRC並反轉在單個0位上計算CRC的動作。 請注意，我們必須插入可能導致異或的低1位，對於這種情況。

我們可以將POLY為： 

crc = crc & 0x80000000 ? (crc << 1) ^ ((POLY << 1) + 1) : crc << 1;

這與使用多項式(POLY << 1) + 1 將 0位附加到非反射CRC的操作完全相同，該多項式只是POLY向左旋轉一位。

然后我們可以編寫一個函數來從標准CRC-32中刪除n零字節： 

#define UNPOLY ((POLY << 1) + 1)

uint32_t crc32_remove_zeros(uint32_t crc, size_t n) {
    crc = ~crc;
    while (n--)
        for (int k = 0; k < 8; k++)
            crc = crc & 0x80000000 ? (crc << 1) ^ UNPOLY : crc << 1;
    crc = ~crc;
    return crc;
}

現在我們可以使用zlib中使用的相同方法，但是使用非反射CRC，編寫一個函數，在O（log（ n ））時間內從CRC-32中刪除n個零。 我們不需要反轉任何矩陣，因為我們已經顛倒了原始操作。

其余部分留給讀者練習。

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