反转 CRC32(a)
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【中文标题】反转 CRC32(a)【英文标题】：Reversing CRC32(a) 【发布时间】：2021-11-15 06:45:43 【问题描述】：我正在尝试将我在本文档中阅读的内容付诸实践： https://sar.informatik.hu-berlin.de/research/publications/SAR-PR-2006-05/SAR-PR-2006-05_.pdf
我使用的是 CRC32，Normal Polynimal = 0x04C11DB7。我建立表格和反向表格。 CRC的表应该是对的，但我不知道反向表是否正确。您将在下面找到生成表格并尝试将 crc 强制到固定位置的代码。
你能给我一些建议吗？我哪里做错了？
#define CRCPOLY 0x04C11DB7
#define CRCPOLYR 0xEDB88320
#define INITXOR 0xFFFFFFFF
#define FINALXOR 0xFFFFFFFF
static uint32_t crc_table[256];
static uint32_t crc_revtable[256];
void make_crc_table()
for (uint32_t byte = 0; byte <= 0xFF; byte++ )
uint32_t crc = (byte << 24);
for (uint8_t bit = 0; bit < 8; bit++ )
if (crc & 0x80000000)
crc = (crc << 1) ^ CRCPOLY;
else
crc = (crc << 1);
crc_table[ byte ] = crc;
void make_crc_revtable()
for (uint32_t byte = 0; byte < 256; byte++)
uint32_t crc = byte;
for (uint8_t bit = 0; bit < 8; bit++)
if ((crc & 1) != 0)
crc = (crc >> 1) ^ CRCPOLYR;
else
crc >>= 1;
crc_revtable[byte] = crc;
int crc32(uint8_t *buffer, size_t length)
uint32_t crcreg = INITXOR;
for (size_t i = 0; i < length; ++i)
crcreg = crc_table[ (buffer[i] ^ (crcreg >> 24)) & 0xFF ] ^ (crcreg << 8);
return crcreg ^ FINALXOR;
void fix_crc_pos(uint8_t *buffer, int length, uint32_t tcrcreg, int fix_pos)
int i;
// make sure fix_pos is within 0..(length-1)
fix_pos = ((fix_pos % length) + length) % length;
// calculate crc register at position fix_pos; this is essentially crc32()
uint32_t crcreg = INITXOR;
for (i = 0; i < fix_pos; ++i)
crcreg = crc_table[ (buffer[i] ^ (crcreg >> 24)) & 0xFF ] ^ (crcreg << 8);
// inject crcreg as content
for (i = 0; i < 4; ++i)
buffer[fix_pos + i] = (crcreg >> i * 8) & 0xFF;
// calculate crc backwards to fix_pos, beginning at the end
tcrcreg ^= FINALXOR;
for (i = length - 1; i >= fix_pos; --i)
tcrcreg = crc_revtable[tcrcreg & 0xFF] ^ buffer[i] ^ (tcrcreg >> 8);
// inject new content
for (i = 0; i < 4; ++i)
buffer[fix_pos + i] = (tcrcreg >> i * 8) & 0xFF;
int main()
make_crc_table();
make_crc_revtable();
// Check valid CRC32(a)
char crc32_check_txt[] = "123456789";
uint32_t crc32_check = 0xFC891918;
uint32_t crc = crc32((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt));
if(crc != crc32_check)
return -1;
// Change 1 byte and restore the previus CRC
crc32_check_txt[0] = '4';
fix_crc_pos((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt), crc32_check, 4);
// Verify CRC32
crc = crc32((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt));
if(crc != crc32_check)
return -1;
return 0;
表格：
CRCa Table
0x00000000, 0x04c11db7, 0x09823b6e, 0x0d4326d9, 0x130476dc, 0x17c56b6b, 0x1a864db2, 0x1e475005,
0x2608edb8, 0x22c9f00f, 0x2f8ad6d6, 0x2b4bcb61, 0x350c9b64, 0x31cd86d3, 0x3c8ea00a, 0x384fbdbd,
0x4c11db70, 0x48d0c6c7, 0x4593e01e, 0x4152fda9, 0x5f15adac, 0x5bd4b01b, 0x569796c2, 0x52568b75,
0x6a1936c8, 0x6ed82b7f, 0x639b0da6, 0x675a1011, 0x791d4014, 0x7ddc5da3, 0x709f7b7a, 0x745e66cd,
0x9823b6e0, 0x9ce2ab57, 0x91a18d8e, 0x95609039, 0x8b27c03c, 0x8fe6dd8b, 0x82a5fb52, 0x8664e6e5,
0xbe2b5b58, 0xbaea46ef, 0xb7a96036, 0xb3687d81, 0xad2f2d84, 0xa9ee3033, 0xa4ad16ea, 0xa06c0b5d,
0xd4326d90, 0xd0f37027, 0xddb056fe, 0xd9714b49, 0xc7361b4c, 0xc3f706fb, 0xceb42022, 0xca753d95,
0xf23a8028, 0xf6fb9d9f, 0xfbb8bb46, 0xff79a6f1, 0xe13ef6f4, 0xe5ffeb43, 0xe8bccd9a, 0xec7dd02d,
0x34867077, 0x30476dc0, 0x3d044b19, 0x39c556ae, 0x278206ab, 0x23431b1c, 0x2e003dc5, 0x2ac12072,
0x128e9dcf, 0x164f8078, 0x1b0ca6a1, 0x1fcdbb16, 0x018aeb13, 0x054bf6a4, 0x0808d07d, 0x0cc9cdca,
0x7897ab07, 0x7c56b6b0, 0x71159069, 0x75d48dde, 0x6b93dddb, 0x6f52c06c, 0x6211e6b5, 0x66d0fb02,
0x5e9f46bf, 0x5a5e5b08, 0x571d7dd1, 0x53dc6066, 0x4d9b3063, 0x495a2dd4, 0x44190b0d, 0x40d816ba,
0xaca5c697, 0xa864db20, 0xa527fdf9, 0xa1e6e04e, 0xbfa1b04b, 0xbb60adfc, 0xb6238b25, 0xb2e29692,
0x8aad2b2f, 0x8e6c3698, 0x832f1041, 0x87ee0df6, 0x99a95df3, 0x9d684044, 0x902b669d, 0x94ea7b2a,
0xe0b41de7, 0xe4750050, 0xe9362689, 0xedf73b3e, 0xf3b06b3b, 0xf771768c, 0xfa325055, 0xfef34de2,
0xc6bcf05f, 0xc27dede8, 0xcf3ecb31, 0xcbffd686, 0xd5b88683, 0xd1799b34, 0xdc3abded, 0xd8fba05a,
0x690ce0ee, 0x6dcdfd59, 0x608edb80, 0x644fc637, 0x7a089632, 0x7ec98b85, 0x738aad5c, 0x774bb0eb,
0x4f040d56, 0x4bc510e1, 0x46863638, 0x42472b8f, 0x5c007b8a, 0x58c1663d, 0x558240e4, 0x51435d53,
0x251d3b9e, 0x21dc2629, 0x2c9f00f0, 0x285e1d47, 0x36194d42, 0x32d850f5, 0x3f9b762c, 0x3b5a6b9b,
0x0315d626, 0x07d4cb91, 0x0a97ed48, 0x0e56f0ff, 0x1011a0fa, 0x14d0bd4d, 0x19939b94, 0x1d528623,
0xf12f560e, 0xf5ee4bb9, 0xf8ad6d60, 0xfc6c70d7, 0xe22b20d2, 0xe6ea3d65, 0xeba91bbc, 0xef68060b,
0xd727bbb6, 0xd3e6a601, 0xdea580d8, 0xda649d6f, 0xc423cd6a, 0xc0e2d0dd, 0xcda1f604, 0xc960ebb3,
0xbd3e8d7e, 0xb9ff90c9, 0xb4bcb610, 0xb07daba7, 0xae3afba2, 0xaafbe615, 0xa7b8c0cc, 0xa379dd7b,
0x9b3660c6, 0x9ff77d71, 0x92b45ba8, 0x9675461f, 0x8832161a, 0x8cf30bad, 0x81b02d74, 0x857130c3,
0x5d8a9099, 0x594b8d2e, 0x5408abf7, 0x50c9b640, 0x4e8ee645, 0x4a4ffbf2, 0x470cdd2b, 0x43cdc09c,
0x7b827d21, 0x7f436096, 0x7200464f, 0x76c15bf8, 0x68860bfd, 0x6c47164a, 0x61043093, 0x65c52d24,
0x119b4be9, 0x155a565e, 0x18197087, 0x1cd86d30, 0x029f3d35, 0x065e2082, 0x0b1d065b, 0x0fdc1bec,
0x3793a651, 0x3352bbe6, 0x3e119d3f, 0x3ad08088, 0x2497d08d, 0x2056cd3a, 0x2d15ebe3, 0x29d4f654,
0xc5a92679, 0xc1683bce, 0xcc2b1d17, 0xc8ea00a0, 0xd6ad50a5, 0xd26c4d12, 0xdf2f6bcb, 0xdbee767c,
0xe3a1cbc1, 0xe760d676, 0xea23f0af, 0xeee2ed18, 0xf0a5bd1d, 0xf464a0aa, 0xf9278673, 0xfde69bc4,
0x89b8fd09, 0x8d79e0be, 0x803ac667, 0x84fbdbd0, 0x9abc8bd5, 0x9e7d9662, 0x933eb0bb, 0x97ffad0c,
0xafb010b1, 0xab710d06, 0xa6322bdf, 0xa2f33668, 0xbcb4666d, 0xb8757bda, 0xb5365d03, 0xb1f740b4,
CRCa Rev Table
0x00000000, 0x77073096, 0xee0e612c, 0x990951ba, 0x076dc419, 0x706af48f, 0xe963a535, 0x9e6495a3,
0x0edb8832, 0x79dcb8a4, 0xe0d5e91e, 0x97d2d988, 0x09b64c2b, 0x7eb17cbd, 0xe7b82d07, 0x90bf1d91,
0x1db71064, 0x6ab020f2, 0xf3b97148, 0x84be41de, 0x1adad47d, 0x6ddde4eb, 0xf4d4b551, 0x83d385c7,
0x136c9856, 0x646ba8c0, 0xfd62f97a, 0x8a65c9ec, 0x14015c4f, 0x63066cd9, 0xfa0f3d63, 0x8d080df5,
0x3b6e20c8, 0x4c69105e, 0xd56041e4, 0xa2677172, 0x3c03e4d1, 0x4b04d447, 0xd20d85fd, 0xa50ab56b,
0x35b5a8fa, 0x42b2986c, 0xdbbbc9d6, 0xacbcf940, 0x32d86ce3, 0x45df5c75, 0xdcd60dcf, 0xabd13d59,
0x26d930ac, 0x51de003a, 0xc8d75180, 0xbfd06116, 0x21b4f4b5, 0x56b3c423, 0xcfba9599, 0xb8bda50f,
0x2802b89e, 0x5f058808, 0xc60cd9b2, 0xb10be924, 0x2f6f7c87, 0x58684c11, 0xc1611dab, 0xb6662d3d,
0x76dc4190, 0x01db7106, 0x98d220bc, 0xefd5102a, 0x71b18589, 0x06b6b51f, 0x9fbfe4a5, 0xe8b8d433,
0x7807c9a2, 0x0f00f934, 0x9609a88e, 0xe10e9818, 0x7f6a0dbb, 0x086d3d2d, 0x91646c97, 0xe6635c01,
0x6b6b51f4, 0x1c6c6162, 0x856530d8, 0xf262004e, 0x6c0695ed, 0x1b01a57b, 0x8208f4c1, 0xf50fc457,
0x65b0d9c6, 0x12b7e950, 0x8bbeb8ea, 0xfcb9887c, 0x62dd1ddf, 0x15da2d49, 0x8cd37cf3, 0xfbd44c65,
0x4db26158, 0x3ab551ce, 0xa3bc0074, 0xd4bb30e2, 0x4adfa541, 0x3dd895d7, 0xa4d1c46d, 0xd3d6f4fb,
0x4369e96a, 0x346ed9fc, 0xad678846, 0xda60b8d0, 0x44042d73, 0x33031de5, 0xaa0a4c5f, 0xdd0d7cc9,
0x5005713c, 0x270241aa, 0xbe0b1010, 0xc90c2086, 0x5768b525, 0x206f85b3, 0xb966d409, 0xce61e49f,
0x5edef90e, 0x29d9c998, 0xb0d09822, 0xc7d7a8b4, 0x59b33d17, 0x2eb40d81, 0xb7bd5c3b, 0xc0ba6cad,
0xedb88320, 0x9abfb3b6, 0x03b6e20c, 0x74b1d29a, 0xead54739, 0x9dd277af, 0x04db2615, 0x73dc1683,
0xe3630b12, 0x94643b84, 0x0d6d6a3e, 0x7a6a5aa8, 0xe40ecf0b, 0x9309ff9d, 0x0a00ae27, 0x7d079eb1,
0xf00f9344, 0x8708a3d2, 0x1e01f268, 0x6906c2fe, 0xf762575d, 0x806567cb, 0x196c3671, 0x6e6b06e7,
0xfed41b76, 0x89d32be0, 0x10da7a5a, 0x67dd4acc, 0xf9b9df6f, 0x8ebeeff9, 0x17b7be43, 0x60b08ed5,
0xd6d6a3e8, 0xa1d1937e, 0x38d8c2c4, 0x4fdff252, 0xd1bb67f1, 0xa6bc5767, 0x3fb506dd, 0x48b2364b,
0xd80d2bda, 0xaf0a1b4c, 0x36034af6, 0x41047a60, 0xdf60efc3, 0xa867df55, 0x316e8eef, 0x4669be79,
0xcb61b38c, 0xbc66831a, 0x256fd2a0, 0x5268e236, 0xcc0c7795, 0xbb0b4703, 0x220216b9, 0x5505262f,
0xc5ba3bbe, 0xb2bd0b28, 0x2bb45a92, 0x5cb36a04, 0xc2d7ffa7, 0xb5d0cf31, 0x2cd99e8b, 0x5bdeae1d,
0x9b64c2b0, 0xec63f226, 0x756aa39c, 0x026d930a, 0x9c0906a9, 0xeb0e363f, 0x72076785, 0x05005713,
0x95bf4a82, 0xe2b87a14, 0x7bb12bae, 0x0cb61b38, 0x92d28e9b, 0xe5d5be0d, 0x7cdcefb7, 0x0bdbdf21,
0x86d3d2d4, 0xf1d4e242, 0x68ddb3f8, 0x1fda836e, 0x81be16cd, 0xf6b9265b, 0x6fb077e1, 0x18b74777,
0x88085ae6, 0xff0f6a70, 0x66063bca, 0x11010b5c, 0x8f659eff, 0xf862ae69, 0x616bffd3, 0x166ccf45,
0xa00ae278, 0xd70dd2ee, 0x4e048354, 0x3903b3c2, 0xa7672661, 0xd06016f7, 0x4969474d, 0x3e6e77db,
0xaed16a4a, 0xd9d65adc, 0x40df0b66, 0x37d83bf0, 0xa9bcae53, 0xdebb9ec5, 0x47b2cf7f, 0x30b5ffe9,
0xbdbdf21c, 0xcabac28a, 0x53b39330, 0x24b4a3a6, 0xbad03605, 0xcdd70693, 0x54de5729, 0x23d967bf,
0xb3667a2e, 0xc4614ab8, 0x5d681b02, 0x2a6f2b94, 0xb40bbe37, 0xc30c8ea1, 0x5a05df1b, 0x2d02ef8d,
【问题讨论】：
【参考方案1】：我修复了问题中的代码。一个问题是 CRC 注入以相反的顺序存储字节。我展开了注入代码的循环以明确排序。生成反向 CRC 表的代码使用正常的 CRC。将 CRC xor 的缓冲区数据反转为 CRC 的最高有效字节的代码。
#include <iostream>
#define CRCPOLY 0x04C11DB7
#define INITXOR 0xFFFFFFFF
#define FINALXOR 0xFFFFFFFF
static uint32_t crc_table[256];
static uint32_t crc_revtable[256];
void make_crc_table()
for (uint32_t byte = 0; byte <= 0xFF; byte++ )
uint32_t crc = (byte << 24);
for (uint8_t bit = 0; bit < 8; bit++ )
if (crc & 0x80000000)
crc = (crc << 1) ^ CRCPOLY;
else
crc = (crc << 1);
crc_table[ byte ] = crc;
void make_crc_revtable()
for (uint32_t byte = 0; byte < 256; byte++)
uint32_t crc = byte;
for (uint8_t bit = 0; bit < 8; bit++)
if ((crc & 1) != 0)
crc = (crc >> 1) ^ (CRCPOLY >> 1) ^ 0x80000000u;
else
crc >>= 1;
crc_revtable[byte] = crc;
int crc32(uint8_t *buffer, size_t length)
uint32_t crcreg = INITXOR;
for (size_t i = 0; i < length; ++i)
crcreg = crc_table[ (buffer[i] ^ (crcreg >> 24)) & 0xFF ] ^ (crcreg << 8);
return crcreg ^ FINALXOR;
void fix_crc_pos(uint8_t *buffer, int length, uint32_t tcrcreg, int fix_pos)
int i;
// make sure fix_pos is within 0..(length-1)
fix_pos = ((fix_pos % length) + length) % length;
// calculate crc register at position fix_pos; this is essentially crc32()
uint32_t crcreg = INITXOR;
for (i = 0; i < fix_pos; ++i)
crcreg = crc_table[(buffer[i]^(crcreg>>24))&0xff]^(crcreg<<8);
// inject crcreg as content
buffer[fix_pos + 0] = (crcreg >> 24) & 0xFF;
buffer[fix_pos + 1] = (crcreg >> 16) & 0xFF;
buffer[fix_pos + 2] = (crcreg >> 8) & 0xFF;
buffer[fix_pos + 3] = (crcreg >> 0) & 0xFF;
// calculate crc backwards to fix_pos, beginning at the end
tcrcreg ^= FINALXOR;
for (i = length - 1; i >= fix_pos; --i)
tcrcreg = crc_revtable[tcrcreg&0xff]^(tcrcreg>>8)^(((uint32_t)buffer[i])<<24);
// inject new content
buffer[fix_pos + 0] = (tcrcreg >> 24) & 0xFF;
buffer[fix_pos + 1] = (tcrcreg >> 16) & 0xFF;
buffer[fix_pos + 2] = (tcrcreg >> 8) & 0xFF;
buffer[fix_pos + 3] = (tcrcreg >> 0) & 0xFF;
int main()
make_crc_table();
make_crc_revtable();
// Check valid CRC32(a)
char crc32_check_txt[] = "123456789";
uint32_t crc32_check = 0xFC891918;
uint32_t crc = crc32((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt));
if(crc != crc32_check)
std::cout << "error on forward crc\n";
return 0;
// Change 1 byte and restore the previus CRC
crc32_check_txt[0] = '4';
fix_crc_pos((uint8_t*)crc32_check_txt, (int) std::strlen(crc32_check_txt), crc32_check, 4);
// Verify CRC32
crc = crc32((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt));
if(crc != crc32_check)
std::cout << "error on backward crc\n";
return 0;
return 0;
这是另一种方法。要向后循环 CRC，您可以将 CRC 乘以 1/pow(2, 要向后循环的位数) == CRC * pow(2, -1-(要向后循环的位数) % CRC_POLY。
#include <iostream>
#define CRCPOLY 0x04C11DB7
#define INITXOR 0xFFFFFFFF
#define FINALXOR 0xFFFFFFFF
static uint32_t crc_table[256];
void make_crc_table()
for (uint32_t byte = 0; byte <= 0xFF; byte++ )
uint32_t crc = (byte << 24);
for (uint8_t bit = 0; bit < 8; bit++ )
if (crc & 0x80000000)
crc = (crc << 1) ^ CRCPOLY;
else
crc = (crc << 1);
crc_table[ byte ] = crc;
int crc32(uint8_t *buffer, size_t length)
uint32_t crcreg = INITXOR;
for (size_t i = 0; i < length; ++i)
crcreg = crc_table[ (buffer[i] ^ (crcreg >> 24)) & 0xFF ] ^ (crcreg << 8);
return crcreg ^ FINALXOR;
// carryless multiply modulo crc polynomial
uint32_t MpyModCrc(uint32_t a, uint32_t b) // (a*b)%crc
uint32_t pd = 0;
uint32_t i;
for(i = 0; i < 32; i++)
pd = (pd<<1)^((0-(pd>>31))&0x04c11db7u);
pd ^= (0-(b>>31))&a;
b <<= 1;
return pd;
// exponentiate by repeated squaring modulo crc
uint32_t PowModCrc(uint32_t p) // pow(2,p)%crc
uint32_t prd = 0x1u; // current product
uint32_t sqr = 0x2u; // current square
while(p)
if(p&1)
prd = MpyModCrc(prd, sqr);
sqr = MpyModCrc(sqr, sqr);
p >>= 1;
return prd;
void fix_crc_pos(uint8_t *buffer, int length, uint32_t tcrcreg, int fix_pos)
uint32_t crc; // crc
uint32_t par; // parities
uint32_t pmr; // 1/pow(2, number of bits to cycle backwards)
size_t i;
for(i = 0; i < 4; i++) // zero out data to be replaced
buffer[fix_pos+i] = 0;
pmr = PowModCrc(-1-((length-fix_pos)*8)); // pmr = 1/pow(2, number of bits to cycle backwards)
crc = crc32(buffer, length); // generate crc
crc ^= tcrcreg; // adjust crc
par = MpyModCrc(crc, pmr); // par = (crc*pmr)%crc
buffer[fix_pos+0] = (uint8_t)(par>>24); // store parities
buffer[fix_pos+1] = (uint8_t)(par>>16);
buffer[fix_pos+2] = (uint8_t)(par>> 8);
buffer[fix_pos+3] = (uint8_t)(par>> 0);
int main()
make_crc_table();
// Check valid CRC32(a)
char crc32_check_txt[] = "123456789";
uint32_t crc32_check = 0xFC891918;
uint32_t crc = crc32((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt));
if(crc != crc32_check)
std::cout << "error on forward crc\n";
return 0;
// Change 1 byte and restore the previus CRC
crc32_check_txt[0] = 'x';
fix_crc_pos((uint8_t*)crc32_check_txt, (int) std::strlen(crc32_check_txt), crc32_check, 4);
// Verify CRC32
crc = crc32((uint8_t*)crc32_check_txt, std::strlen(crc32_check_txt));
if(crc != crc32_check)
std::cout << "error on backward crc\n";
return 0;
return 0;
使用 Visual Studio 内部函数 (PCLMULQDQ) 进行无进位乘法的示例。由于没有无借除法，因此使用多项式的逆（由 GenMPoly() 生成），并进行乘法运算，所得商在 XMM 寄存器的高 64 位中，在下面的代码中为 mt。 m128i_u64[1]，注释为 t[1]。
所以序列是无进位乘以两个 32 位值 p = a · b，得到 63 位结果（没有进位，所以只有 63 位乘积）。乘以 poly 的倒数得到商 t = p/POLY。乘以 poly 得到截断乘积 t = t*POLY。那么乘积模poly = p ^ t。
#define POLY (0x104c11db7ull)
#define POLYM ( 0x04c11db7u)
static __m128i poly; /* poly */
static __m128i invpoly; /* 2^64 / POLY */
void GenMPoly(void) /* generate __m12i8 poly info */
uint64_t N = 0x100000000ull;
uint64_t Q = 0;
for(size_t i = 0; i < 33; i++)
Q <<= 1;
if(N&0x100000000ull)
Q |= 1;
N ^= POLY;
N <<= 1;
poly.m128i_u64[0] = POLY;
invpoly.m128i_u64[0] = Q;
// ...
// carryless multiply modulo crc
uint32_t MpyModCrc(uint32_t a, uint32_t b) // (a*b)%crc
__m128i ma, mb, mp, mt;
ma.m128i_u64[0] = a;
mb.m128i_u64[0] = b;
mp = _mm_clmulepi64_si128(ma, mb, 0x00); // p[0] = a*b
mt = _mm_clmulepi64_si128(mp, invpoly, 0x00); // t[1] = (p[0]*((2^64)/POLY))>>64
mt = _mm_clmulepi64_si128(mt, poly, 0x01); // t[0] = t[1]*POLY
return mp.m128i_u32[0] ^ mt.m128i_u32[0]; // ret = p[0] ^ t[0]
// ...
int main()
GenMPoly();
// ...
【讨论】：
别担心，已删除。我试过你的代码，它正在工作。我对我的代码中的错误感到怀疑。感谢您的帮助 @GiGa - 我更新了我的答案以包含问题代码的固定版本。 你真的很善良 @GiGa - 我与其他项目有关，但我已经有了第一次发布的替代版本。一旦我有时间，我就能够修复问题的代码并更新我的答案。至于替代版本，大多数 PC 都有一个无进位乘法，PCLMULQDQ，但它使用 XMM 寄存器，C++ 代码需要内部函数才能使用它，或者需要在汇编代码中实现。 @GiGa - 我添加了代码来显示替代版本的内在版本。以上是关于反转 CRC32(a)的主要内容，如果未能解决你的问题，请参考以下文章