CRC 实现特定的多项式。多项式与代码中使用的多项式有何关系？

Question

我有以下CRC函数：

#define CRC8INIT    0x00
#define CRC8POLY    0x18              //0X18 = X^8+X^5+X^4+X^0

// ----------------------------------------------------------------------------

uint8_t crc8 (uint8_t *data, uint16_t number_of_bytes_in_data)
{
    uint8_t  crc;
    uint16_t loop_count;
    uint8_t  bit_counter;
    uint8_t  b;
    uint8_t  feedback_bit;

    crc = CRC8INIT;

    for (loop_count = 0; loop_count != number_of_bytes_in_data; loop_count++) {
        b = data[loop_count];
        bit_counter = 8;

        do {
            feedback_bit = (crc ^ b) & 0x01;

            if (feedback_bit == 0x01) {
                crc = crc ^ CRC8POLY;
            }

            crc = (crc >> 1) & 0x7F;

            if (feedback_bit == 0x01) {
                crc = crc | 0x80;
            }

            b = b >> 1;
            bit_counter--;

        } while (bit_counter > 0);
    }

    return crc;
}

0x18与多项式X^8+X^5+X^4+X^0有什么关系？

X^8+X^5+X^4+X^0 = 100110001

0x18 = 00011000

如果我将 CRC8POLY 定义为 0xEA（我已经看到了），那会代表什么多项式呢？

Answer 1

该 CRC 代码写得相当奇怪，多项式应用于两个不同的地方，将位分开。也就是说，在 crc = crc ^ CRCPOLY 和有条件地 crc = crc | 0x80。通常应该这样写：

unsigned crc8(unsigned char const *dat, size_t len) {
    unsigned crc = 0;
    for (size_t i = 0; i < len; i++) {
        crc ^= dat[i];
        for (int k = 0; k < 8; k++)
            crc = crc & 1 ? (crc >> 1) ^ 0x8c : crc >> 1;
    }
    return crc;
}

其中 0x8c 是多项式的反转，没有 x⁸ 项。

Answer 2

How does 0x18 relate to the polynomial X^8+X^5+X^4+X^0?

由于代码是右移CRC，所以每个字节的"most significant bit"是bit 0，而不是bit 7。poly需要从100110001反转为100011001，即右移后的0x119 , 0x119 的第 0 位被移出，因此可以使用 0x118 代替。如果反馈位为 1，代码使用第二个 if 语句 to or in (0x100) >> 1 == 0x80。作为替代方案，由于 feedback_bit 为 0 或 1，则 (0-feeback_bit) 可以用作 poly 的掩码（假设二进制补码数学）而不是使用 if 语句。

typedef unsigned char uint8_t;
typedef unsigned short uint16_t;
#define CRC8INIT 0x00
#define CRC8POLY 0x8c   // 0x119 >> 1

uint8_t crc8 (uint8_t *data, uint16_t number_of_bytes_in_data)
{
    uint8_t  crc;
    uint16_t loop_count;
    uint8_t  bit_counter;
    uint8_t  b;
    uint8_t  feedback_bit;

    crc = CRC8INIT;

    for (loop_count = 0; loop_count != number_of_bytes_in_data; loop_count++) {
        b = data[loop_count];
        bit_counter = 8;
        do {
            feedback_bit = (crc ^ b) & 0x01;
            crc = (crc >> 1) ^ ((0-feedback_bit) & CRC8POLY);
            b = b >> 1;
            bit_counter--;
        } while (bit_counter > 0);
    }

    return crc;
}

Answer 3

参考Wiki article on CRC , it's crc8 realization as used in Dallas 1-wire chips. Note, that polinomial can be represented in Normal, Reversed, Reciprocal and Reversed reciprocal representations (see also Polynomial representations)。看起来，它是省略了最高位的反向倒数表示。

CRC 实现特定的多项式。多项式与代码中使用的多项式有何关系？

CRC implementing a specific polynomial. How does the polynomial relate to the polynomial used in code?

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