Finding polynomial used in CRC-8 calculation table

Question

Here as most rated answer (Implementing CRC8 on Arduino to write to MLX90614) is a good example of CRC-8 calculation/finding using a lookup table. I would like to know what is the polynomial used to generate those table values.

So given the table, how can I recover the polynomial? I tried using this nice calculator of tables, but cannot get the same values.

static const uint8_t crc_table[] = {
 0x00, 0x07, 0x0e, 0x09, 0x1c, 0x1b, 0x12, 0x15, 0x38, 0x3f, 0x36, 0x31,
 0x24, 0x23, 0x2a, 0x2d, 0x70, 0x77, 0x7e, 0x79, 0x6c, 0x6b, 0x62, 0x65,
 0x48, 0x4f, 0x46, 0x41, 0x54, 0x53, 0x5a, 0x5d, 0xe0, 0xe7, 0xee, 0xe9,
 0xfc, 0xfb, 0xf2, 0xf5, 0xd8, 0xdf, 0xd6, 0xd1, 0xc4, 0xc3, 0xca, 0xcd,
 0x90, 0x97, 0x9e, 0x99, 0x8c, 0x8b, 0x82, 0x85, 0xa8, 0xaf, 0xa6, 0xa1,
 0xb4, 0xb3, 0xba, 0xbd, 0xc7, 0xc0, 0xc9, 0xce, 0xdb, 0xdc, 0xd5, 0xd2,
 0xff, 0xf8, 0xf1, 0xf6, 0xe3, 0xe4, 0xed, 0xea, 0xb7, 0xb0, 0xb9, 0xbe,
 0xab, 0xac, 0xa5, 0xa2, 0x8f, 0x88, 0x81, 0x86, 0x93, 0x94, 0x9d, 0x9a,
 0x27, 0x20, 0x29, 0x2e, 0x3b, 0x3c, 0x35, 0x32, 0x1f, 0x18, 0x11, 0x16,
 0x03, 0x04, 0x0d, 0x0a, 0x57, 0x50, 0x59, 0x5e, 0x4b, 0x4c, 0x45, 0x42,
 0x6f, 0x68, 0x61, 0x66, 0x73, 0x74, 0x7d, 0x7a, 0x89, 0x8e, 0x87, 0x80,
 0x95, 0x92, 0x9b, 0x9c, 0xb1, 0xb6, 0xbf, 0xb8, 0xad, 0xaa, 0xa3, 0xa4,
 0xf9, 0xfe, 0xf7, 0xf0, 0xe5, 0xe2, 0xeb, 0xec, 0xc1, 0xc6, 0xcf, 0xc8,
 0xdd, 0xda, 0xd3, 0xd4, 0x69, 0x6e, 0x67, 0x60, 0x75, 0x72, 0x7b, 0x7c,
 0x51, 0x56, 0x5f, 0x58, 0x4d, 0x4a, 0x43, 0x44, 0x19, 0x1e, 0x17, 0x10,
 0x05, 0x02, 0x0b, 0x0c, 0x21, 0x26, 0x2f, 0x28, 0x3d, 0x3a, 0x33, 0x34,
 0x4e, 0x49, 0x40, 0x47, 0x52, 0x55, 0x5c, 0x5b, 0x76, 0x71, 0x78, 0x7f,
 0x6a, 0x6d, 0x64, 0x63, 0x3e, 0x39, 0x30, 0x37, 0x22, 0x25, 0x2c, 0x2b,
 0x06, 0x01, 0x08, 0x0f, 0x1a, 0x1d, 0x14, 0x13, 0xae, 0xa9, 0xa0, 0xa7,
 0xb2, 0xb5, 0xbc, 0xbb, 0x96, 0x91, 0x98, 0x9f, 0x8a, 0x8d, 0x84, 0x83,
 0xde, 0xd9, 0xd0, 0xd7, 0xc2, 0xc5, 0xcc, 0xcb, 0xe6, 0xe1, 0xe8, 0xef,
 0xfa, 0xfd, 0xf4, 0xf3
};

uint8_t
crc8(uint8_t *p, uint8_t len)
{
    uint16_t i;
    uint16_t crc = 0x0;

    while (len--) {
            i = (crc ^ *p++) & 0xFF;
            crc = (crc_table[i] ^ (crc << 8)) & 0xFF;
    }

    return crc & 0xFF;
}

The votes to close this as off-topic due to request a recommendation for off-site resources (which could be opinionated) are inappropriate. It does not request a recommendation. It asks for a factual citation. — Eric Postpischil, Sep 13 '19 at 12:20
A little google-fu is leading here: https://stackoverflow.com/questions/37995208/crc8-with-x8-x2-x-1-working-java-source-code — Eugene Sh., Sep 13 '19 at 13:28
Found the reason an origin of my question, I had enabled tick mark "Show reflected lookup table" in calculator from link above. For that table I could not figure out a polynom. ;) — IKA, Sep 14 '19 at 19:04
Note that the code that calculates this has been blindly copied from crc16 and thus does unnecessary work. For a CRC8 you can replace all the `uint16_t`s with `uint8_t`, remove `& 0xFF`, and delete the `^(crc<<8)`. — Matthias Urlichs, Oct 18 '20 at 15:35
Spent a few hours trying to realize what's wrong with my code. The problem was that byte array with crc8 concatenated didn't give me 0x00. I was looking for an issue with table generator algo, but found that I did extra shift `idx= ((prev << 1) ^ next)` instead `idx= (prev ^ next)`. Omg! — Peter, Aug 11 '22 at 15:02

Marco Bonelli · Accepted Answer · 2022-06-10T15:18:56.437

Well, the link you provided shows the exact same table for the (default) polynomial 0x07, which is (x^8) + x^2 + x + 1. So this should already answer your question.

With the above being said, for a general "target" table, you can very easily brute force the needed polynomial.

I'll let the code speak for itself, here's a working example:

#include <stdio.h>
#include <stdint.h>
#include <string.h>
#include <stdbool.h>

static const uint8_t target[256] = {
    // ... fill in the values here ...
};

void compute_table(uint8_t table[256], uint8_t polynomial) {
    uint8_t crc = 0x80;
    size_t i, j;
    memset(table, 0, 256);

    for (i = 1; i < 256; i <<= 1) {
        if (crc & 0x80)
            crc = (crc << 1) ^ polynomial;
        else
            crc <<= 1;

        for (j = 0; j < i; j++)
            table[i + j] = crc ^ table[j];
    }
}

int main(void) {
    uint8_t table[256];
    uint8_t poly;
    bool found;
    size_t i;

    for (poly = 1; poly != 0; poly++) {
        compute_table(table, poly);

        found = true;

        for (i = 0; i < 256; i++) {
            if (table[i] != target[i]) {
                found = false;
                break;
            }
        }

        if (found)
            printf("Found polynomial: 0x%02x\n", poly);
    }

    return 0;
}

And indeed, plugging in the table you provide, the above code outputs:

Found polynomial: 0x07

If the CRC was a right shifting CRC, the code would need to be modified in order to find the polynomial. — rcgldr, Sep 13 '19 at 15:01
Oh. Apparently I have been mistaken by going through lot of tables. Looks like at the moment of question I have been looking on different polynomial tables. Indeed they are both the same. And polynomial is 0x07 just as expected for CRC-8 CCITT. Anyway the @Marco Bonelli code posted in answer can be useful in other similar problematic cases. — IKA, Sep 14 '19 at 17:26

score 1 · Answer 2 · answered Sep 02 '22 at 03:06

#include <stddef.h>
#include <stdint.h>
#include <stdio.h>

unsigned char table[256] = {0};

void crc_table_create(unsigned char *pdata, unsigned char factor)
{
    unsigned char temp;

    for (int i = 0; i < 256; i++)
    {
        pdata[i] = i;
    }

    for (int i = 0; i < 256; i++)
    {
        for (int j = 7; j >= 0; j--)
        {
            temp = pdata[i] & 0x80; // take the MSB bit

            if (temp) // the MSB bit is 1
            {
                pdata[i] = pdata[i] << 1;
                pdata[i] ^= factor;
            }
            else
            {
                pdata[i] = pdata[i] << 1;
            }
        }
        // printf("table[%i]= 0x%02x", i, table[i]);
        // getchar();
    }
}


int main(void)
{
    int i, j = 0;
    
    uint8_t poly = 0x07;     // X8 + X2 + X1 + 1 (100000111B the MSB bit is mute, so it turn to 0x07)
    


    crc_table_create(table, poly);

    for (i = 0; i < 256; i++)
    {
        if (i % 12 == 0) {
            printf("\n");
        }
        printf("0x%02x, ", table[i]);
    }

    printf("\n");

    return 1;
}

Here is the CRC8 table generater, after run it, the 'table[256]' is the result, enjoy!

Finding polynomial used in CRC-8 calculation table

2 Answers2