gf.c

#include "gf.h"

#define W       0x08
#define PMAX    0x03    //max parity
#define RWIDTH  0x0c    //strip width
#define GFPoly  0x11d
#define GFOrder 0xff    //2^W-1

/*
 * These two tables represent powers and logs of 2 in the Galois field.
 * These values were computed by repeatedly multiplying by 2.
 * The primitive poly is x^8+x^4+x^3+x^2+1 in GF(2^8).
 */
static const uint8_t GF8Pow2[256] = {
    0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80,
    0x1d, 0x3a, 0x74, 0xe8, 0xcd, 0x87, 0x13, 0x26,
    0x4c, 0x98, 0x2d, 0x5a, 0xb4, 0x75, 0xea, 0xc9,
    0x8f, 0x03, 0x06, 0x0c, 0x18, 0x30, 0x60, 0xc0,
    0x9d, 0x27, 0x4e, 0x9c, 0x25, 0x4a, 0x94, 0x35,
    0x6a, 0xd4, 0xb5, 0x77, 0xee, 0xc1, 0x9f, 0x23,
    0x46, 0x8c, 0x05, 0x0a, 0x14, 0x28, 0x50, 0xa0,
    0x5d, 0xba, 0x69, 0xd2, 0xb9, 0x6f, 0xde, 0xa1,
    0x5f, 0xbe, 0x61, 0xc2, 0x99, 0x2f, 0x5e, 0xbc,
    0x65, 0xca, 0x89, 0x0f, 0x1e, 0x3c, 0x78, 0xf0,
    0xfd, 0xe7, 0xd3, 0xbb, 0x6b, 0xd6, 0xb1, 0x7f,
    0xfe, 0xe1, 0xdf, 0xa3, 0x5b, 0xb6, 0x71, 0xe2,
    0xd9, 0xaf, 0x43, 0x86, 0x11, 0x22, 0x44, 0x88,
    0x0d, 0x1a, 0x34, 0x68, 0xd0, 0xbd, 0x67, 0xce,
    0x81, 0x1f, 0x3e, 0x7c, 0xf8, 0xed, 0xc7, 0x93,
    0x3b, 0x76, 0xec, 0xc5, 0x97, 0x33, 0x66, 0xcc,
    0x85, 0x17, 0x2e, 0x5c, 0xb8, 0x6d, 0xda, 0xa9,
    0x4f, 0x9e, 0x21, 0x42, 0x84, 0x15, 0x2a, 0x54,
    0xa8, 0x4d, 0x9a, 0x29, 0x52, 0xa4, 0x55, 0xaa,
    0x49, 0x92, 0x39, 0x72, 0xe4, 0xd5, 0xb7, 0x73,
    0xe6, 0xd1, 0xbf, 0x63, 0xc6, 0x91, 0x3f, 0x7e,
    0xfc, 0xe5, 0xd7, 0xb3, 0x7b, 0xf6, 0xf1, 0xff,
    0xe3, 0xdb, 0xab, 0x4b, 0x96, 0x31, 0x62, 0xc4,
    0x95, 0x37, 0x6e, 0xdc, 0xa5, 0x57, 0xae, 0x41,
    0x82, 0x19, 0x32, 0x64, 0xc8, 0x8d, 0x07, 0x0e,
    0x1c, 0x38, 0x70, 0xe0, 0xdd, 0xa7, 0x53, 0xa6,
    0x51, 0xa2, 0x59, 0xb2, 0x79, 0xf2, 0xf9, 0xef,
    0xc3, 0x9b, 0x2b, 0x56, 0xac, 0x45, 0x8a, 0x09,
    0x12, 0x24, 0x48, 0x90, 0x3d, 0x7a, 0xf4, 0xf5,
    0xf7, 0xf3, 0xfb, 0xeb, 0xcb, 0x8b, 0x0b, 0x16,
    0x2c, 0x58, 0xb0, 0x7d, 0xfa, 0xe9, 0xcf, 0x83,
    0x1b, 0x36, 0x6c, 0xd8, 0xad, 0x47, 0x8e, 0x01
};
static const uint8_t GF8Log2[256] = {
    0x00, 0x00, 0x01, 0x19, 0x02, 0x32, 0x1a, 0xc6,
    0x03, 0xdf, 0x33, 0xee, 0x1b, 0x68, 0xc7, 0x4b,
    0x04, 0x64, 0xe0, 0x0e, 0x34, 0x8d, 0xef, 0x81,
    0x1c, 0xc1, 0x69, 0xf8, 0xc8, 0x08, 0x4c, 0x71,
    0x05, 0x8a, 0x65, 0x2f, 0xe1, 0x24, 0x0f, 0x21,
    0x35, 0x93, 0x8e, 0xda, 0xf0, 0x12, 0x82, 0x45,
    0x1d, 0xb5, 0xc2, 0x7d, 0x6a, 0x27, 0xf9, 0xb9,
    0xc9, 0x9a, 0x09, 0x78, 0x4d, 0xe4, 0x72, 0xa6,
    0x06, 0xbf, 0x8b, 0x62, 0x66, 0xdd, 0x30, 0xfd,
    0xe2, 0x98, 0x25, 0xb3, 0x10, 0x91, 0x22, 0x88,
    0x36, 0xd0, 0x94, 0xce, 0x8f, 0x96, 0xdb, 0xbd,
    0xf1, 0xd2, 0x13, 0x5c, 0x83, 0x38, 0x46, 0x40,
    0x1e, 0x42, 0xb6, 0xa3, 0xc3, 0x48, 0x7e, 0x6e,
    0x6b, 0x3a, 0x28, 0x54, 0xfa, 0x85, 0xba, 0x3d,
    0xca, 0x5e, 0x9b, 0x9f, 0x0a, 0x15, 0x79, 0x2b,
    0x4e, 0xd4, 0xe5, 0xac, 0x73, 0xf3, 0xa7, 0x57,
    0x07, 0x70, 0xc0, 0xf7, 0x8c, 0x80, 0x63, 0x0d,
    0x67, 0x4a, 0xde, 0xed, 0x31, 0xc5, 0xfe, 0x18,
    0xe3, 0xa5, 0x99, 0x77, 0x26, 0xb8, 0xb4, 0x7c,
    0x11, 0x44, 0x92, 0xd9, 0x23, 0x20, 0x89, 0x2e,
    0x37, 0x3f, 0xd1, 0x5b, 0x95, 0xbc, 0xcf, 0xcd,
    0x90, 0x87, 0x97, 0xb2, 0xdc, 0xfc, 0xbe, 0x61,
    0xf2, 0x56, 0xd3, 0xab, 0x14, 0x2a, 0x5d, 0x9e,
    0x84, 0x3c, 0x39, 0x53, 0x47, 0x6d, 0x41, 0xa2,
    0x1f, 0x2d, 0x43, 0xd8, 0xb7, 0x7b, 0xa4, 0x76,
    0xc4, 0x17, 0x49, 0xec, 0x7f, 0x0c, 0x6f, 0xf6,
    0x6c, 0xa1, 0x3b, 0x52, 0x29, 0x9d, 0x55, 0xaa,
    0xfb, 0x60, 0x86, 0xb1, 0xbb, 0xcc, 0x3e, 0x5a,
    0xcb, 0x59, 0x5f, 0xb0, 0x9c, 0xa9, 0xa0, 0x51,
    0x0b, 0xf5, 0x16, 0xeb, 0x7a, 0x75, 0x2c, 0xd7,
    0x4f, 0xae, 0xd5, 0xe9, 0xe6, 0xe7, 0xad, 0xe8,
    0x74, 0xd6, 0xf4, 0xea, 0xa8, 0x50, 0x58, 0xaf,
};


uint8_t GF8Add(uint8_t a, uint8_t b)
{   
    return a^b;
}

uint8_t GF8Multiply(uint8_t a, uint8_t b)
{
    if (a == 0 || b == 0) {
        return 0;
    }
    int s = GF8Log2[a] + GF8Log2[b];
    s %= GFOrder;
    return GF8Pow2[s];
}

uint8_t GF8Invert(uint8_t a)
{
    if (a == 0) return 0;
    return GF8Pow2[GFOrder-GF8Log2[a]];
}
/* This cauchy matrix is generated from
 *  X = {0, 1, 2}
 *  Y = {f4, 47, a7, 7a, ba, ad, dd, 3d, d8, 72, 6c, ed}
 *
 * The matrix design take the AFA frame(12 disk as a group)into count, 
 * according to the (k, m), we use different part of Cauchy Matrix.
 * for example, here k+m = 12, equal to disk number in a disk group
 *  (k,m) = (10,2), we will take CMat[0-1][0-9]
 *  (k,m) = (9, 3), we will take CMat[0-2][0-8]
 *
 * TODO:
 * We can find a sparse matrix(when turn into bit-matrix) to promote
 * the operation in the future. 
 */
static uint8_t CMat[PMAX][RWIDTH] = {
    {0x03, 0x04, 0x05, 0x06, 0x07, 0x08, 0x0a, 0x0c, 0x10, 0x11, 0x20, 0x21},
    {0x8f, 0xa6, 0x46, 0xbb, 0x7b, 0x9c, 0x99, 0xab, 0x73, 0xd9, 0xec, 0x6d},
    {0xd3, 0x4e, 0xc2, 0xdb, 0xc5, 0xb7, 0x91, 0xe1, 0x76, 0x28, 0x2f, 0x6a}};


uint8_t *CauchyMat(uint8_t m, uint8_t k)
{
    uint8_t *mat = (uint8_t *)malloc(sizeof(uint8_t)*(m*k));
    assert(mat != NULL);
    #if 0
    if (k > RWIDTH || m > PMAX || (k+m) != RWIDTH) {
        printf("Wrong arguments\n");
        free(mat);
        return NULL;
    }
    #endif
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < k; j++) {
            mat[i*k+j] = CMat[i][j];
        }
    }
    return mat;
}

/* 
 */
void BitMat(uint8_t a, uint8_t bM[])
{
    for (int i = 0; i < W; i++) {
        bM[i] = GF8Multiply(a, 1<<i);
    }
    return;
}
void uBitMat(uint8_t e, uint8_t w, uint8_t bmat[][w])
{
	assert(w == 8);	// for current version W==8
	for (int i = 0; i < w; ++i) {
		for (int j = 0; j < w; ++j)
			bmat[i][j] = GF8Multiply(e, 1<<j)&(1<<i);
	}
}
uint8_t *ToBitMatrix(uint8_t k, uint8_t m, uint8_t *mat)
{
    uint8_t *bitMat = (uint8_t *)malloc(sizeof(uint8_t)*(k*m*W));
    assert(bitMat != NULL);
    for (int i = 0; i < m; i++) {
        int ind = i*k*W;
        for (int j = 0; j < k; j++) {
            BitMat(*(mat+i*k+j), (bitMat + ind + j*W));
        }
    }   
    return bitMat;
}
uint8_t *uToBitMatrix(uint8_t m, uint8_t k, uint8_t w, uint8_t *mat)
{
	uint8_t *bitMat = (uint8_t *)malloc(sizeof(uint8_t)*k*w*m*w);
	assert(bitMat != NULL);
	for (int i = 0; i < m; ++i) {
		int ii = i*k*w*w;
		for (int j = 0; j < k; ++j) {
			uint8_t e = *(mat+i*k+j);
			int ij = j*w;
			for (int l = 0; l < w; ++l) {
				int il = l*k*w;
				for (int n = 0; n < w; ++n)
					*(bitMat+ii+ij+il+n) = GF8Multiply(e, 1<<n)&(1<<l)?1:0;
			}
		}
	}
	return bitMat;
}
uint8_t *uIdMatrix(uint8_t k)
{
	uint8_t *idmat = (uint8_t *)malloc(sizeof(uint8_t)*k*k);
    assert(idmat != NULL);
    printf("Addr:%p\n", idmat);
	for (int i = 0; i < k; ++i) {
        int indi = i*k;
		for (int j = 0; j < k; ++j) {
            int ind = indi+j;
			if (i == j)
				*(idmat+ind) = 1;
			else
				*(idmat+ind) = 0;
		}
	}
	return idmat;
}
uint8_t *uGenDecodeMatrix(uint8_t k, uint8_t *erased, uint8_t esize, uint8_t *cmat)
{
    uint8_t *idmat = uIdMatrix(k);
    for (int i = 0; i < esize; ++i) {
        int ind = *(erased+i);
        for (int j = 0; j < k; ++j) {
            *(idmat+ind*k+j) = *(cmat+i*k+j);
        }
    }
    return idmat;
}

/* 解码柯西矩阵，
 *
 */
uint8_t *uGenInvertMatrix(uint8_t k, uint8_t *erased, uint8_t esize, uint8_t *cmat)
{
    uint8_t eidx[esize];
    int ind, inde, indi;

    uint8_t *idmat = uIdMatrix(k);

    /* 处理除erased行中所有非erased行数的数,O(sizeof(erased)*k)
     */
    for (int er = 0; er < esize; ++er) {
        ind = eidx[er] = *(erased+er);
        inde = er*k;
        for (int i = 0; i < k; ++i) {
            indi = inde+i;
            if (*(cmat+indi) != 0) {
                int flag = 1;
                for (int j = 0; j < esize; ++j) {
                    if (*(erased+j) == i) {
                        flag = 0;
                        break;
                    }
                }
                if (flag) {
                    *(idmat+ind*k+i) = *(cmat+indi);
                    *(cmat+indi) = 0;
                }
            }
        }
    }
	/* 处理内层与erased行数相关的数，(sizeof(erased))^2的矩阵
	 */
	for (int er = 0; er < esize; ++er) {
        ind = eidx[er];
        inde = er*k;
        uint8_t inv = GF8Invert(*(cmat+inde+ind));
        for (int i = 0; i < esize; ++i) {
            int tmp = *(cmat+inde+eidx[i]);
            if (tmp != 0)
                *(cmat+inde+eidx[i]) = GF8Multiply(inv, tmp);
        }
        for (int j = 0; j < k; ++j) {
            int tmp = *(idmat+ind*k+j);
            if (tmp != 0)
                *(idmat+ind*k+j) = GF8Multiply(inv, tmp);
        }
        for (int i = 0; i < esize; ++i) {
            int idx = i*k;
            if (i != er) {
                uint8_t adde = *(cmat + idx + eidx[er]);
                for (int j = 0; j < esize; ++j) {
                    int tmpa = *(cmat + idx + eidx[j]);
                    int tmpb = *(cmat + er*k + eidx[j]);
                    *(cmat + idx + eidx[j]) = GF8Add(GF8Multiply(adde,tmpb), tmpa);
                }
                for (int l = 0; l < k; ++l) {
                    int tmpa = *(idmat + eidx[i]*k + l);
                    int tmpb = *(idmat + ind*k + l);
                    *(idmat + eidx[i]*k + l) = GF8Add(GF8Multiply(adde, tmpb), tmpa);
                }
            }
        }
    }
    return idmat;		
}
uint8_t *uGenInvertBitMatrix(uint8_t k, uint8_t w, uint8_t *dmat)
{
	return uToBitMatrix(k, k, w, dmat);
}