src/media_tools/reedsolomon.c

/*****************************
 *
 *
 * Multiplication and Arithmetic on Galois Field GF(256)
 *
 * From Mee, Daniel, "Magnetic Recording, Volume III", Ch. 5 by Patel.
 *
 * (c) 1991 Henry Minsky
 *
 *
 ******************************/


#include <stdio.h>
#include <ctype.h>
#include <gpac/tools.h>
#include <gpac/internal/reedsolomon.h>

#ifdef GPAC_ENABLE_MPE


/* This is one of 14 irreducible polynomials
 * of degree 8 and cycle length 255. (Ch 5, pp. 275, Magnetic Recording)
 * The high order 1 bit is implicit */
/* x^8 + x^4 + x^3 + x^2 + 1 */
#define PPOLY 0x1D


int gexp[512];
int glog[256];


static void init_exp_table (void);


void
init_galois_tables (void)
{
	/* initialize the table of powers of alpha */
	init_exp_table();
}


static void
init_exp_table (void)
{
	int i, z;
	int p1,p2,p3,p4,p5,p6,p7,p8;

	p2 = p3 = p4 = p5 = p6 = p7 = p8 = 0;
	p1 = 1;

	gexp[0] = 1;
	gexp[255] = gexp[0];
	glog[0] = 0;			/* shouldn't log[0] be an error? */

	for (i = 1; i < 256; i++) {
		int pinit = p8;
		p8 = p7;
		p7 = p6;
		p6 = p5;
		p5 = p4 ^ pinit;
		p4 = p3 ^ pinit;
		p3 = p2 ^ pinit;
		p2 = p1;
		p1 = pinit;
		gexp[i] = p1 + p2*2 + p3*4 + p4*8 + p5*16 + p6*32 + p7*64 + p8*128;
		gexp[i+255] = gexp[i];
	}

	for (i = 1; i < 256; i++) {
		for (z = 0; z < 256; z++) {
			if (gexp[z] == i) {
				glog[i] = z;
				break;
			}
		}
	}
}

/* multiplication using logarithms */
int gmult(int a, int b)
{
	int i,j;
	if (a==0 || b == 0) return (0);
	i = glog[a];
	j = glog[b];
	return (gexp[i+j]);
}


int ginv (int elt)
{
	return (gexp[255-glog[elt]]);
}


/***********************************************************************
 * Berlekamp-Peterson and Berlekamp-Massey Algorithms for error-location
 *
 * From Cain, Clark, "Error-Correction Coding For Digital Communications", pp. 205.
 *
 * This finds the coefficients of the error locator polynomial.
 *
 * The roots are then found by looking for the values of a^n
 * where evaluating the polynomial yields zero.
 *
 * Error correction is done using the error-evaluator equation  on pp 207.
 *
 * hqm@ai.mit.edu   Henry Minsky
 */


/* The Error Locator Polynomial, also known as Lambda or Sigma. Lambda[0] == 1 */
static int Lambda[MAXDEG];

/* The Error Evaluator Polynomial */
static int Omega[MAXDEG];

/* local ANSI declarations */
static int compute_discrepancy(int lambda[], int S[], int L, int n);
static void init_gamma(int gamma[]);
static void compute_modified_omega (void);
static void mul_z_poly (int src[]);

/* error locations found using Chien's search*/
static int ErrorLocs[256];
static int NErrors;

/* erasure flags */
static int ErasureLocs[256];
static int NErasures;

/* From  Cain, Clark, "Error-Correction Coding For Digital Communications", pp. 216. */
void
Modified_Berlekamp_Massey (void)
{
	int n, L, L2, k, i;
	int psi[MAXDEG], psi2[MAXDEG], D[MAXDEG];
	int gamma[MAXDEG];

	/* initialize Gamma, the erasure locator polynomial */
	init_gamma(gamma);

	/* initialize to z */
	copy_poly(D, gamma);
	mul_z_poly(D);

	copy_poly(psi, gamma);
	k = -1;
	L = NErasures;

	for (n = NErasures; n < NPAR; n++) {

		int d = compute_discrepancy(psi, synBytes, L, n);

		if (d != 0) {

			/* psi2 = psi - d*D */
			for (i = 0; i < MAXDEG; i++) psi2[i] = psi[i] ^ gmult(d, D[i]);


			if (L < (n-k)) {
				L2 = n-k;
				k = n-L;
				/* D = scale_poly(ginv(d), psi); */
				for (i = 0; i < MAXDEG; i++) D[i] = gmult(psi[i], ginv(d));
				L = L2;
			}

			/* psi = psi2 */
			for (i = 0; i < MAXDEG; i++) psi[i] = psi2[i];
		}

		mul_z_poly(D);
	}

	for(i = 0; i < MAXDEG; i++) Lambda[i] = psi[i];
	compute_modified_omega();


}

/* given Psi (called Lambda in Modified_Berlekamp_Massey) and synBytes,
   compute the combined erasure/error evaluator polynomial as
   Psi*S mod z^4
  */
void
compute_modified_omega ()
{
	int i;
	int product[MAXDEG*2];

	mult_polys(product, Lambda, synBytes);
	zero_poly(Omega);
	for(i = 0; i < NPAR; i++) Omega[i] = product[i];

}

/* polynomial multiplication */
void
mult_polys (int dst[], int p1[], int p2[])
{
	int i, j;
	int tmp1[MAXDEG*2];

	for (i=0; i < (MAXDEG*2); i++) dst[i] = 0;

	for (i = 0; i < MAXDEG; i++) {
		for(j=MAXDEG; j<(MAXDEG*2); j++) tmp1[j]=0;

		/* scale tmp1 by p1[i] */
		for(j=0; j<MAXDEG; j++) tmp1[j]=gmult(p2[j], p1[i]);
		/* and mult (shift) tmp1 right by i */
		for (j = (MAXDEG*2)-1; j >= i; j--) tmp1[j] = tmp1[j-i];
		for (j = 0; j < i; j++) tmp1[j] = 0;

		/* add into partial product */
		for(j=0; j < (MAXDEG*2); j++) dst[j] ^= tmp1[j];
	}
}


/* gamma = product (1-z*a^Ij) for erasure locs Ij */
void
init_gamma (int gamma[])
{
	int e, tmp[MAXDEG];

	zero_poly(gamma);
	zero_poly(tmp);
	gamma[0] = 1;

	for (e = 0; e < NErasures; e++) {
		copy_poly(tmp, gamma);
		scale_poly(gexp[ErasureLocs[e]], tmp);
		mul_z_poly(tmp);
		add_polys(gamma, tmp);
	}
}


void
compute_next_omega (int d, int A[], int dst[], int src[])
{
	int i;
	for ( i = 0; i < MAXDEG;  i++) {
		dst[i] = src[i] ^ gmult(d, A[i]);
	}
}


int
compute_discrepancy (int lambda[], int S[], int L, int n)
{
	int i, sum=0;

	for (i = 0; i <= L; i++)
		sum ^= gmult(lambda[i], S[n-i]);
	return (sum);
}

/********** polynomial arithmetic *******************/

void add_polys (int dst[], int src[])
{
	int i;
	for (i = 0; i < MAXDEG; i++) dst[i] ^= src[i];
}

void copy_poly (int dst[], int src[])
{
	int i;
	for (i = 0; i < MAXDEG; i++) dst[i] = src[i];
}

void scale_poly (int k, int poly[])
{
	int i;
	for (i = 0; i < MAXDEG; i++) poly[i] = gmult(k, poly[i]);
}


void zero_poly (int poly[])
{
	int i;
	for (i = 0; i < MAXDEG; i++) poly[i] = 0;
}


/* multiply by z, i.e., shift right by 1 */
static void mul_z_poly (int src[])
{
	int i;
	for (i = MAXDEG-1; i > 0; i--) src[i] = src[i-1];
	src[0] = 0;
}


/* Finds all the roots of an error-locator polynomial with coefficients
 * Lambda[j] by evaluating Lambda at successive values of alpha.
 *
 * This can be tested with the decoder's equations case.
 */


void
Find_Roots (void)
{
	int r, k;
	NErrors = 0;

	for (r = 1; r < 256; r++) {
		int sum = 0;
		/* evaluate lambda at r */
		for (k = 0; k < NPAR+1; k++) {
			sum ^= gmult(gexp[(k*r)%255], Lambda[k]);
		}
		if (sum == 0)
		{
			ErrorLocs[NErrors] = (255-r);
			NErrors++;
			if (RS_DEBUG) fprintf(stderr, "Root found at r = %d, (255-r) = %d\n", r, (255-r));
		}
	}
}

/* Combined Erasure And Error Magnitude Computation
 *
 * Pass in the codeword, its size in bytes, as well as
 * an array of any known erasure locations, along the number
 * of these erasures.
 *
 * Evaluate Omega(actually Psi)/Lambda' at the roots
 * alpha^(-i) for error locs i.
 *
 * Returns 1 if everything ok, or 0 if an out-of-bounds error is found
 *
 */

int
correct_errors_erasures (unsigned char codeword[],
                         int csize,
                         int nerasures,
                         int erasures[])
{
	int i;

	/* If you want to take advantage of erasure correction, be sure to
	   set NErasures and ErasureLocs[] with the locations of erasures.
	   */
	NErasures = nerasures;
	for (i = 0; i < NErasures; i++) ErasureLocs[i] = erasures[i];

	Modified_Berlekamp_Massey();
	Find_Roots();


	if ((NErrors <= NPAR) && NErrors > 0) {
		int r;

		/* first check for illegal error locs */
		for (r = 0; r < NErrors; r++) {
			if (ErrorLocs[r] >= csize) {
				if (RS_DEBUG) fprintf(stderr, "Error loc i=%d outside of codeword length %d\n", i, csize);
				return(0);
			}
		}

		for (r = 0; r < NErrors; r++) {
			int num, denom, err, j;
			i = ErrorLocs[r];
			/* evaluate Omega at alpha^(-i) */

			num = 0;
			for (j = 0; j < MAXDEG; j++)
				num ^= gmult(Omega[j], gexp[((255-i)*j)%255]);

			/* evaluate Lambda' (derivative) at alpha^(-i) ; all odd powers disappear */
			denom = 0;
			for (j = 1; j < MAXDEG; j += 2) {
				denom ^= gmult(Lambda[j], gexp[((255-i)*(j-1)) % 255]);
			}

			err = gmult(num, ginv(denom));
			if (RS_DEBUG) fprintf(stderr, "Error magnitude %#x at loc %d\n", err, csize-i);

			codeword[csize-i-1] ^= err;
		}
		return(1);
	}
	else {
		if (RS_DEBUG && NErrors) fprintf(stderr, "Uncorrectable codeword\n");
		return(0);
	}
}


/*
 * Reed Solomon Encoder/Decoder
 *
 * (c) Henry Minsky (hqm@ua.com), Universal Access 1991-1995
 */

/* Encoder parity bytes */
int pBytes[MAXDEG];

/* Decoder syndrome bytes */
int synBytes[MAXDEG];

/* generator polynomial */
int genPoly[MAXDEG*2];

int RS_DEBUG = FALSE;

static void
compute_genpoly (int nbytes, int genpoly[]);

/* Initialize lookup tables, polynomials, etc. */
void
initialize_ecc ()
{
	/* Initialize the galois field arithmetic tables */
	init_galois_tables();

	/* Compute the encoder generator polynomial */
	compute_genpoly(NPAR, genPoly);
}

void
zero_fill_from (unsigned char buf[], int from, int to)
{
	int i;
	for (i = from; i < to; i++) buf[i] = 0;
}

/* debugging routines */
void
print_parity (void)
{
	int i;
	fprintf(stderr, "Parity Bytes: ");
	for (i = 0; i < NPAR; i++)
		fprintf(stderr, "[%d]:%x, ",i,pBytes[i]);
	fprintf(stderr, "\n");
}


void
print_syndrome (void)
{
	int i;
	fprintf(stderr, "Syndrome Bytes: ");
	for (i = 0; i < NPAR; i++)
		fprintf(stderr, "[%d]:%x, ",i,synBytes[i]);
	fprintf(stderr, "\n");
}

/* Append the parity bytes onto the end of the message */
void
build_codeword (unsigned char msg[], int nbytes, unsigned char dst[])
{
	int i;

	for (i = 0; i < nbytes; i++) dst[i] = msg[i];

	for (i = 0; i < NPAR; i++) {
		dst[i+nbytes] = pBytes[NPAR-1-i];
	}
}

/**********************************************************
 * Reed Solomon Decoder
 *
 * Computes the syndrome of a codeword. Puts the results
 * into the synBytes[] array.
 */

void
decode_data(unsigned char data[], int nbytes)
{
	int i, j;
	for (j = 0; j < NPAR;  j++) {
		int sum = 0;
		for (i = 0; i < nbytes; i++) {
			sum = data[i] ^ gmult(gexp[j+1], sum);
		}
		synBytes[j]  = sum;
	}
}


/* Check if the syndrome is zero */
int
check_syndrome (void)
{
	int i, nz = 0;
	for (i =0 ; i < NPAR; i++) {
		if (synBytes[i] != 0) nz = 1;
	}
	return nz;
}


void
debug_check_syndrome (void)
{
	int i;

	for (i = 0; i < 3; i++) {
		fprintf(stderr, " inv log S[%d]/S[%d] = %d\n", i, i+1,
		        glog[gmult(synBytes[i], ginv(synBytes[i+1]))]);
	}
}


/* Create a generator polynomial for an n byte RS code.
 * The coefficients are returned in the genPoly arg.
 * Make sure that the genPoly array which is passed in is
 * at least n+1 bytes long.
 */

static void
compute_genpoly (int nbytes, int genpoly[])
{
	int i, tp[256], tp1[256];

	/* multiply (x + a^n) for n = 1 to nbytes */

	zero_poly(tp1);
	tp1[0] = 1;

	for (i = 1; i <= nbytes; i++) {
		zero_poly(tp);
		tp[0] = gexp[i];        /* set up x+a^n */
		tp[1] = 1;

		mult_polys(genpoly, tp, tp1);
		copy_poly(tp1, genpoly);
	}
}

/* Simulate a LFSR with generator polynomial for n byte RS code.
 * Pass in a pointer to the data array, and amount of data.
 *
 * The parity bytes are deposited into pBytes[], and the whole message
 * and parity are copied to dest to make a codeword.
 *
 */

void
encode_data (unsigned char msg[], int nbytes, unsigned char dst[])
{
	int i, LFSR[NPAR+1],j;

	for(i=0; i < NPAR+1; i++) LFSR[i]=0;

	for (i = 0; i < nbytes; i++) {
		int dbyte = msg[i] ^ LFSR[NPAR-1];
		for (j = NPAR-1; j > 0; j--) {
			LFSR[j] = LFSR[j-1] ^ gmult(genPoly[j], dbyte);
		}
		LFSR[0] = gmult(genPoly[0], dbyte);
	}

	for (i = 0; i < NPAR; i++)
		pBytes[i] = LFSR[i];

	build_codeword(msg, nbytes, dst);
}

#endif //GPAC_ENABLE_MPE