poly_t poly_gcd(poly_t p1, poly_t p2) {
poly_t a, b, c;
a = poly_copy(p1);
b = poly_copy(p2);
if (poly_deg(a) < poly_deg(b))
c = poly_copy(poly_gcd_aux(b, a));
else
c = poly_copy(poly_gcd_aux(a, b));
poly_free(a);
poly_free(b);
return c;
}
poly_t poly_mul(poly_t p, poly_t q) {
int i,j,dp,dq;
poly_t r;
poly_calcule_deg(p);
poly_calcule_deg(q);
dp = poly_deg(p);
dq = poly_deg(q);
r=poly_alloc(dp+dq);
for (i = 0; i <= dp; ++i)
for (j = 0; j <= dq; ++j)
poly_addto_coeff(r,i+j,gf_mul(poly_coeff(p,i),poly_coeff(q,j)));
poly_calcule_deg(r);
return(r);
}
poly_t * poly_syndrome_init(poly_t generator, gf_t *support, int n)
{
int i,j,t;
gf_t a;
poly_t * F;
F = malloc(n * sizeof (poly_t));
t = poly_deg(generator);
//g(z)=g_t+g_(t-1).z^(t-1)+......+g_1.z+g_0
//f(z)=f_(t-1).z^(t-1)+......+f_1.z+f_0
for(j=0; j<n; j++)
{
F[j] = poly_alloc(t-1);
poly_set_coeff(F[j],t-1,gf_unit());
for(i=t-2; i>=0; i--)
{
poly_set_coeff(F[j],i,gf_add(poly_coeff(generator,i+1),
gf_mul(support[j],poly_coeff(F[j],i+1))));
}
a = gf_add(poly_coeff(generator,0),gf_mul(support[j],poly_coeff(F[j],0)));
for(i=0; i<t; i++)
{
poly_set_coeff(F[j],i, gf_div(poly_coeff(F[j],i),a));
}
}
return F;
}
// retourne le degré du plus petit facteur
int poly_degppf(poly_t g) {
int i, d, res;
poly_t *u, p, r, s;
d = poly_deg(g);
u = malloc(d * sizeof (poly_t *));
for (i = 0; i < d; ++i)
u[i] = poly_alloc(d + 1);
poly_sqmod_init(g, u);
p = poly_alloc(d - 1);
poly_set_deg(p, 1);
poly_set_coeff(p, 1, gf_unit());
r = poly_alloc(d - 1);
res = d;
for (i = 1; i <= (d / 2) * gf_extd(); ++i) {
poly_sqmod(r, p, u, d);
// r = x^(2^i) mod g
if ((i % gf_extd()) == 0) { // donc 2^i = (2^m)^j (m deg d'extension)
poly_addto_coeff(r, 1, gf_unit());
poly_calcule_deg(r); // le degré peut changer
s = poly_gcd(g, r);
if (poly_deg(s) > 0) {
poly_free(s);
res = i / gf_extd();
break;
}
poly_free(s);
poly_addto_coeff(r, 1, gf_unit());
poly_calcule_deg(r); // le degré peut changer
}
// on se sert de s pour l'échange
s = p;
p = r;
r = s;
}
poly_free(p);
poly_free(r);
for (i = 0; i < d; ++i) {
poly_free(u[i]);
}
free(u);
return res;
}
// Returns the degree of the smallest factor
int poly_degppf(poly_t g) {
int i, d, res;
poly_t *u, p, r, s;
d = poly_deg(g);
u = malloc(d * sizeof (poly_t *));
for (i = 0; i < d; ++i)
u[i] = poly_alloc(d + 1);
poly_sqmod_init(g, u);
p = poly_alloc(d - 1);
poly_set_deg(p, 1);
poly_set_coeff(p, 1, gf_unit());
r = poly_alloc(d - 1);
res = d;
for (i = 1; i <= (d / 2) * gf_extd(); ++i) {
poly_sqmod(r, p, u, d);
// r = x^(2^i) mod g
if ((i % gf_extd()) == 0) { // so 2^i = (2^m)^j (m ext. degree)
poly_addto_coeff(r, 1, gf_unit());
poly_calcule_deg(r); // The degree may change
s = poly_gcd(g, r);
if (poly_deg(s) > 0) {
poly_free(s);
res = i / gf_extd();
break;
}
poly_free(s);
poly_addto_coeff(r, 1, gf_unit());
poly_calcule_deg(r); // The degree may change
}
// No need for the exchange s
s = p;
p = r;
r = s;
}
poly_free(p);
poly_free(r);
for (i = 0; i < d; ++i) {
poly_free(u[i]);
}
free(u);
return res;
}
// destructif
poly_t poly_gcd_aux(poly_t p1, poly_t p2) {
if (poly_deg(p2) == -1)
return p1;
else {
poly_rem(p1, p2);
return poly_gcd_aux(p2, p1);
}
}
// p = p * x mod g
// p de degré <= deg(g)-1
void poly_shiftmod(poly_t p, poly_t g) {
int i, t;
gf_t a;
t = poly_deg(g);
a = gf_div(p->coeff[t-1], g->coeff[t]);
for (i = t - 1; i > 0; --i)
p->coeff[i] = gf_add(p->coeff[i - 1], gf_mul(a, g->coeff[i]));
p->coeff[0] = gf_mul(a, g->coeff[0]);
}
void poly_sqmod_init(poly_t g, poly_t * sq) {
int i, d;
d = poly_deg(g);
for (i = 0; i < d / 2; ++i) {
// sq[i] = x^(2i) mod g = x^(2i)
poly_set_to_zero(sq[i]);
poly_set_deg(sq[i], 2 * i);
poly_set_coeff(sq[i], 2 * i, gf_unit());
}
for (; i < d; ++i) {
// sq[i] = x^(2i) mod g = (x^2 * sq[i-1]) mod g
memset(sq[i]->coeff, 0, 2 * sizeof (gf_t));
memcpy(sq[i]->coeff + 2, sq[i - 1]->coeff, d * sizeof (gf_t));
poly_set_deg(sq[i], poly_deg(sq[i - 1]) + 2);
poly_rem(sq[i], g);
}
}
poly_t * poly_sqrtmod_init(poly_t g) {
int i, t;
poly_t * sqrt, aux, p, q, * sq_aux;
t = poly_deg(g);
sq_aux = malloc(t * sizeof (poly_t));
for (i = 0; i < t; ++i)
sq_aux[i] = poly_alloc(t - 1);
poly_sqmod_init(g, sq_aux);
q = poly_alloc(t - 1);
p = poly_alloc(t - 1);
poly_set_deg(p, 1);
poly_set_coeff(p, 1, gf_unit());
// q(z) = 0, p(z) = z
for (i = 0; i < t * gf_extd() - 1; ++i) {
// q(z) <- p(z)^2 mod g(z)
poly_sqmod(q, p, sq_aux, t);
// q(z) <-> p(z)
aux = q;
q = p;
p = aux;
}
// p(z) = z^(2^(tm-1)) mod g(z) = sqrt(z) mod g(z)
sqrt = malloc(t * sizeof (poly_t));
for (i = 0; i < t; ++i)
sqrt[i] = poly_alloc(t - 1);
poly_set(sqrt[1], p);
poly_calcule_deg(sqrt[1]);
for(i = 3; i < t; i += 2) {
poly_set(sqrt[i], sqrt[i - 2]);
poly_shiftmod(sqrt[i], g);
poly_calcule_deg(sqrt[i]);
}
for (i = 0; i < t; i += 2) {
poly_set_to_zero(sqrt[i]);
sqrt[i]->coeff[i / 2] = gf_unit();
sqrt[i]->deg = i / 2;
}
for (i = 0; i < t; ++i)
poly_free(sq_aux[i]);
free(sq_aux);
poly_free(p);
poly_free(q);
return sqrt;
}
// carré de p modulo un certain polynôme g, sq[] contient les carrés
// modulo g de la base canonique des polynômes de degré < d, où d est
// le degré de g. La table sq[] sera calculée par poly_sqmod_init()
void poly_sqmod(poly_t res, poly_t p, poly_t * sq, int d) {
int i, j;
gf_t a;
poly_set_to_zero(res);
// termes de bas degré
for (i = 0; i < d / 2; ++i)
poly_set_coeff(res, i * 2, gf_square(poly_coeff(p, i)));
// termes de haut degré
for (; i < d; ++i) {
if (poly_coeff(p, i) != gf_zero()) {
a = gf_square(poly_coeff(p, i));
for (j = 0; j < d; ++j)
poly_addto_coeff(res, j, gf_mul_fast(a, poly_coeff(sq[i], j)));
}
}
// mise à jour du degré
poly_set_deg(res, d - 1);
while ((poly_deg(res) >= 0) && (poly_coeff(res, poly_deg(res)) == gf_zero()))
poly_set_deg(res, poly_deg(res) - 1);
}
void poly_sqmod(poly_t res, poly_t p, poly_t * sq, int d) {
int i, j;
gf_t a;
poly_set_to_zero(res);
// terms of low degree
for (i = 0; i < d / 2; ++i)
poly_set_coeff(res, i * 2, gf_square(poly_coeff(p, i)));
// terms of high degree
for (; i < d; ++i) {
if (poly_coeff(p, i) != gf_zero()) {
a = gf_square(poly_coeff(p, i));
for (j = 0; j < d; ++j)
poly_addto_coeff(res, j, gf_mul_fast(a, poly_coeff(sq[i], j)));
}
}
// Update degre
poly_set_deg(res, d - 1);
while ((poly_deg(res) >= 0) && (poly_coeff(res, poly_deg(res)) == gf_zero()))
poly_set_deg(res, poly_deg(res) - 1);
}
int root_laguerre(double tol, int maxit, const polyr *p, double x0, double *res) {
polyr pd, pdd;
int deg = poly_deg(p);
double x = x0;
double y = poly_eval(p, x);
int n = 0;
poly_diff(p, &pd);
poly_diff(&pd, &pdd);
#ifdef DEBUG
fprintf(stderr, "tol: %g maxit: %d x0: %g\n", tol, maxit, x0);
poly_print(stderr, p);
poly_print(stderr, &pd);
poly_print(stderr, &pdd);
#endif
while (fabs(y) > tol && n < maxit) {
double g = poly_eval(&pd, x) / poly_eval(p, x);
double h = g*g - poly_eval(&pdd, x) / poly_eval(p, x);
double sqrtarg = (deg-1)*(deg*h - g*g);
double sq = sqrt(sqrtarg);
double denom = (g > 0 ? g+sq : g-sq);
x = x - deg/denom;
y = poly_eval(p, x);
n++;
#ifdef DEBUG
fprintf(stderr, "g: %g h: %g sqa: %g sq: %g x: %g y: %g\n",
g, h, sqrtarg, sq, x, y);
#endif
/* complex roots => EDOM, divide by zero => NAN etc */
if (sqrtarg < 0 || isinf(x) || isnan(x)
|| errno == EDOM || errno == ERANGE)
return 0;
}
if (fabs(y) > tol)
return 0;
*res = x;
return 1;
}
// p contiendra son reste modulo g
void poly_rem(poly_t p, poly_t g) {
int i, j, d;
gf_t a, b;
d = poly_deg(p) - poly_deg(g);
if (d >= 0) {
a = gf_inv(poly_tete(g));
for (i = poly_deg(p); d >= 0; --i, --d) {
if (poly_coeff(p, i) != gf_zero()) {
b = gf_mul_fast(a, poly_coeff(p, i));
for (j = 0; j < poly_deg(g); ++j)
poly_addto_coeff(p, j + d, gf_mul_fast(b, poly_coeff(g, j)));
poly_set_coeff(p, i, gf_zero());
}
}
poly_set_deg(p, poly_deg(g) - 1);
while ((poly_deg(p) >= 0) && (poly_coeff(p, poly_deg(p)) == gf_zero()))
poly_set_deg(p, poly_deg(p) - 1);
}
}
// On suppose deg(g) >= deg(p)
void poly_eeaux(poly_t * u, poly_t * v, poly_t p, poly_t g, int t) {
int i, j, dr, du, delta;
gf_t a;
poly_t aux, r0, r1, u0, u1;
// initialisation des variables locales
// r0 <- g, r1 <- p, u0 <- 0, u1 <- 1
dr = poly_deg(g);
r0 = poly_alloc(dr);
r1 = poly_alloc(dr - 1);
u0 = poly_alloc(dr - 1);
u1 = poly_alloc(dr - 1);
poly_set(r0, g);
poly_set(r1, p);
poly_set_to_zero(u0);
poly_set_to_zero(u1);
poly_set_coeff(u1, 0, gf_unit());
poly_set_deg(u1, 0);
// invariants:
// r1 = u1 * p + v1 * g
// r0 = u0 * p + v0 * g
// et deg(u1) = deg(g) - deg(r0)
// on s'arrête lorsque deg(r1) < t (et deg(r0) >= t)
// et donc deg(u1) = deg(g) - deg(r0) < deg(g) - t
du = 0;
dr = poly_deg(r1);
delta = poly_deg(r0) - dr;
while (dr >= t) {
for (j = delta; j >= 0; --j) {
a = gf_div(poly_coeff(r0, dr + j), poly_coeff(r1, dr));
if (a != gf_zero()) {
// u0(z) <- u0(z) + a * u1(z) * z^j
for (i = 0; i <= du; ++i) {
poly_addto_coeff(u0, i + j, gf_mul_fast(a, poly_coeff(u1, i)));
}
// r0(z) <- r0(z) + a * r1(z) * z^j
for (i = 0; i <= dr; ++i)
poly_addto_coeff(r0, i + j, gf_mul_fast(a, poly_coeff(r1, i)));
}
}
// échanges
aux = r0;
r0 = r1;
r1 = aux;
aux = u0;
u0 = u1;
u1 = aux;
du = du + delta;
delta = 1;
while (poly_coeff(r1, dr - delta) == gf_zero())
delta++;
dr -= delta;
}
poly_set_deg(u1, du);
poly_set_deg(r1, dr);
//return u1 and r1;
*u=u1;
*v=r1;
poly_free(r0);
poly_free(u0);
}
gf_t poly_eval(poly_t p, gf_t a) {
return poly_eval_aux(p->coeff, a, poly_deg(p));
}
