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;
}
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);
}
}
// 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);
}
// 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);
}
