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); }