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;
}
// le corps est déjà défini
// retourne un polynôme unitaire de degré t irreductible dans le corps
poly_t poly_randgen_irred(int t, int (*u8rnd)()) {
int i;
poly_t g;
g = poly_alloc(t);
poly_set_deg(g, t);
poly_set_coeff(g, t, gf_unit());
i = 0;
do
for (i = 0; i < t; ++i)
poly_set_coeff(g, i, gf_rand(u8rnd));
while (poly_degppf(g) < t);
return g;
}
/* Subtract two polynomials */
poly_t *poly_diff(poly_t *p, poly_t *q) {
int n = MAX(p->n, q->n);
poly_t *r = poly_new(n);
int i;
for (i = 0; i < n + 1; i++) {
double c = poly_get_coeff(p, i) - poly_get_coeff(q, i);
poly_set_coeff(r, i, c);
}
return r;
}
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;
}
poly_t poly_quo(poly_t p, poly_t d) {
int i, j, dd, dp;
gf_t a, b;
poly_t quo, rem;
dd = poly_calcule_deg(d);
dp = poly_calcule_deg(p);
rem = poly_copy(p);
quo = poly_alloc(dp - dd);
poly_set_deg(quo, dp - dd);
a = gf_inv(poly_coeff(d, dd));
for (i = dp; i >= dd; --i) {
b = gf_mul_fast(a, poly_coeff(rem, i));
poly_set_coeff(quo, i - dd, b);
if (b != gf_zero()) {
poly_set_coeff(rem, i, gf_zero());
for (j = i - 1; j >= i - dd; --j)
poly_addto_coeff(rem, j, gf_mul_fast(b, poly_coeff(d, dd - i + j)));
}
}
poly_free(rem);
return quo;
}
// 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;
}
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);
}
}
// 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);
}
}
// 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 main(void) {
field_t fp, fx;
mpz_t prime;
darray_t list;
int p = 7;
// Exercise poly_is_irred() with a sieve of Erastosthenes for polynomials.
darray_init(list);
mpz_init(prime);
mpz_set_ui(prime, p);
field_init_fp(fp, prime);
field_init_poly(fx, fp);
element_t e;
element_init(e, fp);
// Enumerate polynomials in F_p[x] up to degree 2.
int a[3], d;
a[0] = a[1] = a[2] = 0;
for(;;) {
element_ptr f = pbc_malloc(sizeof(*f));
element_init(f, fx);
int j;
for(j = 0; j < 3; j++) {
element_set_si(e, a[j]);
poly_set_coeff(f, e, j);
}
// Test poly_degree().
for(j = 2; !a[j] && j >= 0; j--);
EXPECT(poly_degree(f) == j);
// Add monic polynomials to the list.
if (j >= 0 && a[j] == 1) darray_append(list, f);
else {
element_clear(f);
free(f);
}
// Next!
d = 0;
for(;;) {
a[d]++;
if (a[d] >= p) {
a[d] = 0;
d++;
if (d > 2) goto break2;
} else break;
}
}
break2: ;
// Find all composite monic polynomials of degree 3 or less.
darray_t prodlist;
darray_init(prodlist);
void outer(void *data) {
element_ptr f = data;
void inner(void *data2) {
element_ptr g = data2;
if (!poly_degree(f) || !poly_degree(g)) return;
if (poly_degree(f) + poly_degree(g) > 3) return;
element_ptr h = pbc_malloc(sizeof(*h));
element_init(h, fx);
element_mul(h, f, g);
darray_append(prodlist, h);
EXPECT(!poly_is_irred(h));
}
// 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);
}
