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function.c
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function.c
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#include "schoof.h"
/* Fonction prochain nombre premier
*
* ENTREE :
* Entier op tel que op > 2
* SORTIE :
* Entier rop
*
*/
void fmpz_nextprime(fmpz_t rop, fmpz_t op)
{
fmpz_add_ui(rop, op, 2);
while(!fmpz_is_prime(rop))
{
fmpz_add_ui(rop, rop, 2);
}
}
/////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Fonction polynômes de division :
*
* ENTREE :
* Tableau de Fq-polynôme tab et k la taille du tableau
* Entiers a,b tels que E: y² = x³ + ax + b une courbe elliptique sur Fq
* Fq-polynôme eec représentant la courbe elliptique
* Corps fini fq à q éléments
*
* SORTIE :
* Tableau de Fq-polynôme tab rempli de k polynôme de division
*
*/
void division_polynomial(fq_poly_t *tab, fq_t a, fq_t b, fq_poly_t ecc, ulong k, fq_ctx_t fq)
{
// Déclaration
ulong i, n;
fq_t tmp, tmp1, invtwo;
fq_poly_t tmp_poly;
// Initialisation
fq_init(tmp, fq);
fq_init(tmp1, fq);
fq_init(invtwo, fq);
fq_set_ui(invtwo, 2, fq);
fq_inv(invtwo, invtwo, fq);
// On traite à part les cas de f0 à f4
// Initialisation de f0
fq_poly_zero(tab[0], fq); // f0 = 0
fq_poly_one(tab[1], fq); // f1 = 1
fq_set_ui(tmp, 2, fq); // tmp = 2
fq_poly_set_fq(tab[2], tmp, fq); // f2 = 2
// Initialisation de f3
fq_set_ui(tmp, 3, fq); // tmp = 3
fq_poly_set_coeff(tab[3], 4, tmp, fq);
fq_mul_ui(tmp, a, 6, fq); // tmp = 6a
fq_poly_set_coeff(tab[3], 2, tmp, fq);
fq_mul_ui(tmp, b, 12, fq); // tmp = 12b
fq_poly_set_coeff(tab[3], 1, tmp, fq);
fq_sqr(tmp, a, fq); // tmp = a²
fq_neg(tmp, tmp, fq); // tmp = -a²
fq_poly_set_coeff(tab[3], 0, tmp, fq);
// Initialisation de f4
fq_pow_ui(tmp, a, 3, fq); // tmp = a³
fq_sqr(tmp1, b, fq); // tmp1 = b²
fq_mul_ui(tmp1, tmp1, 8, fq); // tmp1 = 8b²
fq_add(tmp, tmp, tmp1, fq); // tmp = a³+8b²
fq_mul_ui(tmp, tmp, 4, fq); // tmp = 4(a³+8b²)
fq_neg(tmp, tmp,fq); // tmp = -4(a³+8b²)
fq_poly_set_coeff(tab[4], 0, tmp, fq);
fq_mul(tmp, a, b, fq); // tmp = ab
fq_mul_ui(tmp, tmp, 16,fq); // tmp = 16ab
fq_neg(tmp, tmp, fq); // tmp = -16ab
fq_poly_set_coeff(tab[4], 1, tmp, fq);
fq_sqr(tmp, a, fq); // tmp = a²
fq_mul_ui(tmp, tmp, 20, fq); // tmp = 20a²
fq_neg(tmp, tmp, fq); // tmp = -20a²
fq_poly_set_coeff(tab[4], 2, tmp, fq);
fq_mul_ui(tmp, b, 80, fq); // tmp = 80b
fq_poly_set_coeff(tab[4], 3, tmp, fq);
fq_mul_ui(tmp, a, 20, fq); // tmp = 20a
fq_poly_set_coeff(tab[4], 4, tmp, fq);
fq_set_ui(tmp, 4, fq); // tmp = 4
fq_poly_set_coeff(tab[4], 6, tmp, fq);
// Libération mémoire
fq_clear(tmp, fq); fq_clear(tmp1, fq);
// Initialisation
fq_poly_init(tmp_poly, fq);
// Remplissage du tableau
for(i = 5; i <= k; i++)
{
// Itération sur le i
if(i & 0x1)
{
// Cas i impair
n = i - 1;
n >>= 1; // i = 2n + 1
if(n & 0x1)
{
// Cas n impair
fq_poly_pow(tmp_poly, tab[n + 1], 3, fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[n - 1], fq);
fq_poly_sqr(tab[i], ecc, fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[i], fq); // c'est la réduction et élimination des y dans le polynôme de division
fq_poly_pow(tab[i], tab[n], 3, fq);
fq_poly_mul(tab[i], tab[i], tab[n + 2], fq);
fq_poly_sub(tab[i], tab[i], tmp_poly, fq);
}
else
{
// Cas n pair
fq_poly_pow(tab[i], tab[n], 3, fq);
fq_poly_mul(tab[i], tab[i], tab[n + 2], fq);
fq_poly_sqr(tmp_poly, ecc, fq);
fq_poly_mul(tab[i], tab[i], tmp_poly, fq); // c'est la réduction et élimination des y dans le polynôme de division
fq_poly_pow(tmp_poly, tab[n + 1], 3, fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[n - 1], fq);
fq_poly_sub(tab[i], tab[i], tmp_poly, fq);
}
}
else
{
// Cas i pair
n = i >> 1; // i = 2n
fq_poly_sqr(tab[i], tab[n - 1], fq);
fq_poly_mul(tab[i], tab [i], tab[n + 2], fq);
fq_poly_sqr(tmp_poly, tab[n + 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[n - 2], fq);
fq_poly_sub(tab[i], tab[i], tmp_poly, fq);
fq_poly_mul(tab[i], tab[i], tab[n], fq);
fq_poly_scalar_mul_fq(tab[i], tab[i], invtwo, fq);
}
}
// Libération de la mémoire
fq_poly_clear(tmp_poly, fq);
fq_clear(invtwo, fq);
}
/////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
/* Algorithme de Schoof :
*
* ENTREE :
* Entier q premier tel que Fq un corps fini à q éléments
* Entiers a,b tels que E: y² = x³ + ax + b une courbe elliptique sur Fq
*
* SORTIE :
* Entier card tel que #E(Fq) = card
*
*/
void schoof(fmpz_t card, fq_t a, fq_t b, fmpz_t q, fq_ctx_t fq)
{
// Déclaration :
ulong lmax, i, k, l, tho;
fmpz_t M, l_fmpz, lmax_fmpz, trace, sqrt, tmp, k_fmpz, tho_fmpz;
fq_t tmp_fq, tmp1_fq, one; // constantes temporaires;
fq_poly_t ecc; // ecc = X³ + aX + b
fq_poly_t gcd_poly; // pgcd
fq_poly_t frob; // frob = X^q - X
fq_poly_t frob2; // frob2 = X^q² - X
fq_poly_t frob3; // frob3 = X^q² + X^q + X
fq_poly_t tmp_poly;
fq_poly_t tmp1_poly;
fq_poly_t alpha;
fq_poly_t beta;
fq_poly_t* tab = NULL;
// Initialisation :
fmpz_init(trace);
fmpz_init(sqrt);
fmpz_init(tmp);
fmpz_init(k_fmpz);
fmpz_init(tho_fmpz);
fmpz_init(lmax_fmpz);
fmpz_init_set_ui(M, 2);
fmpz_init_set_ui(l_fmpz, 3);
fmpz_sqrt(sqrt, q);
fmpz_mul_ui(sqrt, sqrt, 4);
fq_init(tmp_fq, fq);
fq_init(tmp1_fq, fq);
fq_init(one, fq);
fq_one(tmp_fq, fq);
fq_one(one, fq);
fq_poly_init(ecc, fq);
fq_poly_init(gcd_poly, fq);
fq_poly_init(frob, fq);
fq_poly_init(frob2, fq);
fq_poly_init(frob3, fq);
fq_poly_init(tmp_poly, fq);
fq_poly_init(tmp1_poly,fq);
fq_poly_init(alpha, fq);
fq_poly_init(beta, fq);
// Initialisation de l'équation de la courbe elliptique
fq_poly_set_fq(ecc, b, fq);
fq_poly_set_coeff(ecc, 3, one, fq);
fq_poly_set_coeff(ecc, 1, a, fq);
// Initialisation du polynôme X^q-X
fq_poly_set_coeff(frob, fmpz_get_si(q), one, fq); // je transforme q en slong
fq_neg(tmp_fq, one, fq); // tmp_fq = -1
fq_poly_set_coeff(frob, 1, tmp_fq, fq);
// Initialisation du polynôme X^q²-X
fq_poly_set_coeff(frob2, fmpz_get_si(q) * fmpz_get_si(q), one, fq); // je transforme q en slong puissance 2
fq_poly_set_coeff(frob2, 1, tmp_fq, fq);
// Initialisation du polynôme X^q² + X^q + X
fq_poly_set_coeff(frob3, fmpz_get_si(q) * fmpz_get_si(q) , one, fq); // je transforme q en slong puissance 2
fq_poly_set_coeff(frob3, fmpz_get_si(q), one, fq);
fq_poly_set_coeff(frob3, 1, one, fq);
//Initialisation de lmax pour initialiser le tableau
while(fmpz_cmp(M, sqrt) < 0)
{
fmpz_mul(M, M, l_fmpz);
fmpz_set(lmax_fmpz, l_fmpz);
fmpz_nextprime(l_fmpz, l_fmpz);
if(fmpz_equal(l_fmpz, q)) fmpz_nextprime(l_fmpz, l_fmpz);
}
lmax = fmpz_get_ui(lmax_fmpz);
printf("lmax = %lu\n", lmax);
// Construction du tableau de polynôme de division
tab = malloc((lmax + 3) * sizeof(fq_poly_t));
if(tab == NULL)
{
fprintf(stderr, "You need more memory.\n"); exit(EXIT_FAILURE);
}
for(i = 0; i <= lmax + 2; i++) fq_poly_init(tab[i], fq);
// Remplissage du tableau
division_polynomial(tab, a, b, ecc, lmax + 2, fq);
//for(i = 0; i <= lmax + 2 ; i++) { printf("%lu : ", i) ; fq_poly_print_pretty(tab[i], "X", fq) ; printf("\n"); }
// Cas l = 2 :
fq_poly_gcd(gcd_poly, frob, ecc, fq);
if(fq_poly_is_one(gcd_poly, fq)) fmpz_set_ui(trace, 1); // si pgcd = 1 alors t = 1 [M], sinon t = 0 [M] avec ici M = 2
printf("t = "); fmpz_print(trace) ; printf(" [2]\n");
// Cas général :
fmpz_set_ui(M, 2);
fmpz_set_ui(l_fmpz, 3);
while(fmpz_cmp(M, sqrt) < 0)
{
fmpz_mod(k_fmpz, q, l_fmpz); // k = q [l]
k = fmpz_get_ui(k_fmpz);
l = fmpz_get_ui(l_fmpz);
/*
*
* Cas 1 : Test de (phi_l)²P = +-kP
*
*/
// Pré-calculs des polynôme pour le pgcd
if(k & 0x1)
{
// Cas k impair
fq_poly_sqr(gcd_poly, tab[k], fq);
fq_poly_mul(gcd_poly, gcd_poly, frob2, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
}
else
{
// Cas k pair
fq_poly_sqr(gcd_poly, tab[k], fq);
fq_poly_mul(gcd_poly, gcd_poly, frob2, fq);
fq_poly_mul(gcd_poly, gcd_poly, ecc, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
}
// Test du pgcd
fq_poly_gcd(gcd_poly, gcd_poly, tab[l], fq);
fq_poly_print_pretty(gcd_poly, "x", fq); printf("\n");
if(!fq_poly_is_one(gcd_poly, fq))
{
printf("Cas amélioration de Schoof\n");
// Test de q carré modulo l
if(fmpz_jacobi(q, l_fmpz) == -1)
{
fmpz_zero(tmp); // tmp = 0
fmpz_CRT(trace, trace, M, tmp, l_fmpz, 1); // On fait le théorème chinois
printf("t = "); fmpz_print(tmp); printf(" [%lu]\n", l);
printf("trace = "); fmpz_print(trace); printf("\n");
}
else
{
// Calcul de q = w² modulo l
fmpz_sqrtmod(k_fmpz, q, l_fmpz); // On a mis la variable k à la place w pour libérer de la place
k = fmpz_get_ui(k_fmpz);
// Pré-calculs des polynôme pour le pgcd
if(k & 0x1) // On test si w est pair
{
// Cas impair
fq_poly_sqr(gcd_poly, tab[k], fq);
fq_poly_mul(gcd_poly, gcd_poly, frob, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
}
else
{
// Cas pair
fq_poly_sqr(gcd_poly, tab[k], fq);
fq_poly_mul(gcd_poly, gcd_poly, frob, fq);
fq_poly_mul(gcd_poly, gcd_poly, ecc, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
}
// Test du pgcd
fq_poly_gcd(gcd_poly, gcd_poly, tab[l], fq);
if(fq_poly_is_one(gcd_poly, fq))
{
// Cas dans lequel w n'est pas une valeur propre de phi_l
fmpz_zero(tmp); // tmp = 0
fmpz_CRT(trace, trace, M, tmp, l_fmpz, 1); // On fait le théorème chinois
printf("t = "); fmpz_print(tmp); printf(" [%lu]\n", l);
printf("trace = "); fmpz_print(trace); printf("\n");
}
else
{
// Cas dans lequel phi_l(P) = +-wP
// Pré-calculs des polynôme pour le pgcd
fmpz_mul_ui(tmp, k_fmpz, 2); // tmp = 2w
if(k & 0x1) // On test si w est pair
{
// Cas impair
fq_poly_pow(gcd_poly, ecc, (fmpz_get_si(q) + 3) / 2 , fq); // gcd_poly = (X³+aX+b)^((q+3)/2)
}
else
{
// Cas pair
fq_poly_pow(gcd_poly, ecc, (fmpz_get_si(q) - 1) / 2 , fq); // gcd_poly = (X³+aX+b)^((q-1)/2)
}
fq_set_ui(tmp_fq, 4, fq); // tmp_fq = 4
fq_poly_scalar_mul_fq(gcd_poly, gcd_poly, tmp_fq, fq);
fq_poly_pow(tmp_poly, tab[k], 3, fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k + 2], fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[k - 1], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
//printf("k = %lu\n",k);
// Effet de bords
if(k == 0x1)
{
fq_poly_one(tmp_poly, fq);
fq_poly_neg(tmp_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tmp_poly, fq); // k = 1, donc k - 2 = -1 et tab[-1] = -1
}
else
{
fq_poly_sqr(tmp_poly, tab[k - 2], fq);
}
fq_poly_mul(tmp_poly, tmp_poly, tab[k + 1], fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
// Test du pgcd
fq_poly_gcd(gcd_poly, gcd_poly, tab[l], fq);
if(!fq_poly_is_one(gcd_poly, fq)) fmpz_neg(tmp, tmp); // si pgcd = 1 alors tmp = -2w [l], sinon tmp = 2w [l]
// On fait le théorème chinois
fmpz_CRT(trace, trace, M, tmp, l_fmpz, 1);
printf("t = "); fmpz_print(tmp); printf(" [%lu]\n", l);
printf("trace = "); fmpz_print(trace); printf("\n");
}
}
}
/*
*
* Cas 2 - Test de (phi_l)²P != +-kP
*
*/
else
{
printf("Cas de base\n");
if(k & 0x1) // Test de la parité de k
{
//Cas k impair
// Initialisation de alpha
fmpz_mul(tmp, q, q);
fmpz_add_ui(tmp, tmp, 3);
fmpz_cdiv_q_ui(tmp, tmp, 2); //tmp = (q² + 3)/2
fq_set_ui(tmp_fq, 4, fq); // tmp_fq = 4
fq_poly_sqr(alpha, tab[k - 1], fq);
fq_poly_mul(alpha, alpha, tab[k + 2], fq);
fq_poly_sqr(tmp_poly, tab[k + 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[k - 1], fq);
fq_poly_sub(alpha, alpha, tmp_poly, fq);
fq_poly_mul(alpha, alpha, ecc, fq);
fq_poly_pow(tmp_poly, ecc, fmpz_get_si(tmp), fq);
fq_poly_scalar_mul_fq(tmp_poly, tmp_poly, tmp_fq, fq);
fq_poly_pow(tmp1_poly, tab[k], 3, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_sub(alpha, tmp_poly, alpha, fq);
// Initialisation de beta
fq_poly_neg(beta, frob2, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_mul(beta, beta, tmp_poly, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k+1], fq);
fq_poly_sub(beta, beta, tmp_poly, fq);
fq_poly_scalar_mul_fq(beta, beta, tmp_fq, fq);
fq_poly_mul(beta, beta, tab[k], fq);
}
else
{
//Cas k pair
// Initialisation de alpha
fmpz_mul(tmp, q, q);
fmpz_add_ui(tmp, tmp, 1);
fmpz_cdiv_q_ui(tmp, tmp, 2); //tmp = (q² + 1)/2
fq_set_ui(tmp_fq, 4, fq); // tmp_fq = 4
fq_poly_pow(alpha, ecc, fmpz_get_si(tmp), fq);
fq_poly_scalar_mul_fq(alpha, alpha, tmp_fq, fq);
fq_poly_pow(tmp_poly, tab[k], 3, fq);
fq_poly_mul(alpha, alpha, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k + 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[k - 1], fq);
fq_poly_sub(alpha, tmp_poly, alpha, fq);
fq_poly_sqr(tmp_poly, tab[k - 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[k + 2], fq);
fq_poly_sub(alpha, tmp_poly, alpha, fq);
// Initialisation de beta
fq_poly_neg(beta, frob2, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_mul(beta, beta, tmp_poly, fq);
fq_poly_mul(beta, beta, ecc, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k+1], fq);
fq_poly_sub(beta, beta, tmp_poly, fq);
fq_poly_scalar_mul_fq(beta, beta, tmp_fq, fq);
fq_poly_mul(beta, beta, tab[k], fq);
fq_poly_mul(beta, beta, ecc, fq);
}
// On teste tous les tho possible tel que 0 < tho < (l-1)/2
for(tho = 1; tho <= (l - 1) / 2 ; tho++)
{
if(k & 0x1) // Test de la parité de k
{
if(tho & 0x1)
{
printf("Cas k impair et tho impair\n");
fq_poly_mul(gcd_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_mul(gcd_poly, gcd_poly, ecc, fq);
fq_poly_mul(tmp_poly, frob3, tab[k], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); //beta -> y beta
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly, alpha, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_pow(tmp_poly, tab[tho], 2 * fmpz_get_si(q), fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly,beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_pow(tmp1_poly, ecc, fmpz_get_si(q) + 1, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq); // yq et //beta -> y beta
fq_poly_pow(tmp1_poly, tab[tho - 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_pow(tmp1_poly, tab[tho + 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
}
else
{
printf("Cas k impair et tho pair\n");
fq_poly_mul(gcd_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_mul(gcd_poly, gcd_poly, ecc, fq);
fq_poly_mul(tmp_poly, frob3, tab[k], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); //beta -> y beta
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly, alpha, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_pow(tmp_poly, tab[tho], 2 * fmpz_get_si(q), fq);
fq_poly_pow(tmp1_poly, ecc, fmpz_get_si(q), fq); //y2q
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly,beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); // y2
fq_poly_pow(tmp1_poly, tab[tho - 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_pow(tmp1_poly, tab[tho + 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
}
}
else
{
if(tho & 0x1)
{
printf("Cas k pair et tho impair\n");
fq_poly_mul(gcd_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_mul(tmp_poly, frob3, tab[k], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); // alpha -> y alpha
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly, alpha, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); // y2
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_pow(tmp_poly, tab[tho], 2 * fmpz_get_si(q), fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly,beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
/*
fq_poly_sqr(tmp1_poly, ecc, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
*/
fq_poly_pow(tmp1_poly, ecc, fmpz_get_si(q) + 1, fq); // y2q+2
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_pow(tmp1_poly, tab[tho - 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_pow(tmp1_poly, tab[tho + 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
}
else
{
printf("Cas k pair et tho pair\n");
fq_poly_mul(gcd_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_mul(tmp_poly, frob3, tab[k], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, beta, fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly, alpha, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq);
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); // alpha -> y alpha
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_pow(tmp_poly, tab[tho], 2 * fmpz_get_si(q), fq);
fq_poly_pow(tmp1_poly, ecc, fmpz_get_si(q), fq); // y2q
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_sqr(tmp1_poly,beta, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
/*
fq_poly_sqr(tmp1_poly, ecc, fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
*/
fq_poly_mul(tmp_poly, tmp_poly, ecc, fq); // y2
fq_poly_pow(tmp1_poly, tab[tho - 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_pow(tmp1_poly, tab[tho + 1], fmpz_get_si(q), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_add(gcd_poly, gcd_poly, tmp_poly, fq);
}
}
fq_poly_gcd(gcd_poly, gcd_poly, tab[l], fq);
printf("pgcd = "); fq_poly_print_pretty(gcd_poly, "X", fq); printf("\n");
printf("phil = "); fq_poly_print_pretty(tab[l], "X", fq); printf("\n");
if(!fq_poly_is_one(gcd_poly, fq))
{
fq_set_ui(tmp_fq, 2, fq);
fq_poly_set_coeff(gcd_poly, fmpz_get_si(q) * fmpz_get_si(q), tmp_fq, fq);
fq_poly_set_coeff(gcd_poly, 1, one, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_mul(tmp_poly, tab[k - 1], tab[k + 1], fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_mul(gcd_poly, gcd_poly, alpha, fq);
fq_poly_sqr(tmp_poly, tab[k], fq);
fq_poly_pow(tmp1_poly, ecc, fmpz_get_si(tmp), fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_sub(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_pow(tmp_poly, tab[tho], 3 * fmpz_get_si(q), fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_poly_pow(tmp_poly, tab[tho], fmpz_get_si(q) / 2, fq);
fq_poly_mul(gcd_poly, gcd_poly, tmp_poly, fq);
fq_set_ui(tmp_fq, 4, fq);
fq_poly_scalar_mul_fq(gcd_poly, gcd_poly, tmp_fq, fq);
fq_poly_sqr(tmp_poly, tab[tho - 1], fq);
fq_poly_mul(tmp_poly, tmp_poly, tab[tho + 2], fq);
fq_poly_sqr(tmp1_poly, tab[tho + 1], fq);
// Effet de bords
if(tho == 0x1)
{
fq_poly_add(tmp_poly, tmp_poly, tmp1_poly, fq); // tho = 1 , donc tho - 2 = -1 et tab[-1] = -1
}
else
{
fq_poly_mul(tmp1_poly, tmp1_poly, tab[tho - 2], fq);
fq_poly_sub(tmp_poly, tmp_poly, tmp1_poly, fq);
}
fq_poly_pow(tmp_poly, tmp_poly, fmpz_get_si(q), fq);
fq_poly_sqr(tmp1_poly, tab[k], fq);
fq_poly_mul(tmp_poly, tmp_poly, tmp1_poly, fq);
fq_poly_mul(tmp_poly, tmp_poly, beta, fq);
fmpz_set_ui(tmp, tho);
fq_poly_print_pretty(tmp_poly, "X", fq);printf("\n");
fq_poly_gcd(gcd_poly, gcd_poly, tab[l], fq);
printf("pgcd = "); fq_poly_print_pretty(gcd_poly, "X", fq); printf("\n");
printf("phil = "); fq_poly_print_pretty(tab[l], "X", fq); printf("\n");
if(!fq_poly_is_one(gcd_poly, fq)) fmpz_neg(tmp, tmp);
fmpz_CRT(trace, trace, M, tmp, l_fmpz, 1);
printf("t = "); fmpz_print(tmp); printf(" [%lu]\n", l);
printf("trace = "); fmpz_print(trace); printf("\n");
break;
}
}
}
printfmpz("l", l_fmpz);
// Incrémentation de la boucle
fmpz_mul(M, M, l_fmpz);
// On fait la boucle jusqu'à obtenir un nombre premier, il devrait il y avoir des améliorations possible
fmpz_nextprime(l_fmpz, l_fmpz);
if(fmpz_equal(l_fmpz, q)) fmpz_nextprime(l_fmpz, l_fmpz);
printf("\n");
}
printf("trace = "); fmpz_print(trace); printf("\n");
// Retourne le résultat dans card
fmpz_set_ui(card, 1);
fmpz_add(card, card, q);
fmpz_sub(card, card, trace);
// Libération de la mémoire du tableau :
for(i = 0; i <= lmax + 2; i++) fq_poly_clear(tab[i], fq);
free(tab);
// Libération de la mémoire des variables :
fq_poly_clear(ecc, fq); fq_poly_clear(gcd_poly, fq); fq_poly_clear(frob, fq); fq_poly_clear(frob2, fq); fq_poly_clear(frob3, fq);
fq_poly_clear(tmp_poly, fq); fq_poly_clear(tmp1_poly, fq); fq_poly_clear(alpha, fq); fq_poly_clear(beta, fq);
fq_clear(tmp_fq, fq); fq_clear(tmp1_fq, fq); fq_clear(one, fq);
fmpz_clear(M); fmpz_clear(l_fmpz); fmpz_clear(sqrt); fmpz_clear(trace); fmpz_clear(tmp); fmpz_clear(k_fmpz); fmpz_clear(tho_fmpz);
}