void pbc_param_init_d_gen(pbc_param_ptr p, pbc_cm_ptr cm) { d_param_init(p); d_param_ptr param = p->data; field_t Fq, Fqx, Fqd; element_t irred, nqr; int d = cm->k / 2; int i; compute_cm_curve(param, cm); field_init_fp(Fq, param->q); field_init_poly(Fqx, Fq); element_init(irred, Fqx); do { poly_random_monic(irred, d); } while (!poly_is_irred(irred)); field_init_polymod(Fqd, irred); // Find a quadratic nonresidue of Fqd lying in Fq. element_init(nqr, Fqd); do { element_random(((element_t *) nqr->data)[0]); } while (element_is_sqr(nqr)); param->coeff = pbc_realloc(param->coeff, sizeof(mpz_t) * d); for (i=0; i<d; i++) { mpz_init(param->coeff[i]); element_to_mpz(param->coeff[i], element_item(irred, i)); } element_to_mpz(param->nqr, ((element_t *) nqr->data)[0]); element_clear(nqr); element_clear(irred); field_clear(Fqx); field_clear(Fqd); field_clear(Fq); }
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)); }
void pbc_param_init_f_gen(pbc_param_t p, int bits) { f_init(p); f_param_ptr fp = p->data; //36 is a 6-bit number int xbit = (bits - 6) / 4; //TODO: use binary search to find smallest appropriate x mpz_t x, t; mpz_ptr q = fp->q; mpz_ptr r = fp->r; mpz_ptr b = fp->b; field_t Fq, Fq2, Fq2x; element_t e1; element_t f; field_t c; element_t P; mpz_init(x); mpz_init(t); mpz_setbit(x, xbit); for (;;) { mpz_mul(t, x, x); mpz_mul_ui(t, t, 6); mpz_add_ui(t, t, 1); tryminusx(q, x); mpz_sub(r, q, t); mpz_add_ui(r, r, 1); if (mpz_probab_prime_p(q, 10) && mpz_probab_prime_p(r, 10)) break; tryplusx(q, x); mpz_sub(r, q, t); mpz_add_ui(r, r, 1); if (mpz_probab_prime_p(q, 10) && mpz_probab_prime_p(r, 10)) break; mpz_add_ui(x, x, 1); } field_init_fp(Fq, q); element_init(e1, Fq); for (;;) { element_random(e1); field_init_curve_b(c, e1, r, NULL); element_init(P, c); element_random(P); element_mul_mpz(P, P, r); if (element_is0(P)) break; element_clear(P); field_clear(c); } element_to_mpz(b, e1); element_clear(e1); field_init_quadratic(Fq2, Fq); element_to_mpz(fp->beta, field_get_nqr(Fq)); field_init_poly(Fq2x, Fq2); element_init(f, Fq2x); // Find an irreducible polynomial of the form f = x^6 + alpha. // Call poly_set_coeff1() first so we can use element_item() for the other // coefficients. poly_set_coeff1(f, 6); for (;;) { element_random(element_item(f, 0)); if (poly_is_irred(f)) break; } //extend F_q^2 using f = x^6 + alpha //see if sextic twist contains a subgroup of order r //if not, it's the wrong twist: replace alpha with alpha^5 { field_t ctest; element_t Ptest; mpz_t z0, z1; mpz_init(z0); mpz_init(z1); element_init(e1, Fq2); element_set_mpz(e1, fp->b); element_mul(e1, e1, element_item(f, 0)); element_neg(e1, e1); field_init_curve_b(ctest, e1, r, NULL); element_init(Ptest, ctest); element_random(Ptest); //I'm not sure what the #E'(F_q^2) is, but //it definitely divides n_12 = #E(F_q^12). It contains a //subgroup of order r if and only if //(n_12 / r^2)P != O for some (in fact most) P in E'(F_q^6) mpz_pow_ui(z0, q, 12); mpz_add_ui(z0, z0, 1); pbc_mpz_trace_n(z1, q, t, 12); mpz_sub(z1, z0, z1); mpz_mul(z0, r, r); mpz_divexact(z1, z1, z0); element_mul_mpz(Ptest, Ptest, z1); if (element_is0(Ptest)) { mpz_set_ui(z0, 5); element_pow_mpz(element_item(f, 0), element_item(f, 0), z0); } element_clear(e1); element_clear(Ptest); field_clear(ctest); mpz_clear(z0); mpz_clear(z1); } element_to_mpz(fp->alpha0, element_x(element_item(f, 0))); element_to_mpz(fp->alpha1, element_y(element_item(f, 0))); element_clear(f); field_clear(Fq2x); field_clear(Fq2); field_clear(Fq); mpz_clear(t); mpz_clear(x); }