// Existing points are invalidated as this mangles c.
void field_reinit_curve_twist(field_ptr c) {
curve_data_ptr cdp = (curve_data_ptr)c->data;
element_ptr nqr = field_get_nqr(cdp->field);
element_mul(cdp->a, cdp->a, nqr);
element_mul(cdp->a, cdp->a, nqr);
element_mul(cdp->b, cdp->b, nqr);
element_mul(cdp->b, cdp->b, nqr);
element_mul(cdp->b, cdp->b, nqr);
// Recompute generators.
curve_random_no_cofac_solvefory(cdp->gen_no_cofac);
if (cdp->cofac) {
element_mul_mpz(cdp->gen, cdp->gen_no_cofac, cdp->cofac);
} else{
element_set(cdp->gen, cdp->gen_no_cofac);
}
}
// Return the quadratic nonresidue used to build this field.
// Should only be called from routines used exclusively by the generic quadratic
// extension code.
static inline element_ptr fq_nqr(field_ptr f) {
return field_get_nqr((field_ptr) f->data);
}
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);
}
static void d_init_pairing(pairing_ptr pairing, void *data) {
d_param_ptr param = data;
pptr p;
element_t a, b;
element_t irred;
int d = param->k / 2;
int i;
if (param->k % 2) pbc_die("k must be even");
mpz_init(pairing->r);
mpz_set(pairing->r, param->r);
field_init_fp(pairing->Zr, pairing->r);
pairing->map = cc_pairing;
pairing->prod_pairings = cc_pairings_affine;
pairing->is_almost_coddh = cc_is_almost_coddh;
p = pairing->data = pbc_malloc(sizeof(*p));
field_init_fp(p->Fq, param->q);
element_init(a, p->Fq);
element_init(b, p->Fq);
element_set_mpz(a, param->a);
element_set_mpz(b, param->b);
field_init_curve_ab(p->Eq, a, b, pairing->r, param->h);
field_init_poly(p->Fqx, p->Fq);
element_init(irred, p->Fqx);
poly_set_coeff1(irred, d);
for (i = 0; i < d; i++) {
element_set_mpz(element_item(irred, i), param->coeff[i]);
}
field_init_polymod(p->Fqd, irred);
element_clear(irred);
p->Fqd->nqr = pbc_malloc(sizeof(element_t));
element_init(p->Fqd->nqr, p->Fqd);
element_set_mpz(((element_t *) p->Fqd->nqr->data)[0], param->nqr);
field_init_quadratic(p->Fqk, p->Fqd);
// Compute constants involved in the final powering.
if (param->k == 6) {
mpz_ptr q = param->q;
mpz_ptr z = pairing->phikonr;
mpz_init(z);
mpz_mul(z, q, q);
mpz_sub(z, z, q);
mpz_add_ui(z, z, 1);
mpz_divexact(z, z, pairing->r);
element_ptr e = p->xpowq;
element_init(e, p->Fqd);
element_set1(((element_t *) e->data)[1]);
element_pow_mpz(e, e, q);
element_init(p->xpowq2, p->Fqd);
element_square(p->xpowq2, e);
} else {
mpz_init(p->tateexp);
mpz_sub_ui(p->tateexp, p->Fqk->order, 1);
mpz_divexact(p->tateexp, p->tateexp, pairing->r);
}
field_init_curve_ab_map(p->Etwist, p->Eq, element_field_to_polymod, p->Fqd, pairing->r, NULL);
field_reinit_curve_twist(p->Etwist);
mpz_t ndonr;
mpz_init(ndonr);
// ndonr temporarily holds the trace.
mpz_sub(ndonr, param->q, param->n);
mpz_add_ui(ndonr, ndonr, 1);
// Negate it because we want the trace of the twist.
mpz_neg(ndonr, ndonr);
pbc_mpz_curve_order_extn(ndonr, param->q, ndonr, d);
mpz_divexact(ndonr, ndonr, param->r);
field_curve_set_quotient_cmp(p->Etwist, ndonr);
mpz_clear(ndonr);
element_init(p->nqrinv, p->Fqd);
element_invert(p->nqrinv, field_get_nqr(p->Fqd));
element_init(p->nqrinv2, p->Fqd);
element_square(p->nqrinv2, p->nqrinv);
pairing->G1 = p->Eq;
pairing->G2 = p->Etwist;
p->k = param->k;
pairing_GT_init(pairing, p->Fqk);
pairing->finalpow = cc_finalpow;
// By default use affine coordinates.
cc_miller_no_denom_fn = cc_miller_no_denom_affine;
pairing->option_set = d_pairing_option_set;
pairing->pp_init = d_pairing_pp_init;
pairing->pp_clear = d_pairing_pp_clear;
pairing->pp_apply = d_pairing_pp_apply;
pairing->clear_func = d_pairing_clear;
element_clear(a);
element_clear(b);
}