static void test_gf33m_inverse(void) {
element_random(a3);
element_invert(b3, a3);
element_mul(a3, a3, b3);
element_ptr a0 = element_item(a3, 0);
EXPECT(!element_cmp(a0, e1));
}
static val_ptr run_item(val_ptr v[]) {
mpz_t z;
mpz_init(z);
element_to_mpz(z, v[1]->elem);
int i = mpz_get_si(z);
mpz_clear(z);
element_ptr a = element_item(v[0]->elem, i);
element_ptr e = (element_ptr)pbc_malloc(sizeof(*e));
element_init_same_as(e, a);
element_set(e, a);
return val_new_element(e);
}
// Define l = aX + bY + c where a, b, c are in Fq.
// Compute e0 = l(Q) specialized for the case when Q has the form
// (Qx, Qy * sqrt(v)) where Qx, Qy are in Fqd and v is the quadratic nonresidue
// used to construct the quadratic field extension Fqk of Fqd.
static inline void d_miller_evalfn(element_t e0,
element_t a, element_t b, element_t c, element_t Qx, element_t Qy) {
element_ptr re_out = element_x(e0);
element_ptr im_out = element_y(e0);
int i;
int d = polymod_field_degree(re_out->field);
for (i = 0; i < d; i++) {
element_mul(element_item(re_out, i), element_item(Qx, i), a);
element_mul(element_item(im_out, i), element_item(Qy, i), b);
}
element_add(element_item(re_out, 0), element_item(re_out, 0), c);
}
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);
}
static element_ptr mulg_item(element_ptr e, int i) {
return element_item(e->data, i);
}
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 f_init_pairing(pairing_t pairing, void *data) {
f_param_ptr param = data;
f_pairing_data_ptr p;
element_t irred;
element_t e0, e1, e2;
p = pairing->data = pbc_malloc(sizeof(f_pairing_data_t));
mpz_init(pairing->r);
mpz_set(pairing->r, param->r);
field_init_fp(pairing->Zr, pairing->r);
field_init_fp(p->Fq, param->q);
p->Fq->nqr = pbc_malloc(sizeof(element_t));
element_init(p->Fq->nqr, p->Fq);
element_set_mpz(p->Fq->nqr, param->beta);
field_init_quadratic(p->Fq2, p->Fq);
field_init_poly(p->Fq2x, p->Fq2);
element_init(irred, p->Fq2x);
// Call poly_set_coeff1() first so we can use element_item() for the other
// coefficients.
poly_set_coeff1(irred, 6);
element_init(p->negalpha, p->Fq2);
element_init(p->negalphainv, p->Fq2);
element_set_mpz(element_x(p->negalpha), param->alpha0);
element_set_mpz(element_y(p->negalpha), param->alpha1);
element_set(element_item(irred, 0), p->negalpha);
field_init_polymod(p->Fq12, irred);
element_neg(p->negalpha, p->negalpha);
element_invert(p->negalphainv, p->negalpha);
element_clear(irred);
element_init(e0, p->Fq);
element_init(e1, p->Fq);
element_init(e2, p->Fq2);
// Initialize the curve Y^2 = X^3 + b.
element_set_mpz(e1, param->b);
field_init_curve_ab(p->Eq, e0, e1, pairing->r, NULL);
// Initialize the curve Y^2 = X^3 - alpha0 b - alpha1 sqrt(beta) b.
element_set_mpz(e0, param->alpha0);
element_neg(e0, e0);
element_mul(element_x(e2), e0, e1);
element_set_mpz(e0, param->alpha1);
element_neg(e0, e0);
element_mul(element_y(e2), e0, e1);
element_clear(e0);
element_init(e0, p->Fq2);
field_init_curve_ab(p->Etwist, e0, e2, pairing->r, NULL);
element_clear(e0);
element_clear(e1);
element_clear(e2);
mpz_t ndonr;
mpz_init(ndonr);
// ndonr temporarily holds the trace.
mpz_sub(ndonr, param->q, param->r);
mpz_add_ui(ndonr, ndonr, 1);
// TODO: We can use a smaller quotient_cmp, but I have to figure out
// BN curves again.
pbc_mpz_curve_order_extn(ndonr, param->q, ndonr, 12);
mpz_divexact(ndonr, ndonr, param->r);
mpz_divexact(ndonr, ndonr, param->r);
field_curve_set_quotient_cmp(p->Etwist, ndonr);
mpz_clear(ndonr);
pairing->G1 = p->Eq;
pairing->G2 = p->Etwist;
pairing_GT_init(pairing, p->Fq12);
pairing->finalpow = f_finalpow;
pairing->map = f_pairing;
pairing->clear_func = f_pairing_clear;
mpz_init(p->tateexp);
/* unoptimized tate exponent
mpz_pow_ui(p->tateexp, param->q, 12);
mpz_sub_ui(p->tateexp, p->tateexp, 1);
mpz_divexact(p->tateexp, p->tateexp, param->r);
*/
mpz_ptr z = p->tateexp;
mpz_mul(z, param->q, param->q);
mpz_sub_ui(z, z, 1);
mpz_mul(z, z, param->q);
mpz_mul(z, z, param->q);
mpz_add_ui(z, z, 1);
mpz_divexact(z, z, param->r);
element_init(p->xpowq2, p->Fq2);
element_init(p->xpowq6, p->Fq2);
element_init(p->xpowq8, p->Fq2);
element_t xpowq;
element_init(xpowq, p->Fq12);
//there are smarter ways since we know q = 1 mod 6
//and that x^6 = -alpha
//but this is fast enough
element_set1(element_item(xpowq, 1));
element_pow_mpz(xpowq, xpowq, param->q);
element_pow_mpz(xpowq, xpowq, param->q);
element_set(p->xpowq2, element_item(xpowq, 1));
element_pow_mpz(xpowq, xpowq, param->q);
element_pow_mpz(xpowq, xpowq, param->q);
element_pow_mpz(xpowq, xpowq, param->q);
element_pow_mpz(xpowq, xpowq, param->q);
element_set(p->xpowq6, element_item(xpowq, 1));
element_pow_mpz(xpowq, xpowq, param->q);
element_pow_mpz(xpowq, xpowq, param->q);
element_set(p->xpowq8, element_item(xpowq, 1));
element_clear(xpowq);
}
/* computation of $c <- U ^ {3^{3m} - 1}$
* This is the algorithm 6 in the paper above. */
static void algorithm6(element_t c, element_t u) {
element_ptr u0 = ITEM(u,0,0), u1 = ITEM(u,0,1), u2 = ITEM(u,1,0), u3 =
ITEM(u,1,1), u4 = ITEM(u,2,0), u5 = ITEM(u,2,1);
field_ptr f = FIELD(u0); /*GF(3^m)*/
field_t f3; /*GF(3^{3*m})*/
field_init_gf33m(f3, f);
element_t v0, v1, m0, m1, m2, a0, a1, i;
element_init(v0, f3);
element_init(v1, f3);
element_init(m0, f3);
element_init(m1, f3);
element_init(m2, f3);
element_init(a0, f3);
element_init(a1, f3);
element_init(i, f3);
element_set(element_item(v0, 0), u0);
element_set(element_item(v0, 1), u2);
element_set(element_item(v0, 2), u4);
element_set(element_item(v1, 0), u1);
element_set(element_item(v1, 1), u3);
element_set(element_item(v1, 2), u5);
element_mul(m0, v0, v0);
element_mul(m1, v1, v1);
element_mul(m2, v0, v1);
element_sub(a0, m0, m1);
element_add(a1, m0, m1);
element_invert(i, a1);
element_mul(v0, a0, i);
element_mul(v1, m2, i);
element_set(ITEM(c,0,0), element_item(v0, 0));
element_set(ITEM(c,1,0), element_item(v0, 1));
element_set(ITEM(c,2,0), element_item(v0, 2));
element_set(ITEM(c,0,1), element_item(v1, 0));
element_set(ITEM(c,1,1), element_item(v1, 1));
element_set(ITEM(c,2,1), element_item(v1, 2));
element_clear(v0);
element_clear(v1);
element_clear(m0);
element_clear(m1);
element_clear(m2);
element_clear(a0);
element_clear(a1);
element_clear(i);
field_clear(f3);
}
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);
}
// Computes a curve and sets fp to the field it is defined over using the
// complex multiplication method, where cm holds the appropriate information
// (e.g. discriminant, field order).
static void compute_cm_curve(d_param_ptr param, pbc_cm_ptr cm) {
element_t hp, root;
field_t fp, fpx;
field_t cc;
field_init_fp(fp, cm->q);
field_init_poly(fpx, fp);
element_init(hp, fpx);
mpz_t *coefflist;
int n = (int)pbc_hilbert(&coefflist, cm->D);
// Temporarily set the coefficient of x^{n-1} to 1 so hp has degree n - 1,
// allowing us to use poly_coeff().
poly_set_coeff1(hp, n - 1);
int i;
for (i = 0; i < n; i++) {
element_set_mpz(element_item(hp, i), coefflist[i]);
}
pbc_hilbert_free(coefflist, n);
// TODO: Remove x = 0, 1728 roots.
// TODO: What if there are no roots?
//printf("hp ");
//element_out_str(stdout, 0, hp);
//printf("\n");
element_init(root, fp);
poly_findroot(root, hp);
//printf("root = ");
//element_out_str(stdout, 0, root);
//printf("\n");
element_clear(hp);
field_clear(fpx);
// The root is the j-invariant of the desired curve.
field_init_curve_j(cc, root, cm->n, NULL);
element_clear(root);
// We may need to twist it.
{
// Pick a random point P and twist the curve if it has the wrong order.
element_t P;
element_init(P, cc);
element_random(P);
element_mul_mpz(P, P, cm->n);
if (!element_is0(P)) field_reinit_curve_twist(cc);
element_clear(P);
}
mpz_set(param->q, cm->q);
mpz_set(param->n, cm->n);
mpz_set(param->h, cm->h);
mpz_set(param->r, cm->r);
element_to_mpz(param->a, curve_field_a_coeff(cc));
element_to_mpz(param->b, curve_field_b_coeff(cc));
param->k = cm->k;
{
mpz_t z;
mpz_init(z);
// Compute order of curve in F_q^k.
// n = q - t + 1 hence t = q - n + 1
mpz_sub(z, param->q, param->n);
mpz_add_ui(z, z, 1);
pbc_mpz_trace_n(z, param->q, z, param->k);
mpz_pow_ui(param->nk, param->q, param->k);
mpz_sub_ui(z, z, 1);
mpz_sub(param->nk, param->nk, z);
mpz_mul(z, param->r, param->r);
mpz_divexact(param->hk, param->nk, z);
mpz_clear(z);
}
field_clear(cc);
field_clear(fp);
}
void KSET0(element_t out){
element_set0(out);
element_ptr re_out = element_x(out);
element_set0(element_item(re_out,0));
}
