void pbc_param_init_e_gen(pbc_param_t par, int rbits, int qbits) {
e_init(par);
e_param_ptr p = par->data;
//3 takes 2 bits to represent
int hbits = (qbits - 2) / 2 - rbits;
mpz_ptr q = p->q;
mpz_ptr r = p->r;
mpz_ptr h = p->h;
mpz_t n;
field_t Fq;
field_t cc;
element_t j;
int found = 0;
//won't find any curves is hbits is too low
if (hbits < 3) hbits = 3;
mpz_init(n);
do {
int i;
mpz_set_ui(r, 0);
if (rand() % 2) {
p->exp2 = rbits - 1;
p->sign1 = 1;
} else {
p->exp2 = rbits;
p->sign1 = -1;
}
mpz_setbit(r, p->exp2);
p->exp1 = (rand() % (p->exp2 - 1)) + 1;
//use q as a temp variable
mpz_set_ui(q, 0);
mpz_setbit(q, p->exp1);
if (p->sign1 > 0) {
mpz_add(r, r, q);
} else {
mpz_sub(r, r, q);
}
if (rand() % 2) {
p->sign0 = 1;
mpz_add_ui(r, r, 1);
} else {
p->sign0 = -1;
mpz_sub_ui(r, r, 1);
}
if (!mpz_probab_prime_p(r, 10)) continue;
for (i=0; i<10; i++) {
//use q as a temp variable
mpz_set_ui(q, 0);
mpz_setbit(q, hbits + 1);
pbc_mpz_random(h, q);
mpz_mul(h, h, h);
mpz_mul_ui(h, h, 3);
//finally q takes the value it should
mpz_mul(n, r, r);
mpz_mul(n, n, h);
mpz_add_ui(q, n, 1);
if (mpz_probab_prime_p(q, 10)) {
found = 1;
break;
}
}
} while (!found);
/*
do {
mpz_set_ui(r, 0);
mpz_setbit(r, rbits);
pbc_mpz_random(r, r);
mpz_nextprime(r, r);
mpz_mul(n, r, r);
mpz_mul_ui(n, n, 3);
mpz_add_ui(q, n, 1);
} while (!mpz_probab_prime_p(q, 10));
*/
field_init_fp(Fq, q);
element_init(j, Fq);
element_set_si(j, 1);
field_init_curve_b(cc, j, n, NULL);
element_clear(j);
// We may need to twist it.
{
// Pick a random point P and twist the curve if P has the wrong order.
element_t P;
element_init(P, cc);
element_random(P);
element_mul_mpz(P, P, n);
if (!element_is0(P)) field_reinit_curve_twist(cc);
element_clear(P);
}
element_to_mpz(p->a, curve_field_a_coeff(cc));
element_to_mpz(p->b, curve_field_b_coeff(cc));
mpz_clear(n);
}
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);
}
