static int fi_is_sqr(element_ptr e) {
// x + yi is a square <=> x^2 + y^2 is (in the base field).
// Proof: (=>) if x+yi = (a+bi)^2, then a^2 - b^2 = x, 2ab = y,
// thus (a^2 + b^2)^2 = (a^2 - b^2)^2 + (2ab)^2 = x^2 + y^2
// (<=) Suppose A^2 = x^2 + y^2. If there exist a, b satisfying:
// a^2 = (+-A + x)/2, b^2 = (+-A - x)/2
// then (a + bi)^2 = x + yi.
//
// We show that exactly one of (A + x)/2, (-A + x)/2 is a quadratic residue
// (thus a, b do exist). Suppose not. Then the product (x^2 - A^2) / 4 is
// some quadratic residue, a contradiction since this would imply x^2 - A^2 =
// -y^2 is also a quadratic residue, but we know -1 is not a quadratic
// residue. QED.
eptr p = e->data;
element_t e0, e1;
int result;
element_init(e0, p->x->field);
element_init(e1, e0->field);
element_square(e0, p->x);
element_square(e1, p->y);
element_add(e0, e0, e1);
result = element_is_sqr(e0);
element_clear(e0);
element_clear(e1);
return result;
}
static void fi_sqrt(element_ptr n, element_ptr e) {
eptr p = e->data;
eptr r = n->data;
element_t e0, e1, e2;
// If (a+bi)^2 = x+yi then 2a^2 = x +- sqrt(x^2 + y^2)
// where we choose the sign so that a exists, and 2ab = y.
// Thus 2b^2 = - (x -+ sqrt(x^2 + y^2)).
element_init(e0, p->x->field);
element_init(e1, e0->field);
element_init(e2, e0->field);
element_square(e0, p->x);
element_square(e1, p->y);
element_add(e0, e0, e1);
element_sqrt(e0, e0);
// e0 = sqrt(x^2 + y^2)
element_add(e1, p->x, e0);
element_set_si(e2, 2);
element_invert(e2, e2);
element_mul(e1, e1, e2);
// e1 = (x + sqrt(x^2 + y^2))/2
if (!element_is_sqr(e1)) {
element_sub(e1, e1, e0);
// e1 should be a square.
}
element_sqrt(e0, e1);
element_add(e1, e0, e0);
element_invert(e1, e1);
element_mul(r->y, p->y, e1);
element_set(r->x, e0);
element_clear(e0);
element_clear(e1);
element_clear(e2);
}
static void curve_from_hash(element_t a, void *data, int len) {
element_t t, t1;
point_ptr p = (point_ptr)a->data;
curve_data_ptr cdp = (curve_data_ptr)a->field->data;
element_init(t, cdp->field);
element_init(t1, cdp->field);
p->inf_flag = 0;
element_from_hash(p->x, data, len);
for(;;) {
element_square(t, p->x);
element_add(t, t, cdp->a);
element_mul(t, t, p->x);
element_add(t, t, cdp->b);
if (element_is_sqr(t)) break;
// Compute x <- x^2 + 1 and try again.
element_square(p->x, p->x);
element_set1(t);
element_add(p->x, p->x, t);
}
element_sqrt(p->y, t);
if (element_sgn(p->y) < 0) element_neg(p->y, p->y);
if (cdp->cofac) element_mul_mpz(a, a, cdp->cofac);
element_clear(t);
element_clear(t1);
}
//--------------------------------------------------
// square root in extended Fp
//--------------------------------------------------
int bn254_fp2_sqrt(Element z, const Element x)
{
mpz_t _v;
int m, r, i;
field_precomp_sqrt_p ps;
Element *t = field(z)->tmp;
if (!element_is_sqr(x)) {
return FALSE;
}
ps = ((field_precomp_p)(field(x)->precomp))->ps;
element_set(t[0], ps->n_v); // t0 = n^v
r = ps->e; // r = e
mpz_init_set(_v, ps->v);
mpz_sub_ui(_v, _v, 1);
mpz_tdiv_q_2exp(_v, _v, 1);
element_pow(t[1], x, _v); // t1 = x^{(v-1)/2}
element_sqr(t[2], t[1]);
element_mul(t[2], t[2], x); // t2 = x*t1^2
element_mul(t[1], x, t[1]); // t1 = x*t1
mpz_clear(_v);
while (!element_is_one(t[2]))
{
m = 0;
element_set(t[3], t[2]);
do {
element_sqr(t[3], t[3]);
m++;
}
while (!element_is_one(t[3]) && m < r);
r = r - m - 1;
element_set(t[3], t[0]);
for (i = 1; i <= r; i++) {
element_sqr(t[3], t[3]);
} // t3 = t2^{r-m-1}
element_sqr(t[0], t[3]); // t0 = t3^2
r = m;
element_mul(t[1], t[1], t[3]);// t1 = t1*t3
element_mul(t[2], t[2], t[0]);// t2 = t2*t0
}
element_set(z, t[1]);
return TRUE;
}
//============================================
// 平方根計算のテスト
//============================================
void test_sqrt(Field f)
{
int i;
unsigned long long int t1, t2;
Element a, b, c, d;
element_init(a, f);
element_init(b, f);
element_init(c, f);
element_init(d, f);
for (i = 0; i < 50; i++)
{
element_random(a);
element_sqr(b, a);
assert(element_is_sqr(b));
element_sqrt(c, b);
element_sqr(d, c);
assert(element_cmp(d, b) == 0);
}
t1 = rdtsc();
for (i = 0; i < N; i++) { element_is_sqr(b); }
t2 = rdtsc();
printf("element is sqr: %.2lf [clock]\n", (double)(t2 - t1) / N);
t1 = rdtsc();
for (i = 0; i < M; i++) { element_sqrt(c, b); }
t2 = rdtsc();
printf("element sqrt: %.2lf [clock]\n", (double)(t2 - t1) / M);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(d);
}
static int fq_is_sqr(element_ptr e) {
//x + y sqrt(nqr) is a square iff x^2 - nqr y^2 is (in the base field)
eptr p = e->data;
element_t e0, e1;
element_ptr nqr = fq_nqr(e->field);
int result;
element_init(e0, p->x->field);
element_init(e1, e0->field);
element_square(e0, p->x);
element_square(e1, p->y);
element_mul(e1, e1, nqr);
element_sub(e0, e0, e1);
result = element_is_sqr(e0);
element_clear(e0);
element_clear(e1);
return result;
}
//singular with node: y^2 = x^3 + x^2
static void sn_random(element_t a) {
point_ptr p = a->data;
element_t t;
element_init(t, p->x->field);
p->inf_flag = 0;
do {
element_random(p->x);
if (element_is0(p->x)) continue;
element_square(t, p->x);
element_add(t, t, p->x);
element_mul(t, t, p->x);
} while (!element_is_sqr(t));
element_sqrt(p->y, t);
element_clear(t);
}
static void curve_random_no_cofac_solvefory(element_ptr a) {
//TODO: with 0.5 probability negate y-coord
curve_data_ptr cdp = (curve_data_ptr)a->field->data;
point_ptr p = (point_ptr)a->data;
element_t t;
element_init(t, cdp->field);
p->inf_flag = 0;
do {
element_random(p->x);
element_square(t, p->x);
element_add(t, t, cdp->a);
element_mul(t, t, p->x);
element_add(t, t, cdp->b);
} while (!element_is_sqr(t));
element_sqrt(p->y, t);
element_clear(t);
}
//---------------------------------------------------------
// precomputation for sqrt
//---------------------------------------------------------
void bn254_fp2_precomp_sqrt(field_precomp_sqrt_p ps, const Field f)
{
//-----------------------------
// decompose of value
// (p^2-1) = 2^e * v
//-----------------------------
mpz_init_set(ps->v, f->order);
mpz_sub_ui(ps->v, ps->v, 1);
ps->e = (int)mpz_scan1(ps->v, 0);
mpz_fdiv_q_2exp(ps->v, ps->v, ps->e);
//----------------------------------
// n_v = n^v :
// n is some integer (n/p) = -1
//----------------------------------
element_init(ps->n_v, f);
do { element_random(ps->n_v); }
while (element_is_sqr(ps->n_v));
element_pow(ps->n_v, ps->n_v, ps->v);
}
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 void fq_sqrt(element_ptr n, element_ptr e) {
eptr p = e->data;
eptr r = n->data;
element_ptr nqr = fq_nqr(n->field);
element_t e0, e1, e2;
//if (a+b sqrt(nqr))^2 = x+y sqrt(nqr) then
//2a^2 = x +- sqrt(x^2 - nqr y^2)
//(take the sign which allows a to exist)
//and 2ab = y
element_init(e0, p->x->field);
element_init(e1, e0->field);
element_init(e2, e0->field);
element_square(e0, p->x);
element_square(e1, p->y);
element_mul(e1, e1, nqr);
element_sub(e0, e0, e1);
element_sqrt(e0, e0);
//e0 = sqrt(x^2 - nqr y^2)
element_add(e1, p->x, e0);
element_set_si(e2, 2);
element_invert(e2, e2);
element_mul(e1, e1, e2);
//e1 = (x + sqrt(x^2 - nqr y^2))/2
if (!element_is_sqr(e1)) {
element_sub(e1, e1, e0);
//e1 should be a square
}
element_sqrt(e0, e1);
element_add(e1, e0, e0);
element_invert(e1, e1);
element_mul(r->y, p->y, e1);
element_set(r->x, e0);
element_clear(e0);
element_clear(e1);
element_clear(e2);
}