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/*
* ecgen, tool for generating Elliptic curve domain parameters
* Copyright (C) 2017-2018 J08nY
*/
#include "cm_any.h"
#include "exhaustive/arg.h"
#include "io/output.h"
#include "obj/curve.h"
#include "util/memory.h"
/**
* @brief Slightly adapted algorithm from section 4.2 of
* Constructing elliptic curves of prescribed order,
* Reiner Broker
* @param order
*/
static void good_qdisc_minimal(cm_any_qdisc_t *qdisc, GEN order) {
pari_sp ltop = avma;
GEN d = stoi(2);
size_t j = 0;
while (true) {
++j;
if (!issquarefree(d)) {
d = addis(d, 1);
continue;
}
if (j % 100 == 0) {
debug_log("D: %Ps", d);
}
GEN D = quaddisc(negi(d));
GEN K = Buchall(quadpoly(D), 0, DEFAULTPREC);
GEN alphas = bnfisintnorm(K, order);
long len = glength(alphas);
if (len != 0) {
debug_log("Got some elems of norm N: %Ps", alphas);
for (long i = 1; i <= len; ++i) {
GEN alpha = gel(alphas, i);
GEN trace = nftrace(K, alpha);
GEN p = subii(addis(order, 1), trace);
if (isprime(p)) {
debug_log(
"Got an elem of prime trace: %Pi, d = %Pi, D = %Pi", p,
d, D);
qdisc->p = p;
qdisc->d = D;
gerepileall(ltop, 2, &qdisc->p, &qdisc->d);
return;
}
}
}
d = gerepileupto(ltop, addis(d, 1));
}
}
/**
* @brief Find a fundamental quadratic discriminant < d_range, start looking
* at the sides of the inverted Hasse interval around order, upto p_range
*
* order + 1 - 2*sqrt(order) order + 1 + 2*sqrt(order)
* | |
* |-----|----------|----------|-----|
* >_____> | <_____<
* | order |
* p_range p_range
* Inspired by a method from:
* Constructing elliptic curves of prescribed order,
* Reiner Broker
*
* @param order
* @param p_range
* @param d_range
* @return
*//*
static cm_any_qdisc_t *good_qdisc_brute_range(GEN order, GEN p_range, GEN d_range) {
pari_sp ltop = avma;
GEN tsqrt_ord = mulis(sqrti(order), 2);
GEN left_p = subii(addis(order, 1), tsqrt_ord);
GEN right_p = addii(addis(order, 1), tsqrt_ord);
GEN left_max_p = addii(left_p, p_range);
GEN right_min_p = subii(right_p, p_range);
GEN min_d = stoi(0);
bool left = true;
while (true) {
GEN p;
if (left) {
left_p = nextprime(addis(left_p, 1));
p = left_p;
} else {
right_p = precprime(subis(right_p, 1));
p = right_p;
}
left = !left;
GEN t = subii(addis(p, 1), order);
GEN D = subii(sqri(t), mulis(p, 4));
GEN d = coredisc(D);
if (gequal0(min_d) || cmpii(d, min_d) > 0) {
debug_log("New min D = %Pi", d);
min_d = d;
}
if (cmpii(absi(d), d_range) < 0) {
debug_log("Good min D = %Pi", d);
} else if (cmpii(left_p, left_max_p) > 0 || cmpii(right_p, right_min_p) < 0) {
debug_log("Over p_range, D = %Pi", d);
} else {
continue;
}
cm_any_qdisc_t *result = try_calloc(sizeof(cm_any_qdisc_t));
result->p = p;
result->d = d;
gerepileall(ltop, 2, &result->p, &result->d);
return result;
};
}
*/
/**
* @brief Find a fundamental quadratic discriminant < order^beta, start looking
* at the sides of the inverted Hasse interval around order, upto
* order^alpha width.
* @param order
* @param alpha
* @param beta
* @return
*//*
static cm_any_qdisc_t *good_qdisc_brute(GEN order, GEN alpha, GEN beta) {
GEN ord_a = ground(gpow(order, alpha, DEFAULTPREC));
GEN ord_b = ground(gpow(order, beta, DEFAULTPREC));
return good_qdisc_brute_range(order, ord_a, ord_b);
}
*/
GEN cm_construct_curve(GEN order, GEN d, GEN p, bool ord_prime) {
debug_log("Constructing a curve with N = %Pi, d = %Pi, p = %Pi", order, d,
p);
pari_sp ltop = avma;
GEN H = polclass(d, 0, 0);
debug_log("H = %Ps", H);
GEN r = FpX_roots(H, p);
debug_log("roots = %Ps", r);
if (gequal(r, gtovec(gen_0))) {
return NULL;
}
long rlen = glength(r);
for (long i = 1; i <= rlen; ++i) {
GEN root = gel(r, i);
debug_log("trying root = %Pi", root);
GEN e = ellinit(ellfromj(mkintmod(root, p)), p, 0);
pari_CATCH(e_TYPE) { continue; }
pari_TRY { checkell(e); };
pari_ENDCATCH{};
if (ord_prime) {
// Easy part, the requested order is prime so
// [order]G = 0 iff the curve has exactly order points, for any G on
// it. otherwise it is the twist
GEN g = genrand(e);
if (ell_is_inf(ellmul(e, g, order))) {
debug_log("Got curve.");
return gerepilecopy(ltop, e);
} else {
debug_log("Got curve twist.");
return gerepilecopy(ltop, ellinit(elltwist(e, NULL), p, 0));
}
} else {
// Hard part, requested order is composite, so it might share a
// factor with the order of the twist, which means [order]G = 0
// might be true for a point on the twist as well as a point o the
// right curve.
//
// We calculate what the twist order is, then compute gcd of the
// orders which leads to the product of the factors that the orders
// do not share. By multiplying a random point by this product on
// some curve, we can determine that that curve has that number of
// points.
GEN twist_order = subii(addis(p, 1), subii(order, addis(p, 1)));
GEN twist = ellinit(elltwist(e, NULL), p, 0);
GEN gcd = gcdii(order, twist_order);
GEN orig_mul = divii(order, gcd);
GEN twist_mul = divii(twist_order, gcd);
while (true) {
GEN orig_point = genrand(e);
if (ell_is_inf(ellmul(e, orig_point, orig_mul))) {
debug_log("Got curve.");
return gerepilecopy(ltop, e);
}
if (ell_is_inf(ellmul(e, orig_point, twist_mul))) {
debug_log("Got curve twist.");
return gerepilecopy(ltop, twist);
}
GEN twist_point = genrand(twist);
if (ell_is_inf(ellmul(e, twist_point, twist_mul))) {
debug_log("Got curve.");
return gerepilecopy(ltop, e);
}
if (ell_is_inf(ellmul(e, twist_point, orig_mul))) {
debug_log("Got curve twist.");
return gerepilecopy(ltop, twist);
}
}
}
}
return NULL;
}
GENERATOR(cm_gen_curve_any) {
HAS_ARG(args);
pari_sp ltop = avma;
const char *order_s = (const char *)args->args;
GEN order = strtoi(order_s);
cm_any_qdisc_t min_disc = {0};
good_qdisc_minimal(&min_disc, order);
debug_log("Got min D = %Pi", min_disc.d);
GEN e = cm_construct_curve(order, min_disc.d, min_disc.p, false);
if (e == NULL) {
fprintf(err, "Could not construct curve.");
avma = ltop;
return -3;
}
curve->field = min_disc.p;
curve->a = ell_get_a4(e);
curve->b = ell_get_a6(e);
curve->curve = e;
return 1;
}
GENERATOR(cm_gen_order) {
HAS_ARG(args);
const char *order_s = (const char *)args->args;
curve->order = strtoi(order_s);
return 1;
}
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