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openssl/crypto/ec/curve448/curve448.c
Matt Caswell 4de88fe6da Implement a stricter ECX_KEY type 
Add ref counting and control how we allocate storage for the private key.
We will need this type in following commits where we move the ecx code
to be provider aware.

Reviewed-by: Patrick Steuer <patrick.steuer@de.ibm.com>
Reviewed-by: Shane Lontis <shane.lontis@oracle.com>
(Merged from https://github.com/openssl/openssl/pull/10964)
2020-02-11 22:32:47 +00:00

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/*
* Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
* Copyright 2015-2016 Cryptography Research, Inc.
*
* Licensed under the Apache License 2.0 (the "License"). You may not use
* this file except in compliance with the License. You can obtain a copy
* in the file LICENSE in the source distribution or at
* https://www.openssl.org/source/license.html
*
* Originally written by Mike Hamburg
*/
#include <openssl/crypto.h>
#include "word.h"
#include "field.h"
#include "point_448.h"
#include "ed448.h"
#include "crypto/ecx.h"
#include "curve448_local.h"
#define COFACTOR 4
#define C448_WNAF_FIXED_TABLE_BITS 5
#define C448_WNAF_VAR_TABLE_BITS 3
#define EDWARDS_D (-39081)
static const curve448_scalar_t precomputed_scalarmul_adjustment = {
{
{
SC_LIMB(0xc873d6d54a7bb0cfULL), SC_LIMB(0xe933d8d723a70aadULL),
SC_LIMB(0xbb124b65129c96fdULL), SC_LIMB(0x00000008335dc163ULL)
}
}
};
#define TWISTED_D (EDWARDS_D - 1)
#define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
/* Inverse. */
static void gf_invert(gf y, const gf x, int assert_nonzero)
{
mask_t ret;
gf t1, t2;
gf_sqr(t1, x); /* o^2 */
ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
(void)ret;
if (assert_nonzero)
assert(ret);
gf_sqr(t1, t2);
gf_mul(t2, t1, x); /* not direct to y in case of alias. */
gf_copy(y, t2);
}
/** identity = (0,1) */
const curve448_point_t curve448_point_identity =
{ {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
static void point_double_internal(curve448_point_t p, const curve448_point_t q,
int before_double)
{
gf a, b, c, d;
gf_sqr(c, q->x);
gf_sqr(a, q->y);
gf_add_nr(d, c, a); /* 2+e */
gf_add_nr(p->t, q->y, q->x); /* 2+e */
gf_sqr(b, p->t);
gf_subx_nr(b, b, d, 3); /* 4+e */
gf_sub_nr(p->t, a, c); /* 3+e */
gf_sqr(p->x, q->z);
gf_add_nr(p->z, p->x, p->x); /* 2+e */
gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
if (GF_HEADROOM == 5)
gf_weak_reduce(a); /* or 1+e */
gf_mul(p->x, a, b);
gf_mul(p->z, p->t, a);
gf_mul(p->y, p->t, d);
if (!before_double)
gf_mul(p->t, b, d);
}
void curve448_point_double(curve448_point_t p, const curve448_point_t q)
{
point_double_internal(p, q, 0);
}
/* Operations on [p]niels */
static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
{
gf_cond_swap(n->a, n->b, neg);
gf_cond_neg(n->c, neg);
}
static void pt_to_pniels(pniels_t b, const curve448_point_t a)
{
gf_sub(b->n->a, a->y, a->x);
gf_add(b->n->b, a->x, a->y);
gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
gf_add(b->z, a->z, a->z);
}
static void pniels_to_pt(curve448_point_t e, const pniels_t d)
{
gf eu;
gf_add(eu, d->n->b, d->n->a);
gf_sub(e->y, d->n->b, d->n->a);
gf_mul(e->t, e->y, eu);
gf_mul(e->x, d->z, e->y);
gf_mul(e->y, d->z, eu);
gf_sqr(e->z, d->z);
}
static void niels_to_pt(curve448_point_t e, const niels_t n)
{
gf_add(e->y, n->b, n->a);
gf_sub(e->x, n->b, n->a);
gf_mul(e->t, e->y, e->x);
gf_copy(e->z, ONE);
}
static void add_niels_to_pt(curve448_point_t d, const niels_t e,
int before_double)
{
gf a, b, c;
gf_sub_nr(b, d->y, d->x); /* 3+e */
gf_mul(a, e->a, b);
gf_add_nr(b, d->x, d->y); /* 2+e */
gf_mul(d->y, e->b, b);
gf_mul(d->x, e->c, d->t);
gf_add_nr(c, a, d->y); /* 2+e */
gf_sub_nr(b, d->y, a); /* 3+e */
gf_sub_nr(d->y, d->z, d->x); /* 3+e */
gf_add_nr(a, d->x, d->z); /* 2+e */
gf_mul(d->z, a, d->y);
gf_mul(d->x, d->y, b);
gf_mul(d->y, a, c);
if (!before_double)
gf_mul(d->t, b, c);
}
static void sub_niels_from_pt(curve448_point_t d, const niels_t e,
int before_double)
{
gf a, b, c;
gf_sub_nr(b, d->y, d->x); /* 3+e */
gf_mul(a, e->b, b);
gf_add_nr(b, d->x, d->y); /* 2+e */
gf_mul(d->y, e->a, b);
gf_mul(d->x, e->c, d->t);
gf_add_nr(c, a, d->y); /* 2+e */
gf_sub_nr(b, d->y, a); /* 3+e */
gf_add_nr(d->y, d->z, d->x); /* 2+e */
gf_sub_nr(a, d->z, d->x); /* 3+e */
gf_mul(d->z, a, d->y);
gf_mul(d->x, d->y, b);
gf_mul(d->y, a, c);
if (!before_double)
gf_mul(d->t, b, c);
}
static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
int before_double)
{
gf L0;
gf_mul(L0, p->z, pn->z);
gf_copy(p->z, L0);
add_niels_to_pt(p, pn->n, before_double);
}
static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
int before_double)
{
gf L0;
gf_mul(L0, p->z, pn->z);
gf_copy(p->z, L0);
sub_niels_from_pt(p, pn->n, before_double);
}
c448_bool_t curve448_point_eq(const curve448_point_t p,
const curve448_point_t q)
{
mask_t succ;
gf a, b;
/* equality mod 2-torsion compares x/y */
gf_mul(a, p->y, q->x);
gf_mul(b, q->y, p->x);
succ = gf_eq(a, b);
return mask_to_bool(succ);
}
c448_bool_t curve448_point_valid(const curve448_point_t p)
{
mask_t out;
gf a, b, c;
gf_mul(a, p->x, p->y);
gf_mul(b, p->z, p->t);
out = gf_eq(a, b);
gf_sqr(a, p->x);
gf_sqr(b, p->y);
gf_sub(a, b, a);
gf_sqr(b, p->t);
gf_mulw(c, b, TWISTED_D);
gf_sqr(b, p->z);
gf_add(b, b, c);
out &= gf_eq(a, b);
out &= ~gf_eq(p->z, ZERO);
return mask_to_bool(out);
}
static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
const niels_t * table,
int nelts, int idx)
{
constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
}
void curve448_precomputed_scalarmul(curve448_point_t out,
const curve448_precomputed_s * table,
const curve448_scalar_t scalar)
{
unsigned int i, j, k;
const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
niels_t ni;
curve448_scalar_t scalar1x;
curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
curve448_scalar_halve(scalar1x, scalar1x);
for (i = s; i > 0; i--) {
if (i != s)
point_double_internal(out, out, 0);
for (j = 0; j < n; j++) {
int tab = 0;
mask_t invert;
for (k = 0; k < t; k++) {
unsigned int bit = (i - 1) + s * (k + j * t);
if (bit < C448_SCALAR_BITS)
tab |=
(scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
}
invert = (tab >> (t - 1)) - 1;
tab ^= invert;
tab &= (1 << (t - 1)) - 1;
constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
1 << (t - 1), tab);
cond_neg_niels(ni, invert);
if ((i != s) || j != 0)
add_niels_to_pt(out, ni, j == n - 1 && i != 1);
else
niels_to_pt(out, ni);
}
}
OPENSSL_cleanse(ni, sizeof(ni));
OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
}
void curve448_point_mul_by_ratio_and_encode_like_eddsa(
uint8_t enc[EDDSA_448_PUBLIC_BYTES],
const curve448_point_t p)
{
gf x, y, z, t;
curve448_point_t q;
/* The point is now on the twisted curve. Move it to untwisted. */
curve448_point_copy(q, p);
{
/* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
gf u;
gf_sqr(x, q->x);
gf_sqr(t, q->y);
gf_add(u, x, t);
gf_add(z, q->y, q->x);
gf_sqr(y, z);
gf_sub(y, y, u);
gf_sub(z, t, x);
gf_sqr(x, q->z);
gf_add(t, x, x);
gf_sub(t, t, z);
gf_mul(x, t, y);
gf_mul(y, z, u);
gf_mul(z, u, t);
OPENSSL_cleanse(u, sizeof(u));
}
/* Affinize */
gf_invert(z, z, 1);
gf_mul(t, x, z);
gf_mul(x, y, z);
/* Encode */
enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
gf_serialize(enc, x, 1);
enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
OPENSSL_cleanse(x, sizeof(x));
OPENSSL_cleanse(y, sizeof(y));
OPENSSL_cleanse(z, sizeof(z));
OPENSSL_cleanse(t, sizeof(t));
curve448_point_destroy(q);
}
c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
curve448_point_t p,
const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
{
uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
mask_t low;
mask_t succ;
memcpy(enc2, enc, sizeof(enc2));
low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
succ = gf_deserialize(p->y, enc2, 1, 0);
succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
gf_sqr(p->x, p->y);
gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
gf_mul(p->x, p->z, p->t);
succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
gf_copy(p->z, ONE);
{
gf a, b, c, d;
/* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
gf_sqr(c, p->x);
gf_sqr(a, p->y);
gf_add(d, c, a);
gf_add(p->t, p->y, p->x);
gf_sqr(b, p->t);
gf_sub(b, b, d);
gf_sub(p->t, a, c);
gf_sqr(p->x, p->z);
gf_add(p->z, p->x, p->x);
gf_sub(a, p->z, d);
gf_mul(p->x, a, b);
gf_mul(p->z, p->t, a);
gf_mul(p->y, p->t, d);
gf_mul(p->t, b, d);
OPENSSL_cleanse(a, sizeof(a));
OPENSSL_cleanse(b, sizeof(b));
OPENSSL_cleanse(c, sizeof(c));
OPENSSL_cleanse(d, sizeof(d));
}
OPENSSL_cleanse(enc2, sizeof(enc2));
assert(curve448_point_valid(p) || ~succ);
return c448_succeed_if(mask_to_bool(succ));
}
c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES],
const uint8_t base[X_PUBLIC_BYTES],
const uint8_t scalar[X_PRIVATE_BYTES])
{
gf x1, x2, z2, x3, z3, t1, t2;
int t;
mask_t swap = 0;
mask_t nz;
(void)gf_deserialize(x1, base, 1, 0);
gf_copy(x2, ONE);
gf_copy(z2, ZERO);
gf_copy(x3, x1);
gf_copy(z3, ONE);
for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
uint8_t sb = scalar[t / 8];
mask_t k_t;
/* Scalar conditioning */
if (t / 8 == 0)
sb &= -(uint8_t)COFACTOR;
else if (t == X_PRIVATE_BITS - 1)
sb = -1;
k_t = (sb >> (t % 8)) & 1;
k_t = 0 - k_t; /* set to all 0s or all 1s */
swap ^= k_t;
gf_cond_swap(x2, x3, swap);
gf_cond_swap(z2, z3, swap);
swap = k_t;
/*
* The "_nr" below skips coefficient reduction. In the following
* comments, "2+e" is saying that the coefficients are at most 2+epsilon
* times the reduction limit.
*/
gf_add_nr(t1, x2, z2); /* A = x2 + z2 */ /* 2+e */
gf_sub_nr(t2, x2, z2); /* B = x2 - z2 */ /* 3+e */
gf_sub_nr(z2, x3, z3); /* D = x3 - z3 */ /* 3+e */
gf_mul(x2, t1, z2); /* DA */
gf_add_nr(z2, z3, x3); /* C = x3 + z3 */ /* 2+e */
gf_mul(x3, t2, z2); /* CB */
gf_sub_nr(z3, x2, x3); /* DA-CB */ /* 3+e */
gf_sqr(z2, z3); /* (DA-CB)^2 */
gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
gf_add_nr(z2, x2, x3); /* (DA+CB) */ /* 2+e */
gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
gf_sqr(z2, t1); /* AA = A^2 */
gf_sqr(t1, t2); /* BB = B^2 */
gf_mul(x2, z2, t1); /* x2 = AA*BB */
gf_sub_nr(t2, z2, t1); /* E = AA-BB */ /* 3+e */
gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
gf_add_nr(t1, t1, z2); /* AA + a24*E */ /* 2+e */
gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
}
/* Finish */
gf_cond_swap(x2, x3, swap);
gf_cond_swap(z2, z3, swap);
gf_invert(z2, z2, 0);
gf_mul(x1, x2, z2);
gf_serialize(out, x1, 1);
nz = ~gf_eq(x1, ZERO);
OPENSSL_cleanse(x1, sizeof(x1));
OPENSSL_cleanse(x2, sizeof(x2));
OPENSSL_cleanse(z2, sizeof(z2));
OPENSSL_cleanse(x3, sizeof(x3));
OPENSSL_cleanse(z3, sizeof(z3));
OPENSSL_cleanse(t1, sizeof(t1));
OPENSSL_cleanse(t2, sizeof(t2));
return c448_succeed_if(mask_to_bool(nz));
}
void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
out[X_PUBLIC_BYTES],
const curve448_point_t p)
{
curve448_point_t q;
curve448_point_copy(q, p);
gf_invert(q->t, q->x, 0); /* 1/x */
gf_mul(q->z, q->t, q->y); /* y/x */
gf_sqr(q->y, q->z); /* (y/x)^2 */
gf_serialize(out, q->y, 1);
curve448_point_destroy(q);
}
void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
const uint8_t scalar[X_PRIVATE_BYTES])
{
/* Scalar conditioning */
uint8_t scalar2[X_PRIVATE_BYTES];
curve448_scalar_t the_scalar;
curve448_point_t p;
unsigned int i;
memcpy(scalar2, scalar, sizeof(scalar2));
scalar2[0] &= -(uint8_t)COFACTOR;
scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
/* Compensate for the encoding ratio */
for (i = 1; i < X448_ENCODE_RATIO; i <<= 1)
curve448_scalar_halve(the_scalar, the_scalar);
curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
curve448_point_destroy(p);
}
/* Control for variable-time scalar multiply algorithms. */
struct smvt_control {
int power, addend;
};
#if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ > 3))
# define NUMTRAILINGZEROS __builtin_ctz
#else
# define NUMTRAILINGZEROS numtrailingzeros
static uint32_t numtrailingzeros(uint32_t i)
{
uint32_t tmp;
uint32_t num = 31;
if (i == 0)
return 32;
tmp = i << 16;
if (tmp != 0) {
i = tmp;
num -= 16;
}
tmp = i << 8;
if (tmp != 0) {
i = tmp;
num -= 8;
}
tmp = i << 4;
if (tmp != 0) {
i = tmp;
num -= 4;
}
tmp = i << 2;
if (tmp != 0) {
i = tmp;
num -= 2;
}
tmp = i << 1;
if (tmp != 0)
num--;
return num;
}
#endif
static int recode_wnaf(struct smvt_control *control,
/* [nbits/(table_bits + 1) + 3] */
const curve448_scalar_t scalar,
unsigned int table_bits)
{
unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
int position = table_size - 1; /* at the end */
uint64_t current = scalar->limb[0] & 0xFFFF;
uint32_t mask = (1 << (table_bits + 1)) - 1;
unsigned int w;
const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
unsigned int n, i;
/* place the end marker */
control[position].power = -1;
control[position].addend = 0;
position--;
/*
* PERF: Could negate scalar if it's large. But then would need more cases
* in the actual code that uses it, all for an expected reduction of like
* 1/5 op. Probably not worth it.
*/
for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
/* Refill the 16 high bits of current */
current += (uint32_t)((scalar->limb[w / B_OVER_16]
>> (16 * (w % B_OVER_16))) << 16);
}
while (current & 0xFFFF) {
uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
uint32_t odd = (uint32_t)current >> pos;
int32_t delta = odd & mask;
assert(position >= 0);
if (odd & (1 << (table_bits + 1)))
delta -= (1 << (table_bits + 1));
current -= delta * (1 << pos);
control[position].power = pos + 16 * (w - 1);
control[position].addend = delta;
position--;
}
current >>= 16;
}
assert(current == 0);
position++;
n = table_size - position;
for (i = 0; i < n; i++)
control[i] = control[i + position];
return n - 1;
}
static void prepare_wnaf_table(pniels_t * output,
const curve448_point_t working,
unsigned int tbits)
{
curve448_point_t tmp;
int i;
pniels_t twop;
pt_to_pniels(output[0], working);
if (tbits == 0)
return;
curve448_point_double(tmp, working);
pt_to_pniels(twop, tmp);
add_pniels_to_pt(tmp, output[0], 0);
pt_to_pniels(output[1], tmp);
for (i = 2; i < 1 << tbits; i++) {
add_pniels_to_pt(tmp, twop, 0);
pt_to_pniels(output[i], tmp);
}
curve448_point_destroy(tmp);
OPENSSL_cleanse(twop, sizeof(twop));
}
void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
const curve448_scalar_t scalar1,
const curve448_point_t base2,
const curve448_scalar_t scalar2)
{
const int table_bits_var = C448_WNAF_VAR_TABLE_BITS;
const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
struct smvt_control control_var[C448_SCALAR_BITS /
(C448_WNAF_VAR_TABLE_BITS + 1) + 3];
struct smvt_control control_pre[C448_SCALAR_BITS /
(C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
int contp = 0, contv = 0, i;
prepare_wnaf_table(precmp_var, base2, table_bits_var);
i = control_var[0].power;
if (i < 0) {
curve448_point_copy(combo, curve448_point_identity);
return;
}
if (i > control_pre[0].power) {
pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
contv++;
} else if (i == control_pre[0].power && i >= 0) {
pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
i);
contv++;
contp++;
} else {
i = control_pre[0].power;
niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
contp++;
}
for (i--; i >= 0; i--) {
int cv = (i == control_var[contv].power);
int cp = (i == control_pre[contp].power);
point_double_internal(combo, combo, i && !(cv || cp));
if (cv) {
assert(control_var[contv].addend);
if (control_var[contv].addend > 0)
add_pniels_to_pt(combo,
precmp_var[control_var[contv].addend >> 1],
i && !cp);
else
sub_pniels_from_pt(combo,
precmp_var[(-control_var[contv].addend)
>> 1], i && !cp);
contv++;
}
if (cp) {
assert(control_pre[contp].addend);
if (control_pre[contp].addend > 0)
add_niels_to_pt(combo,
curve448_wnaf_base[control_pre[contp].addend
>> 1], i);
else
sub_niels_from_pt(combo,
curve448_wnaf_base[(-control_pre
[contp].addend) >> 1], i);
contp++;
}
}
/* This function is non-secret, but whatever this is cheap. */
OPENSSL_cleanse(control_var, sizeof(control_var));
OPENSSL_cleanse(control_pre, sizeof(control_pre));
OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
assert(contv == ncb_var);
(void)ncb_var;
assert(contp == ncb_pre);
(void)ncb_pre;
}
void curve448_point_destroy(curve448_point_t point)
{
OPENSSL_cleanse(point, sizeof(curve448_point_t));
}
int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
const uint8_t peer_public_value[56])
{
return x448_int(out_shared_key, peer_public_value, private_key)
== C448_SUCCESS;
}
void X448_public_from_private(uint8_t out_public_value[56],
const uint8_t private_key[56])
{
x448_derive_public_key(out_public_value, private_key);
}
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