Files
poulpy/core/src/ggsw_ciphertext.rs
2025-05-30 09:05:07 +02:00

618 lines
22 KiB
Rust

use backend::{
Backend, FFT64, MatZnxDft, MatZnxDftAlloc, MatZnxDftOps, MatZnxDftScratch, Module, ScalarZnx, Scratch, VecZnxAlloc,
VecZnxBigAlloc, VecZnxBigOps, VecZnxBigScratch, VecZnxDft, VecZnxDftAlloc, VecZnxDftOps, VecZnxOps, VecZnxToMut, ZnxInfos,
ZnxZero,
};
use sampling::source::Source;
use crate::{
ScratchCore,
automorphism::AutomorphismKey,
elem::{GetRow, Infos, SetRow},
glwe_ciphertext::GLWECiphertext,
glwe_ciphertext_fourier::GLWECiphertextFourier,
keys::SecretKeyFourier,
keyswitch_key::GLWESwitchingKey,
tensor_key::TensorKey,
utils::derive_size,
};
pub struct GGSWCiphertext<C, B: Backend> {
pub data: MatZnxDft<C, B>,
pub basek: usize,
pub k: usize,
}
impl<B: Backend> GGSWCiphertext<Vec<u8>, B> {
pub fn alloc(module: &Module<B>, basek: usize, k: usize, rows: usize, rank: usize) -> Self {
Self {
data: module.new_mat_znx_dft(rows, rank + 1, rank + 1, derive_size(basek, k)),
basek: basek,
k: k,
}
}
pub fn bytes_of(module: &Module<FFT64>, basek: usize, k: usize, rows: usize, rank: usize) -> usize {
module.bytes_of_mat_znx_dft(rows, rank + 1, rank + 1, derive_size(basek, k))
}
}
impl<T, B: Backend> Infos for GGSWCiphertext<T, B> {
type Inner = MatZnxDft<T, B>;
fn inner(&self) -> &Self::Inner {
&self.data
}
fn basek(&self) -> usize {
self.basek
}
fn k(&self) -> usize {
self.k
}
}
impl<T, B: Backend> GGSWCiphertext<T, B> {
pub fn rank(&self) -> usize {
self.data.cols_out() - 1
}
}
impl GGSWCiphertext<Vec<u8>, FFT64> {
pub fn encrypt_sk_scratch_space(module: &Module<FFT64>, basek: usize, k: usize, rank: usize) -> usize {
let size = derive_size(basek, k);
GLWECiphertext::encrypt_sk_scratch_space(module, basek, k)
+ module.bytes_of_vec_znx(rank + 1, size)
+ module.bytes_of_vec_znx(1, size)
+ module.bytes_of_vec_znx_dft(rank + 1, size)
}
pub(crate) fn expand_row_scratch_space(
module: &Module<FFT64>,
basek: usize,
self_k: usize,
tsk_k: usize,
rank: usize,
) -> usize {
let tsk_size: usize = derive_size(basek, tsk_k);
let self_size: usize = derive_size(basek, self_k);
let tmp_dft_i: usize = module.bytes_of_vec_znx_dft(rank + 1, tsk_size);
let tmp_dft_col_data: usize = module.bytes_of_vec_znx_dft(1, self_size);
let vmp: usize = tmp_dft_col_data + module.vmp_apply_tmp_bytes(self_size, self_size, self_size, rank, rank, tsk_size);
let tmp_idft: usize = module.bytes_of_vec_znx_big(1, tsk_size);
let norm: usize = module.vec_znx_big_normalize_tmp_bytes();
tmp_dft_i + ((tmp_dft_col_data + vmp) | (tmp_idft + norm))
}
pub(crate) fn keyswitch_internal_col0_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
in_k: usize,
ksk_k: usize,
rank: usize,
) -> usize {
GLWECiphertext::keyswitch_from_fourier_scratch_space(module, basek, out_k, rank, in_k, rank, ksk_k)
+ module.bytes_of_vec_znx_dft(rank + 1, derive_size(basek, in_k))
}
pub fn keyswitch_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
in_k: usize,
ksk_k: usize,
tsk_k: usize,
rank: usize,
) -> usize {
let out_size: usize = derive_size(basek, out_k);
let res_znx: usize = module.bytes_of_vec_znx(rank + 1, out_size);
let ci_dft: usize = module.bytes_of_vec_znx_dft(rank + 1, out_size);
let ks: usize = GGSWCiphertext::keyswitch_internal_col0_scratch_space(module, basek, out_k, in_k, ksk_k, rank);
let expand_rows: usize = GGSWCiphertext::expand_row_scratch_space(module, basek, out_k, tsk_k, rank);
let res_dft: usize = module.bytes_of_vec_znx_dft(rank + 1, out_size);
res_znx + ci_dft + (ks | expand_rows | res_dft)
}
pub fn keyswitch_inplace_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
ksk_k: usize,
tsk_k: usize,
rank: usize,
) -> usize {
GGSWCiphertext::keyswitch_scratch_space(module, basek, out_k, out_k, ksk_k, tsk_k, rank)
}
pub fn automorphism_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
in_k: usize,
atk_k: usize,
tsk_k: usize,
rank: usize,
) -> usize {
let cols: usize = rank + 1;
let out_size: usize = derive_size(basek, out_k);
let res: usize = module.bytes_of_vec_znx(cols, out_size);
let res_dft: usize = module.bytes_of_vec_znx_dft(cols, out_size);
let ci_dft: usize = module.bytes_of_vec_znx_dft(cols, out_size);
let ks_internal: usize = GGSWCiphertext::keyswitch_internal_col0_scratch_space(module, basek, out_k, in_k, atk_k, rank);
let expand: usize = GGSWCiphertext::expand_row_scratch_space(module, basek, out_k, tsk_k, rank);
res + ci_dft + (ks_internal | expand | res_dft)
}
pub fn automorphism_inplace_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
atk_k: usize,
tsk_k: usize,
rank: usize,
) -> usize {
GGSWCiphertext::automorphism_scratch_space(module, basek, out_k, out_k, atk_k, tsk_k, rank)
}
pub fn external_product_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
in_k: usize,
ggsw_k: usize,
rank: usize,
) -> usize {
let tmp_in: usize = GLWECiphertextFourier::bytes_of(module, basek, in_k, rank);
let tmp_out: usize = GLWECiphertextFourier::bytes_of(module, basek, out_k, rank);
let ggsw: usize = GLWECiphertextFourier::external_product_scratch_space(module, basek, out_k, in_k, ggsw_k, rank);
tmp_in + tmp_out + ggsw
}
pub fn external_product_inplace_scratch_space(
module: &Module<FFT64>,
basek: usize,
out_k: usize,
ggsw_k: usize,
rank: usize,
) -> usize {
let tmp: usize = GLWECiphertextFourier::bytes_of(module, basek, out_k, rank);
let ggsw: usize = GLWECiphertextFourier::external_product_inplace_scratch_space(module, basek, out_k, ggsw_k, rank);
tmp + ggsw
}
}
impl<DataSelf: AsMut<[u8]> + AsRef<[u8]>> GGSWCiphertext<DataSelf, FFT64> {
pub fn encrypt_sk<DataPt: AsRef<[u8]>, DataSk: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
pt: &ScalarZnx<DataPt>,
sk_dft: &SecretKeyFourier<DataSk, FFT64>,
source_xa: &mut Source,
source_xe: &mut Source,
sigma: f64,
scratch: &mut Scratch,
) {
#[cfg(debug_assertions)]
{
assert_eq!(self.rank(), sk_dft.rank());
assert_eq!(self.n(), module.n());
assert_eq!(pt.n(), module.n());
assert_eq!(sk_dft.n(), module.n());
}
let basek: usize = self.basek();
let k: usize = self.k();
let rank: usize = self.rank();
let (mut tmp_pt, scratch1) = scratch.tmp_glwe_pt(module, basek, k);
let (mut tmp_ct, scratch2) = scratch1.tmp_glwe_ct(module, basek, k, rank);
(0..self.rows()).for_each(|row_i| {
tmp_pt.data.zero();
// Adds the scalar_znx_pt to the i-th limb of the vec_znx_pt
module.vec_znx_add_scalar_inplace(&mut tmp_pt.data, 0, row_i, pt, 0);
module.vec_znx_normalize_inplace(basek, &mut tmp_pt.data, 0, scratch2);
(0..rank + 1).for_each(|col_j| {
// rlwe encrypt of vec_znx_pt into vec_znx_ct
tmp_ct.encrypt_sk_private(
module,
Some((&tmp_pt, col_j)),
sk_dft,
source_xa,
source_xe,
sigma,
scratch2,
);
// Switch vec_znx_ct into DFT domain
{
let (mut tmp_ct_dft, _) = scratch2.tmp_glwe_fourier(module, basek, k, rank);
tmp_ct.dft(module, &mut tmp_ct_dft);
self.set_row(module, row_i, col_j, &tmp_ct_dft);
}
});
});
}
pub(crate) fn expand_row<R, DataCi: AsRef<[u8]>, DataTsk: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
col_j: usize,
res: &mut R,
ci_dft: &VecZnxDft<DataCi, FFT64>,
tsk: &TensorKey<DataTsk, FFT64>,
scratch: &mut Scratch,
) where
R: VecZnxToMut,
{
let cols: usize = self.rank() + 1;
assert!(
scratch.available() >= GGSWCiphertext::expand_row_scratch_space(module, self.basek(), self.k(), tsk.k(), self.rank())
);
// Example for rank 3:
//
// Note: M is a vector (m, Bm, B^2m, B^3m, ...), so each column is
// actually composed of that many rows and we focus on a specific row here
// implicitely given ci_dft.
//
// # Input
//
// col 0: (-(a0s0 + a1s1 + a2s2) + M[i], a0 , a1 , a2 )
// col 1: (0, 0, 0, 0)
// col 2: (0, 0, 0, 0)
// col 3: (0, 0, 0, 0)
//
// # Output
//
// col 0: (-(a0s0 + a1s1 + a2s2) + M[i], a0 , a1 , a2 )
// col 1: (-(b0s0 + b1s1 + b2s2) , b0 + M[i], b1 , b2 )
// col 2: (-(c0s0 + c1s1 + c2s2) , c0 , c1 + M[i], c2 )
// col 3: (-(d0s0 + d1s1 + d2s2) , d0 , d1 , d2 + M[i])
let (mut tmp_dft_i, scratch1) = scratch.tmp_vec_znx_dft(module, cols, tsk.size());
{
let (mut tmp_dft_col_data, scratch2) = scratch1.tmp_vec_znx_dft(module, 1, self.size());
// Performs a key-switch for each combination of s[i]*s[j], i.e. for a0, a1, a2
//
// # Example for col=1
//
// a0 * (-(f0s0 + f1s1 + f1s2) + s0^2, f0, f1, f2) = (-(a0f0s0 + a0f1s1 + a0f1s2) + a0s0^2, a0f0, a0f1, a0f2)
// +
// a1 * (-(g0s0 + g1s1 + g1s2) + s0s1, g0, g1, g2) = (-(a1g0s0 + a1g1s1 + a1g1s2) + a1s0s1, a1g0, a1g1, a1g2)
// +
// a2 * (-(h0s0 + h1s1 + h1s2) + s0s2, h0, h1, h2) = (-(a2h0s0 + a2h1s1 + a2h1s2) + a2s0s2, a2h0, a2h1, a2h2)
// =
// (-(x0s0 + x1s1 + x2s2) + s0(a0s0 + a1s1 + a2s2), x0, x1, x2)
(1..cols).for_each(|col_i| {
// Extracts a[i] and multipies with Enc(s[i]s[j])
tmp_dft_col_data.extract_column(0, ci_dft, col_i);
if col_i == 1 {
module.vmp_apply(
&mut tmp_dft_i,
&tmp_dft_col_data,
&tsk.at(col_i - 1, col_j - 1).0.data, // Selects Enc(s[i]s[j])
scratch2,
);
} else {
module.vmp_apply_add(
&mut tmp_dft_i,
&tmp_dft_col_data,
&tsk.at(col_i - 1, col_j - 1).0.data, // Selects Enc(s[i]s[j])
scratch2,
);
}
});
}
// Adds -(sum a[i] * s[i]) + m) on the i-th column of tmp_idft_i
//
// (-(x0s0 + x1s1 + x2s2) + a0s0s0 + a1s0s1 + a2s0s2, x0, x1, x2)
// +
// (0, -(a0s0 + a1s1 + a2s2) + M[i], 0, 0)
// =
// (-(x0s0 + x1s1 + x2s2) + s0(a0s0 + a1s1 + a2s2), x0 -(a0s0 + a1s1 + a2s2) + M[i], x1, x2)
// =
// (-(x0s0 + x1s1 + x2s2), x0 + M[i], x1, x2)
module.vec_znx_dft_add_inplace(&mut tmp_dft_i, col_j, ci_dft, 0);
let (mut tmp_idft, scratch2) = scratch1.tmp_vec_znx_big(module, 1, tsk.size());
(0..cols).for_each(|i| {
module.vec_znx_idft_tmp_a(&mut tmp_idft, 0, &mut tmp_dft_i, i);
module.vec_znx_big_normalize(self.basek(), res, i, &tmp_idft, 0, scratch2);
});
}
pub fn keyswitch<DataLhs: AsRef<[u8]>, DataKsk: AsRef<[u8]>, DataTsk: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
lhs: &GGSWCiphertext<DataLhs, FFT64>,
ksk: &GLWESwitchingKey<DataKsk, FFT64>,
tsk: &TensorKey<DataTsk, FFT64>,
scratch: &mut Scratch,
) {
let rank: usize = self.rank();
let cols: usize = rank + 1;
let basek: usize = self.basek();
let (mut tmp_res, scratch1) = scratch.tmp_glwe_ct(module, basek, self.k(), rank);
let (mut ci_dft, scratch2) = scratch1.tmp_vec_znx_dft(module, cols, lhs.size());
// Keyswitch the j-th row of the col 0
(0..lhs.rows()).for_each(|row_i| {
// Key-switch column 0, i.e.
// col 0: (-(a0s0 + a1s1 + a2s2) + M[i], a0, a1, a2) -> (-(a0s0' + a1s1' + a2s2') + M[i], a0, a1, a2)
lhs.keyswitch_internal_col0(module, row_i, &mut tmp_res, ksk, scratch2);
// Isolates DFT(a[i])
(0..cols).for_each(|col_i| {
module.vec_znx_dft(&mut ci_dft, col_i, &tmp_res.data, col_i);
});
module.vmp_prepare_row(&mut self.data, row_i, 0, &ci_dft);
// Generates
//
// col 1: (-(b0s0' + b1s1' + b2s2') , b0 + M[i], b1 , b2 )
// col 2: (-(c0s0' + c1s1' + c2s2') , c0 , c1 + M[i], c2 )
// col 3: (-(d0s0' + d1s1' + d2s2') , d0 , d1 , d2 + M[i])
(1..cols).for_each(|col_j| {
self.expand_row(module, col_j, &mut tmp_res.data, &ci_dft, tsk, scratch2);
let (mut tmp_res_dft, _) = scratch2.tmp_glwe_fourier(module, basek, self.k(), rank);
tmp_res.dft(module, &mut tmp_res_dft);
self.set_row(module, row_i, col_j, &tmp_res_dft);
});
})
}
pub fn keyswitch_inplace<DataKsk: AsRef<[u8]>, DataTsk: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
ksk: &GLWESwitchingKey<DataKsk, FFT64>,
tsk: &TensorKey<DataTsk, FFT64>,
scratch: &mut Scratch,
) {
unsafe {
let self_ptr: *mut GGSWCiphertext<DataSelf, FFT64> = self as *mut GGSWCiphertext<DataSelf, FFT64>;
self.keyswitch(module, &*self_ptr, ksk, tsk, scratch);
}
}
pub fn automorphism<DataLhs: AsRef<[u8]>, DataAk: AsRef<[u8]>, DataTsk: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
lhs: &GGSWCiphertext<DataLhs, FFT64>,
auto_key: &AutomorphismKey<DataAk, FFT64>,
tensor_key: &TensorKey<DataTsk, FFT64>,
scratch: &mut Scratch,
) {
#[cfg(debug_assertions)]
{
assert_eq!(
self.rank(),
lhs.rank(),
"ggsw_out rank: {} != ggsw_in rank: {}",
self.rank(),
lhs.rank()
);
assert_eq!(
self.rank(),
auto_key.rank(),
"ggsw_in rank: {} != auto_key rank: {}",
self.rank(),
auto_key.rank()
);
assert_eq!(
self.rank(),
tensor_key.rank(),
"ggsw_in rank: {} != tensor_key rank: {}",
self.rank(),
tensor_key.rank()
);
assert!(
scratch.available()
>= GGSWCiphertext::automorphism_scratch_space(
module,
self.basek(),
self.k(),
lhs.k(),
auto_key.k(),
tensor_key.k(),
self.rank()
)
)
};
let rank: usize = self.rank();
let cols: usize = rank + 1;
let basek: usize = self.basek();
let (mut tmp_res, scratch1) = scratch.tmp_glwe_ct(module, basek, self.k(), rank);
let (mut ci_dft, scratch2) = scratch1.tmp_vec_znx_dft(module, cols, lhs.size());
// Keyswitch the j-th row of the col 0
(0..lhs.rows()).for_each(|row_i| {
// Key-switch column 0, i.e.
// col 0: (-(a0s0 + a1s1 + a2s2) + M[i], a0, a1, a2) -> (-(a0pi^-1(s0) + a1pi^-1(s1) + a2pi^-1(s2)) + M[i], a0, a1, a2)
lhs.keyswitch_internal_col0(module, row_i, &mut tmp_res, &auto_key.key, scratch2);
// Isolates DFT(AUTO(a[i]))
(0..cols).for_each(|col_i| {
// (-(a0pi^-1(s0) + a1pi^-1(s1) + a2pi^-1(s2)) + M[i], a0, a1, a2) -> (-(a0s0 + a1s1 + a2s2) + pi(M[i]), a0, a1, a2)
module.vec_znx_automorphism_inplace(auto_key.p(), &mut tmp_res.data, col_i);
module.vec_znx_dft(&mut ci_dft, col_i, &tmp_res.data, col_i);
});
module.vmp_prepare_row(&mut self.data, row_i, 0, &ci_dft);
// Generates
//
// col 1: (-(b0s0 + b1s1 + b2s2) , b0 + pi(M[i]), b1 , b2 )
// col 2: (-(c0s0 + c1s1 + c2s2) , c0 , c1 + pi(M[i]), c2 )
// col 3: (-(d0s0 + d1s1 + d2s2) , d0 , d1 , d2 + pi(M[i]))
(1..cols).for_each(|col_j| {
self.expand_row(
module,
col_j,
&mut tmp_res.data,
&ci_dft,
tensor_key,
scratch2,
);
let (mut tmp_res_dft, _) = scratch2.tmp_glwe_fourier(module, basek, self.k(), rank);
tmp_res.dft(module, &mut tmp_res_dft);
self.set_row(module, row_i, col_j, &tmp_res_dft);
});
})
}
pub fn automorphism_inplace<DataKsk: AsRef<[u8]>, DataTsk: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
auto_key: &AutomorphismKey<DataKsk, FFT64>,
tensor_key: &TensorKey<DataTsk, FFT64>,
scratch: &mut Scratch,
) {
unsafe {
let self_ptr: *mut GGSWCiphertext<DataSelf, FFT64> = self as *mut GGSWCiphertext<DataSelf, FFT64>;
self.automorphism(module, &*self_ptr, auto_key, tensor_key, scratch);
}
}
pub fn external_product<DataLhs: AsRef<[u8]>, DataRhs: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
lhs: &GGSWCiphertext<DataLhs, FFT64>,
rhs: &GGSWCiphertext<DataRhs, FFT64>,
scratch: &mut Scratch,
) {
#[cfg(debug_assertions)]
{
assert_eq!(
self.rank(),
lhs.rank(),
"ggsw_out rank: {} != ggsw_in rank: {}",
self.rank(),
lhs.rank()
);
assert_eq!(
self.rank(),
rhs.rank(),
"ggsw_in rank: {} != ggsw_apply rank: {}",
self.rank(),
rhs.rank()
);
}
let (mut tmp_ct_in, scratch1) = scratch.tmp_glwe_fourier(module, lhs.basek(), lhs.k(), lhs.rank());
let (mut tmp_ct_out, scratch2) = scratch1.tmp_glwe_fourier(module, self.basek(), self.k(), self.rank());
(0..self.rank() + 1).for_each(|col_i| {
(0..self.rows()).for_each(|row_j| {
lhs.get_row(module, row_j, col_i, &mut tmp_ct_in);
tmp_ct_out.external_product(module, &tmp_ct_in, rhs, scratch2);
self.set_row(module, row_j, col_i, &tmp_ct_out);
});
});
tmp_ct_out.data.zero();
(self.rows().min(lhs.rows())..self.rows()).for_each(|row_i| {
(0..self.rank() + 1).for_each(|col_j| {
self.set_row(module, row_i, col_j, &tmp_ct_out);
});
});
}
pub fn external_product_inplace<DataRhs: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
rhs: &GGSWCiphertext<DataRhs, FFT64>,
scratch: &mut Scratch,
) {
#[cfg(debug_assertions)]
{
assert_eq!(
self.rank(),
rhs.rank(),
"ggsw_out rank: {} != ggsw_apply: {}",
self.rank(),
rhs.rank()
);
}
let (mut tmp_ct, scratch1) = scratch.tmp_glwe_fourier(module, self.basek(), self.k(), self.rank());
(0..self.rank() + 1).for_each(|col_i| {
(0..self.rows()).for_each(|row_j| {
self.get_row(module, row_j, col_i, &mut tmp_ct);
tmp_ct.external_product_inplace(module, rhs, scratch1);
self.set_row(module, row_j, col_i, &tmp_ct);
});
});
}
}
impl<DataSelf: AsRef<[u8]>> GGSWCiphertext<DataSelf, FFT64> {
pub(crate) fn keyswitch_internal_col0<DataRes: AsMut<[u8]> + AsRef<[u8]>, DataKsk: AsRef<[u8]>>(
&self,
module: &Module<FFT64>,
row_i: usize,
res: &mut GLWECiphertext<DataRes>,
ksk: &GLWESwitchingKey<DataKsk, FFT64>,
scratch: &mut Scratch,
) {
#[cfg(debug_assertions)]
{
assert_eq!(self.rank(), ksk.rank());
assert_eq!(res.rank(), ksk.rank());
assert!(
scratch.available()
>= GGSWCiphertext::keyswitch_internal_col0_scratch_space(
module,
self.basek(),
res.k(),
self.k(),
ksk.k(),
ksk.rank()
)
)
}
let (mut tmp_dft_dft, scratch1) = scratch.tmp_glwe_fourier(module, self.basek(), self.k(), self.rank());
self.get_row(module, row_i, 0, &mut tmp_dft_dft);
res.keyswitch_from_fourier(module, &tmp_dft_dft, ksk, scratch1);
}
}
impl<DataSelf: AsRef<[u8]>> GetRow<FFT64> for GGSWCiphertext<DataSelf, FFT64> {
fn get_row<R: AsMut<[u8]> + AsRef<[u8]>>(
&self,
module: &Module<FFT64>,
row_i: usize,
col_j: usize,
res: &mut GLWECiphertextFourier<R, FFT64>,
) {
module.vmp_extract_row(&mut res.data, &self.data, row_i, col_j);
}
}
impl<DataSelf: AsMut<[u8]> + AsRef<[u8]>> SetRow<FFT64> for GGSWCiphertext<DataSelf, FFT64> {
fn set_row<R: AsRef<[u8]>>(
&mut self,
module: &Module<FFT64>,
row_i: usize,
col_j: usize,
a: &GLWECiphertextFourier<R, FFT64>,
) {
module.vmp_prepare_row(&mut self.data, row_i, col_j, &a.data);
}
}