monero/tests/unit_tests/ringct.cpp

667 lines
20 KiB
C++

// Copyright (c) 2014-2016, The Monero Project
//
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without modification, are
// permitted provided that the following conditions are met:
//
// 1. Redistributions of source code must retain the above copyright notice, this list of
// conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright notice, this list
// of conditions and the following disclaimer in the documentation and/or other
// materials provided with the distribution.
//
// 3. Neither the name of the copyright holder nor the names of its contributors may be
// used to endorse or promote products derived from this software without specific
// prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
// MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL
// THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
// STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF
// THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Parts of this file are originally copyright (c) 2012-2013 The Cryptonote developers
#include "gtest/gtest.h"
#include <cstdint>
#include <algorithm>
#include "ringct/rctTypes.h"
#include "ringct/rctSigs.h"
#include "ringct/rctOps.h"
using namespace crypto;
using namespace rct;
TEST(ringct, SNL)
{
key x, P1;
skpkGen(x, P1);
key P2 = pkGen();
key P3 = pkGen();
key L1, s1, s2;
GenSchnorrNonLinkable(L1, s1, s2, x, P1, P2, 0);
// a valid one
// an invalid one
ASSERT_TRUE(VerSchnorrNonLinkable(P1, P2, L1, s1, s2));
ASSERT_FALSE(VerSchnorrNonLinkable(P1, P3, L1, s1, s2));
}
TEST(ringct, ASNL)
{
int j = 0;
//Tests for ASNL
//#ASNL true one, false one, C != sum Ci, and one out of the range..
int N = 64;
key64 xv;
key64 P1v;
key64 P2v;
bits indi;
for (j = 0 ; j < N ; j++) {
indi[j] = (int)randXmrAmount(2);
xv[j] = skGen();
if ( (int)indi[j] == 0 ) {
P1v[j] = scalarmultBase(xv[j]);
P2v[j] = pkGen();
} else {
P2v[j] = scalarmultBase(xv[j]);
P1v[j] = pkGen();
}
}
//#true one
asnlSig L1s2s = GenASNL(xv, P1v, P2v, indi);
ASSERT_TRUE(VerASNL(P1v, P2v, L1s2s));
//#false one
indi[3] = (indi[3] + 1) % 2;
L1s2s = GenASNL(xv, P1v, P2v, indi);
ASSERT_FALSE(VerASNL(P1v, P2v, L1s2s));
//#true one again
indi[3] = (indi[3] + 1) % 2;
L1s2s = GenASNL(xv, P1v, P2v, indi);
ASSERT_TRUE(VerASNL(P1v, P2v, L1s2s));
//#false one
L1s2s = GenASNL(xv, P2v, P1v, indi);
ASSERT_FALSE(VerASNL(P1v, P2v, L1s2s));
}
TEST(ringct, MG_sigs)
{
int j = 0;
int N = 0;
//Tests for MG Sigs
//#MG sig: true one
N = 3;// #cols
int R = 3;// #rows
keyV xtmp = skvGen(R);
keyM xm = keyMInit(R, N);// = [[None]*N] #just used to generate test public keys
keyV sk = skvGen(R);
keyM P = keyMInit(R, N);// = keyM[[None]*N] #stores the public keys;
int ind = 2;
int i = 0;
for (j = 0 ; j < R ; j++) {
for (i = 0 ; i < N ; i++)
{
xm[i][j] = skGen();
P[i][j] = scalarmultBase(xm[i][j]);
}
}
for (j = 0 ; j < R ; j++) {
sk[j] = xm[ind][j];
}
key message = identity();
mgSig IIccss = MLSAG_Gen(message, P, sk, ind);
ASSERT_TRUE(MLSAG_Ver(message, P, IIccss, IIccss.II));
//#MG sig: false one
N = 3;// #cols
R = 3;// #rows
xtmp = skvGen(R);
keyM xx(N, xtmp);// = [[None]*N] #just used to generate test public keys
sk = skvGen(R);
//P (N, xtmp);// = keyM[[None]*N] #stores the public keys;
ind = 2;
for (j = 0 ; j < R ; j++) {
for (i = 0 ; i < N ; i++)
{
xx[i][j] = skGen();
P[i][j] = scalarmultBase(xx[i][j]);
}
sk[j] = xx[ind][j];
}
sk[2] = skGen();//asume we don't know one of the private keys..
IIccss = MLSAG_Gen(message, P, sk, ind);
ASSERT_FALSE(MLSAG_Ver(message, P, IIccss, IIccss.II));
}
TEST(ringct, range_proofs)
{
//Ring CT Stuff
//ct range proofs
ctkeyV sc, pc;
ctkey sctmp, pctmp;
//add fake input 5000
tie(sctmp, pctmp) = ctskpkGen(6000);
sc.push_back(sctmp);
pc.push_back(pctmp);
tie(sctmp, pctmp) = ctskpkGen(7000);
sc.push_back(sctmp);
pc.push_back(pctmp);
vector<xmr_amount >amounts;
//add output 500
amounts.push_back(500);
keyV destinations;
key Sk, Pk;
skpkGen(Sk, Pk);
destinations.push_back(Pk);
//add output for 12500
amounts.push_back(12500);
skpkGen(Sk, Pk);
destinations.push_back(Pk);
//compute rct data with mixin 500
rctSig s = genRct(sc, pc, destinations, amounts, rct::zero(), 3);
//verify rct data
ASSERT_TRUE(verRct(s));
//decode received amount
ASSERT_TRUE(decodeRct(s, Sk, 1));
// Ring CT with failing MG sig part should not verify!
// Since sum of inputs != outputs
amounts[1] = 12501;
skpkGen(Sk, Pk);
destinations[1] = Pk;
//compute rct data with mixin 500
s = genRct(sc, pc, destinations, amounts, rct::zero(), 3);
//verify rct data
ASSERT_FALSE(verRct(s));
//decode received amount
ASSERT_TRUE(decodeRct(s, Sk, 1));
}
TEST(ringct, range_proofs_with_fee)
{
//Ring CT Stuff
//ct range proofs
ctkeyV sc, pc;
ctkey sctmp, pctmp;
//add fake input 5000
tie(sctmp, pctmp) = ctskpkGen(6001);
sc.push_back(sctmp);
pc.push_back(pctmp);
tie(sctmp, pctmp) = ctskpkGen(7000);
sc.push_back(sctmp);
pc.push_back(pctmp);
vector<xmr_amount >amounts;
//add output 500
amounts.push_back(500);
keyV destinations;
key Sk, Pk;
skpkGen(Sk, Pk);
destinations.push_back(Pk);
//add txn fee for 1
//has no corresponding destination..
amounts.push_back(1);
//add output for 12500
amounts.push_back(12500);
skpkGen(Sk, Pk);
destinations.push_back(Pk);
//compute rct data with mixin 500
rctSig s = genRct(sc, pc, destinations, amounts, rct::zero(), 3);
//verify rct data
ASSERT_TRUE(verRct(s));
//decode received amount
ASSERT_TRUE(decodeRct(s, Sk, 1));
// Ring CT with failing MG sig part should not verify!
// Since sum of inputs != outputs
amounts[1] = 12501;
skpkGen(Sk, Pk);
destinations[1] = Pk;
//compute rct data with mixin 500
s = genRct(sc, pc, destinations, amounts, rct::zero(), 3);
//verify rct data
ASSERT_FALSE(verRct(s));
//decode received amount
ASSERT_TRUE(decodeRct(s, Sk, 1));
}
static rct::rctSig make_sample_rct_sig(int n_inputs, const uint64_t input_amounts[], int n_outputs, const uint64_t output_amounts[], bool last_is_fee)
{
ctkeyV sc, pc;
ctkey sctmp, pctmp;
vector<xmr_amount >amounts;
keyV destinations;
key Sk, Pk;
for (int n = 0; n < n_inputs; ++n) {
tie(sctmp, pctmp) = ctskpkGen(input_amounts[n]);
sc.push_back(sctmp);
pc.push_back(pctmp);
}
for (int n = 0; n < n_outputs; ++n) {
amounts.push_back(output_amounts[n]);
skpkGen(Sk, Pk);
if (n < n_outputs - 1 || !last_is_fee)
destinations.push_back(Pk);
}
return genRct(sc, pc, destinations, amounts, rct::zero(), 3);;
}
static bool range_proof_test(bool expected_valid,
int n_inputs, const uint64_t input_amounts[], int n_outputs, const uint64_t output_amounts[], bool last_is_fee)
{
//compute rct data
bool valid;
try {
rctSig s = make_sample_rct_sig(n_inputs, input_amounts, n_outputs, output_amounts, last_is_fee);
valid = verRct(s);
}
catch (const std::exception &e) {
valid = false;
}
if (valid == expected_valid) {
return testing::AssertionSuccess();
}
else {
return testing::AssertionFailure();
}
}
#define NELTS(array) (sizeof(array)/sizeof(array[0]))
TEST(ringct, range_proofs_reject_empty_outs)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_empty_ins)
{
const uint64_t inputs[] = {};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_all_empty)
{
const uint64_t inputs[] = {};
const uint64_t outputs[] = {};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_empty)
{
const uint64_t inputs[] = {0};
const uint64_t outputs[] = {};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_empty_zero)
{
const uint64_t inputs[] = {};
const uint64_t outputs[] = {0};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_zero)
{
const uint64_t inputs[] = {0};
const uint64_t outputs[] = {0};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_out_first)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {0, 5000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_out_last)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {5000, 0};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_out_middle)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {2500, 0, 2500};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_in_first)
{
const uint64_t inputs[] = {0, 5000};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_in_last)
{
const uint64_t inputs[] = {5000, 0};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_zero_in_middle)
{
const uint64_t inputs[] = {2500, 0, 2500};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_single_lower)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {1};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_single_higher)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {5001};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_single_out_negative)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {(uint64_t)-1000ll};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_out_negative_first)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {(uint64_t)-1000ll, 6000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_out_negative_last)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {6000, (uint64_t)-1000ll};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_out_negative_middle)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {3000, (uint64_t)-1000ll, 3000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_single_in_negative)
{
const uint64_t inputs[] = {(uint64_t)-1000ll};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_in_negative_first)
{
const uint64_t inputs[] = {(uint64_t)-1000ll, 6000};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_in_negative_last)
{
const uint64_t inputs[] = {6000, (uint64_t)-1000ll};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_in_negative_middle)
{
const uint64_t inputs[] = {3000, (uint64_t)-1000ll, 3000};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_reject_higher_list)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {1000, 1000, 1000, 1000, 1000, 1000};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_1_to_1)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_1_to_N)
{
const uint64_t inputs[] = {5000};
const uint64_t outputs[] = {1000, 1000, 1000, 1000, 1000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_N_to_1)
{
const uint64_t inputs[] = {1000, 1000, 1000, 1000, 1000};
const uint64_t outputs[] = {5000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_N_to_N)
{
const uint64_t inputs[] = {1000, 1000, 1000, 1000, 1000};
const uint64_t outputs[] = {1000, 1000, 1000, 1000, 1000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, range_proofs_accept_very_long)
{
const size_t N=64;
uint64_t inputs[N];
uint64_t outputs[N];
for (size_t n = 0; n < N; ++n) {
inputs[n] = n;
outputs[n] = n;
}
std::random_shuffle(inputs, inputs + N);
std::random_shuffle(outputs, outputs + N);
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, false));
}
TEST(ringct, HPow2)
{
key G = scalarmultBase(d2h(1));
key H = hashToPointSimple(G);
for (int j = 0 ; j < ATOMS ; j++) {
ASSERT_TRUE(equalKeys(H, H2[j]));
addKeys(H, H, H);
}
}
static const xmr_amount test_amounts[]={0, 1, 2, 3, 4, 5, 10000, 10000000000000000000ull, 10203040506070809000ull, 123456789123456789};
TEST(ringct, ecdh_roundtrip)
{
key k, P1;
ecdhTuple t0, t1;
for (auto amount: test_amounts) {
skpkGen(k, P1);
t0.mask = skGen();
t0.amount = d2h(amount);
t1 = t0;
ecdhEncode(t1, P1);
ecdhDecode(t1, k);
ASSERT_TRUE(t0.mask == t1.mask);
ASSERT_TRUE(equalKeys(t0.mask, t1.mask));
ASSERT_TRUE(t0.amount == t1.amount);
ASSERT_TRUE(equalKeys(t0.amount, t1.amount));
}
}
TEST(ringct, d2h)
{
key k, P1;
skpkGen(k, P1);
for (auto amount: test_amounts) {
d2h(k, amount);
ASSERT_TRUE(amount == h2d(k));
}
}
TEST(ringct, d2b)
{
for (auto amount: test_amounts) {
bits b;
d2b(b, amount);
ASSERT_TRUE(amount == b2d(b));
}
}
TEST(ringct, prooveRange_is_non_deterministic)
{
key C[2], mask[2];
for (int n = 0; n < 2; ++n)
proveRange(C[n], mask[n], 80);
ASSERT_TRUE(memcmp(C[0].bytes, C[1].bytes, sizeof(C[0].bytes)));
ASSERT_TRUE(memcmp(mask[0].bytes, mask[1].bytes, sizeof(mask[0].bytes)));
}
TEST(ringct, fee_0_valid)
{
const uint64_t inputs[] = {1000, 1000};
const uint64_t outputs[] = {2000, 0};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, true));
}
TEST(ringct, fee_non_0_valid)
{
const uint64_t inputs[] = {1000, 1000};
const uint64_t outputs[] = {1900, 100};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, true));
}
TEST(ringct, fee_non_0_invalid_higher)
{
const uint64_t inputs[] = {1000, 1000};
const uint64_t outputs[] = {1990, 100};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, true));
}
TEST(ringct, fee_non_0_invalid_lower)
{
const uint64_t inputs[] = {1000, 1000};
const uint64_t outputs[] = {1000, 100};
EXPECT_TRUE(range_proof_test(false, NELTS(inputs), inputs, NELTS(outputs), outputs, true));
}
TEST(ringct, fee_burn_valid_one_out)
{
const uint64_t inputs[] = {1000, 1000};
const uint64_t outputs[] = {0, 2000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, true));
}
TEST(ringct, fee_burn_valid_zero_out)
{
const uint64_t inputs[] = {1000, 1000};
const uint64_t outputs[] = {2000};
EXPECT_TRUE(range_proof_test(true, NELTS(inputs), inputs, NELTS(outputs), outputs, true));
}
#define TEST_rctSig_elements(name, op) \
TEST(ringct, rctSig_##name) \
{ \
const uint64_t inputs[] = {1000, 1000}; \
const uint64_t outputs[] = {1000, 1000}; \
rct::rctSig sig = make_sample_rct_sig(NELTS(inputs), inputs, NELTS(outputs), outputs, true); \
ASSERT_TRUE(rct::verRct(sig)); \
op; \
ASSERT_FALSE(rct::verRct(sig)); \
}
TEST_rctSig_elements(rangeSigs_empty, sig.rangeSigs.resize(0));
TEST_rctSig_elements(rangeSigs_too_many, sig.rangeSigs.push_back(sig.rangeSigs.back()));
TEST_rctSig_elements(rangeSigs_too_few, sig.rangeSigs.pop_back());
TEST_rctSig_elements(mgSig_ss_empty, sig.MG.ss.resize(0));
TEST_rctSig_elements(mgSig_ss_too_many, sig.MG.ss.push_back(sig.MG.ss.back()));
TEST_rctSig_elements(mgSig_ss_too_few, sig.MG.ss.pop_back());
TEST_rctSig_elements(mgSig_ss0_empty, sig.MG.ss[0].resize(0));
TEST_rctSig_elements(mgSig_ss0_too_many, sig.MG.ss[0].push_back(sig.MG.ss[0].back()));
TEST_rctSig_elements(mgSig_ss0_too_few, sig.MG.ss[0].pop_back());
TEST_rctSig_elements(mgSig_II_empty, sig.MG.II.resize(0));
TEST_rctSig_elements(mgSig_II_too_many, sig.MG.II.push_back(sig.MG.II.back()));
TEST_rctSig_elements(mgSig_II_too_few, sig.MG.II.pop_back());
TEST_rctSig_elements(mgSig_mixRing_empty, sig.mixRing.resize(0));
TEST_rctSig_elements(mgSig_mixRing_too_many, sig.mixRing.push_back(sig.mixRing.back()));
TEST_rctSig_elements(mgSig_mixRing_too_few, sig.mixRing.pop_back());
TEST_rctSig_elements(mgSig_mixRing0_empty, sig.mixRing[0].resize(0));
TEST_rctSig_elements(mgSig_mixRing0_too_many, sig.mixRing[0].push_back(sig.mixRing[0].back()));
TEST_rctSig_elements(mgSig_mixRing0_too_few, sig.mixRing[0].pop_back());
TEST_rctSig_elements(ecdhInfo_empty, sig.ecdhInfo.resize(0));
TEST_rctSig_elements(ecdhInfo_too_many, sig.ecdhInfo.push_back(sig.ecdhInfo.back()));
TEST_rctSig_elements(ecdhInfo_too_few, sig.ecdhInfo.pop_back());
TEST_rctSig_elements(outPk_empty, sig.outPk.resize(0));
TEST_rctSig_elements(outPk_too_many, sig.outPk.push_back(sig.outPk.back()));
TEST_rctSig_elements(outPk_too_few, sig.outPk.pop_back());