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// Copyright (c) 2017-2023, 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.
// Implements the Bulletproofs+ prover and verifier algorithms
//
// Preprint: https://eprint.iacr.org/2020/735, version 17 Jun 2020
//
// NOTE ON NOTATION:
// In the signature constructions used in Monero, commitments to zero are treated as
// public keys against the curve group generator `G`. This means that amount
// commitments must use another generator `H` for values in order to show balance.
// The result is that the roles of `g` and `h` in the preprint are effectively swapped
// in this code, taking on the roles of `H` and `G`, respectively. Read carefully!
#include <stdlib.h>
#include <boost/thread/mutex.hpp>
#include <boost/thread/lock_guard.hpp>
#include "misc_log_ex.h"
#include "span.h"
#include "cryptonote_config.h"
extern "C"
{
#include "crypto/crypto-ops.h"
}
#include "rctOps.h"
#include "multiexp.h"
#include "bulletproofs_plus.h"
#undef MONERO_DEFAULT_LOG_CATEGORY
#define MONERO_DEFAULT_LOG_CATEGORY "bulletproof_plus"
#define STRAUS_SIZE_LIMIT 232
#define PIPPENGER_SIZE_LIMIT 0
namespace rct
{
// Vector functions
static rct::key vector_exponent(const rct::keyV &a, const rct::keyV &b);
static rct::keyV vector_of_scalar_powers(const rct::key &x, size_t n);
// Proof bounds
static constexpr size_t maxN = 64; // maximum number of bits in range
static constexpr size_t maxM = BULLETPROOF_PLUS_MAX_OUTPUTS; // maximum number of outputs to aggregate into a single proof
// Cached public generators
static ge_p3 Hi_p3[maxN*maxM], Gi_p3[maxN*maxM];
static std::shared_ptr<straus_cached_data> straus_HiGi_cache;
static std::shared_ptr<pippenger_cached_data> pippenger_HiGi_cache;
// Useful scalar constants
static const constexpr rct::key ZERO = { {0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 } }; // 0
static const constexpr rct::key ONE = { {0x01, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 } }; // 1
static const constexpr rct::key TWO = { {0x02, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 , 0x00, 0x00, 0x00,0x00 } }; // 2
static const constexpr rct::key MINUS_ONE = { { 0xec, 0xd3, 0xf5, 0x5c, 0x1a, 0x63, 0x12, 0x58, 0xd6, 0x9c, 0xf7, 0xa2, 0xde, 0xf9, 0xde, 0x14, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10 } }; // -1
static const constexpr rct::key MINUS_INV_EIGHT = { { 0x74, 0xa4, 0x19, 0x7a, 0xf0, 0x7d, 0x0b, 0xf7, 0x05, 0xc2, 0xda, 0x25, 0x2b, 0x5c, 0x0b, 0x0d, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0a } }; // -(8**(-1))
static rct::key TWO_SIXTY_FOUR_MINUS_ONE; // 2**64 - 1
// Initial transcript hash
static rct::key initial_transcript;
static boost::mutex init_mutex;
// Use the generator caches to compute a multiscalar multiplication
static inline rct::key multiexp(const std::vector<MultiexpData> &data, size_t HiGi_size)
{
if (HiGi_size > 0)
{
static_assert(232 <= STRAUS_SIZE_LIMIT, "Straus in precalc mode can only be calculated till STRAUS_SIZE_LIMIT");
return HiGi_size <= 232 && data.size() == HiGi_size ? straus(data, straus_HiGi_cache, 0) : pippenger(data, pippenger_HiGi_cache, HiGi_size, get_pippenger_c(data.size()));
}
else
{
return data.size() <= 95 ? straus(data, NULL, 0) : pippenger(data, NULL, 0, get_pippenger_c(data.size()));
}
}
// Confirm that a scalar is properly reduced
static inline bool is_reduced(const rct::key &scalar)
{
return sc_check(scalar.bytes) == 0;
}
// Use hashed values to produce indexed public generators
static ge_p3 get_exponent(const rct::key &base, size_t idx)
{
std::string hashed = std::string((const char*)base.bytes, sizeof(base)) + config::HASH_KEY_BULLETPROOF_PLUS_EXPONENT + tools::get_varint_data(idx);
rct::key generator;
ge_p3 generator_p3;
rct::hash_to_p3(generator_p3, rct::hash2rct(crypto::cn_fast_hash(hashed.data(), hashed.size())));
ge_p3_tobytes(generator.bytes, &generator_p3);
CHECK_AND_ASSERT_THROW_MES(!(generator == rct::identity()), "Exponent is point at infinity");
return generator_p3;
}
// Construct public generators
static void init_exponents()
{
boost::lock_guard<boost::mutex> lock(init_mutex);
// Only needs to be done once
static bool init_done = false;
if (init_done)
return;
std::vector<MultiexpData> data;
data.reserve(maxN*maxM*2);
for (size_t i = 0; i < maxN*maxM; ++i)
{
Hi_p3[i] = get_exponent(rct::H, i * 2);
Gi_p3[i] = get_exponent(rct::H, i * 2 + 1);
data.push_back({rct::zero(), Gi_p3[i]});
data.push_back({rct::zero(), Hi_p3[i]});
}
straus_HiGi_cache = straus_init_cache(data, STRAUS_SIZE_LIMIT);
pippenger_HiGi_cache = pippenger_init_cache(data, 0, PIPPENGER_SIZE_LIMIT);
// Compute 2**64 - 1 for later use in simplifying verification
TWO_SIXTY_FOUR_MINUS_ONE = TWO;
for (size_t i = 0; i < 6; i++)
{
sc_mul(TWO_SIXTY_FOUR_MINUS_ONE.bytes, TWO_SIXTY_FOUR_MINUS_ONE.bytes, TWO_SIXTY_FOUR_MINUS_ONE.bytes);
}
sc_sub(TWO_SIXTY_FOUR_MINUS_ONE.bytes, TWO_SIXTY_FOUR_MINUS_ONE.bytes, ONE.bytes);
// Generate the initial Fiat-Shamir transcript hash, which is constant across all proofs
const std::string domain_separator(config::HASH_KEY_BULLETPROOF_PLUS_TRANSCRIPT);
ge_p3 initial_transcript_p3;
rct::hash_to_p3(initial_transcript_p3, rct::hash2rct(crypto::cn_fast_hash(domain_separator.data(), domain_separator.size())));
ge_p3_tobytes(initial_transcript.bytes, &initial_transcript_p3);
init_done = true;
}
// Given two scalar arrays, construct a vector pre-commitment:
//
// a = (a_0, ..., a_{n-1})
// b = (b_0, ..., b_{n-1})
//
// Outputs a_0*Gi_0 + ... + a_{n-1}*Gi_{n-1} +
// b_0*Hi_0 + ... + b_{n-1}*Hi_{n-1}
static rct::key vector_exponent(const rct::keyV &a, const rct::keyV &b)
{
CHECK_AND_ASSERT_THROW_MES(a.size() == b.size(), "Incompatible sizes of a and b");
CHECK_AND_ASSERT_THROW_MES(a.size() <= maxN*maxM, "Incompatible sizes of a and maxN");
std::vector<MultiexpData> multiexp_data;
multiexp_data.reserve(a.size()*2);
for (size_t i = 0; i < a.size(); ++i)
{
multiexp_data.emplace_back(a[i], Gi_p3[i]);
multiexp_data.emplace_back(b[i], Hi_p3[i]);
}
return multiexp(multiexp_data, 2 * a.size());
}
// Helper function used to compute the L and R terms used in the inner-product round function
static rct::key compute_LR(size_t size, const rct::key &y, const std::vector<ge_p3> &G, size_t G0, const std::vector<ge_p3> &H, size_t H0, const rct::keyV &a, size_t a0, const rct::keyV &b, size_t b0, const rct::key &c, const rct::key &d)
{
CHECK_AND_ASSERT_THROW_MES(size + G0 <= G.size(), "Incompatible size for G");
CHECK_AND_ASSERT_THROW_MES(size + H0 <= H.size(), "Incompatible size for H");
CHECK_AND_ASSERT_THROW_MES(size + a0 <= a.size(), "Incompatible size for a");
CHECK_AND_ASSERT_THROW_MES(size + b0 <= b.size(), "Incompatible size for b");
CHECK_AND_ASSERT_THROW_MES(size <= maxN*maxM, "size is too large");
std::vector<MultiexpData> multiexp_data;
multiexp_data.resize(size*2 + 2);
rct::key temp;
for (size_t i = 0; i < size; ++i)
{
sc_mul(temp.bytes, a[a0+i].bytes, y.bytes);
sc_mul(multiexp_data[i*2].scalar.bytes, temp.bytes, INV_EIGHT.bytes);
multiexp_data[i*2].point = G[G0+i];
sc_mul(multiexp_data[i*2+1].scalar.bytes, b[b0+i].bytes, INV_EIGHT.bytes);
multiexp_data[i*2+1].point = H[H0+i];
}
sc_mul(multiexp_data[2*size].scalar.bytes, c.bytes, INV_EIGHT.bytes);
ge_p3 H_p3;
ge_frombytes_vartime(&H_p3, rct::H.bytes);
multiexp_data[2*size].point = H_p3;
sc_mul(multiexp_data[2*size+1].scalar.bytes, d.bytes, INV_EIGHT.bytes);
ge_p3 G_p3;
ge_frombytes_vartime(&G_p3, rct::G.bytes);
multiexp_data[2*size+1].point = G_p3;
return multiexp(multiexp_data, 0);
}
// Given a scalar, construct a vector of its powers:
//
// Output (1,x,x**2,...,x**{n-1})
static rct::keyV vector_of_scalar_powers(const rct::key &x, size_t n)
{
CHECK_AND_ASSERT_THROW_MES(n != 0, "Need n > 0");
rct::keyV res(n);
res[0] = rct::identity();
if (n == 1)
return res;
res[1] = x;
for (size_t i = 2; i < n; ++i)
{
sc_mul(res[i].bytes, res[i-1].bytes, x.bytes);
}
return res;
}
// Given a scalar, construct the sum of its powers from 2 to n (where n is a power of 2):
//
// Output x**2 + x**4 + x**6 + ... + x**n
static rct::key sum_of_even_powers(const rct::key &x, size_t n)
{
CHECK_AND_ASSERT_THROW_MES((n & (n - 1)) == 0, "Need n to be a power of 2");
CHECK_AND_ASSERT_THROW_MES(n != 0, "Need n > 0");
rct::key x1 = copy(x);
sc_mul(x1.bytes, x1.bytes, x1.bytes);
rct::key res = copy(x1);
while (n > 2)
{
sc_muladd(res.bytes, x1.bytes, res.bytes, res.bytes);
sc_mul(x1.bytes, x1.bytes, x1.bytes);
n /= 2;
}
return res;
}
// Given a scalar, return the sum of its powers from 1 to n
//
// Output x**1 + x**2 + x**3 + ... + x**n
static rct::key sum_of_scalar_powers(const rct::key &x, size_t n)
{
CHECK_AND_ASSERT_THROW_MES(n != 0, "Need n > 0");
rct::key res = ONE;
if (n == 1)
return x;
n += 1;
rct::key x1 = copy(x);
const bool is_power_of_2 = (n & (n - 1)) == 0;
if (is_power_of_2)
{
sc_add(res.bytes, res.bytes, x1.bytes);
while (n > 2)
{
sc_mul(x1.bytes, x1.bytes, x1.bytes);
sc_muladd(res.bytes, x1.bytes, res.bytes, res.bytes);
n /= 2;
}
}
else
{
rct::key prev = x1;
for (size_t i = 1; i < n; ++i)
{
if (i > 1)
sc_mul(prev.bytes, prev.bytes, x1.bytes);
sc_add(res.bytes, res.bytes, prev.bytes);
}
}
sc_sub(res.bytes, res.bytes, ONE.bytes);
return res;
}
// Given two scalar arrays, construct the weighted inner product against another scalar
//
// Output a_0*b_0*y**1 + a_1*b_1*y**2 + ... + a_{n-1}*b_{n-1}*y**n
static rct::key weighted_inner_product(const epee::span<const rct::key> &a, const epee::span<const rct::key> &b, const rct::key &y)
{
CHECK_AND_ASSERT_THROW_MES(a.size() == b.size(), "Incompatible sizes of a and b");
rct::key res = rct::zero();
rct::key y_power = ONE;
rct::key temp;
for (size_t i = 0; i < a.size(); ++i)
{
sc_mul(temp.bytes, a[i].bytes, b[i].bytes);
sc_mul(y_power.bytes, y_power.bytes, y.bytes);
sc_muladd(res.bytes, temp.bytes, y_power.bytes, res.bytes);
}
return res;
}
static rct::key weighted_inner_product(const rct::keyV &a, const epee::span<const rct::key> &b, const rct::key &y)
{
CHECK_AND_ASSERT_THROW_MES(a.size() == b.size(), "Incompatible sizes of a and b");
return weighted_inner_product(epee::to_span(a), b, y);
}
// Fold inner-product point vectors
static void hadamard_fold(std::vector<ge_p3> &v, const rct::key &a, const rct::key &b)
{
CHECK_AND_ASSERT_THROW_MES((v.size() & 1) == 0, "Vector size should be even");
const size_t sz = v.size() / 2;
for (size_t n = 0; n < sz; ++n)
{
ge_dsmp c[2];
ge_dsm_precomp(c[0], &v[n]);
ge_dsm_precomp(c[1], &v[sz + n]);
ge_double_scalarmult_precomp_vartime2_p3(&v[n], a.bytes, c[0], b.bytes, c[1]);
}
v.resize(sz);
}
// Add vectors componentwise
static rct::keyV vector_add(const rct::keyV &a, const rct::keyV &b)
{
CHECK_AND_ASSERT_THROW_MES(a.size() == b.size(), "Incompatible sizes of a and b");
rct::keyV res(a.size());
for (size_t i = 0; i < a.size(); ++i)
{
sc_add(res[i].bytes, a[i].bytes, b[i].bytes);
}
return res;
}
// Add a scalar to all elements of a vector
static rct::keyV vector_add(const rct::keyV &a, const rct::key &b)
{
rct::keyV res(a.size());
for (size_t i = 0; i < a.size(); ++i)
{
sc_add(res[i].bytes, a[i].bytes, b.bytes);
}
return res;
}
// Subtract a scalar from all elements of a vector
static rct::keyV vector_subtract(const rct::keyV &a, const rct::key &b)
{
rct::keyV res(a.size());
for (size_t i = 0; i < a.size(); ++i)
{
sc_sub(res[i].bytes, a[i].bytes, b.bytes);
}
return res;
}
// Multiply a scalar by all elements of a vector
static rct::keyV vector_scalar(const epee::span<const rct::key> &a, const rct::key &x)
{
rct::keyV res(a.size());
for (size_t i = 0; i < a.size(); ++i)
{
sc_mul(res[i].bytes, a[i].bytes, x.bytes);
}
return res;
}
// Inversion helper function
static rct::key sm(rct::key y, int n, const rct::key &x)
{
while (n--)
sc_mul(y.bytes, y.bytes, y.bytes);
sc_mul(y.bytes, y.bytes, x.bytes);
return y;
}
// Compute the inverse of a nonzero
static rct::key invert(const rct::key &x)
{
CHECK_AND_ASSERT_THROW_MES(!(x == ZERO), "Cannot invert zero!");
rct::key _1, _10, _100, _11, _101, _111, _1001, _1011, _1111;
_1 = x;
sc_mul(_10.bytes, _1.bytes, _1.bytes);
sc_mul(_100.bytes, _10.bytes, _10.bytes);
sc_mul(_11.bytes, _10.bytes, _1.bytes);
sc_mul(_101.bytes, _10.bytes, _11.bytes);
sc_mul(_111.bytes, _10.bytes, _101.bytes);
sc_mul(_1001.bytes, _10.bytes, _111.bytes);
sc_mul(_1011.bytes, _10.bytes, _1001.bytes);
sc_mul(_1111.bytes, _100.bytes, _1011.bytes);
rct::key inv;
sc_mul(inv.bytes, _1111.bytes, _1.bytes);
inv = sm(inv, 123 + 3, _101);
inv = sm(inv, 2 + 2, _11);
inv = sm(inv, 1 + 4, _1111);
inv = sm(inv, 1 + 4, _1111);
inv = sm(inv, 4, _1001);
inv = sm(inv, 2, _11);
inv = sm(inv, 1 + 4, _1111);
inv = sm(inv, 1 + 3, _101);
inv = sm(inv, 3 + 3, _101);
inv = sm(inv, 3, _111);
inv = sm(inv, 1 + 4, _1111);
inv = sm(inv, 2 + 3, _111);
inv = sm(inv, 2 + 2, _11);
inv = sm(inv, 1 + 4, _1011);
inv = sm(inv, 2 + 4, _1011);
inv = sm(inv, 6 + 4, _1001);
inv = sm(inv, 2 + 2, _11);
inv = sm(inv, 3 + 2, _11);
inv = sm(inv, 3 + 2, _11);
inv = sm(inv, 1 + 4, _1001);
inv = sm(inv, 1 + 3, _111);
inv = sm(inv, 2 + 4, _1111);
inv = sm(inv, 1 + 4, _1011);
inv = sm(inv, 3, _101);
inv = sm(inv, 2 + 4, _1111);
inv = sm(inv, 3, _101);
inv = sm(inv, 1 + 2, _11);
return inv;
}
// Invert a batch of scalars, all of which _must_ be nonzero
static rct::keyV invert(rct::keyV x)
{
rct::keyV scratch;
scratch.reserve(x.size());
rct::key acc = rct::identity();
for (size_t n = 0; n < x.size(); ++n)
{
CHECK_AND_ASSERT_THROW_MES(!(x[n] == ZERO), "Cannot invert zero!");
scratch.push_back(acc);
if (n == 0)
acc = x[0];
else
sc_mul(acc.bytes, acc.bytes, x[n].bytes);
}
acc = invert(acc);
rct::key tmp;
for (int i = x.size(); i-- > 0; )
{
sc_mul(tmp.bytes, acc.bytes, x[i].bytes);
sc_mul(x[i].bytes, acc.bytes, scratch[i].bytes);
acc = tmp;
}
return x;
}
// Compute the slice of a vector
static epee::span<const rct::key> slice(const rct::keyV &a, size_t start, size_t stop)
{
CHECK_AND_ASSERT_THROW_MES(start < a.size(), "Invalid start index");
CHECK_AND_ASSERT_THROW_MES(stop <= a.size(), "Invalid stop index");
CHECK_AND_ASSERT_THROW_MES(start < stop, "Invalid start/stop indices");
return epee::span<const rct::key>(&a[start], stop - start);
}
// Update the transcript
static rct::key transcript_update(rct::key &transcript, const rct::key &update_0)
{
rct::key data[2];
data[0] = transcript;
data[1] = update_0;
rct::hash_to_scalar(transcript, data, sizeof(data));
return transcript;
}
static rct::key transcript_update(rct::key &transcript, const rct::key &update_0, const rct::key &update_1)
{
rct::key data[3];
data[0] = transcript;
data[1] = update_0;
data[2] = update_1;
rct::hash_to_scalar(transcript, data, sizeof(data));
return transcript;
}
// Given a value v [0..2**N) and a mask gamma, construct a range proof
BulletproofPlus bulletproof_plus_PROVE(const rct::key &sv, const rct::key &gamma)
{
return bulletproof_plus_PROVE(rct::keyV(1, sv), rct::keyV(1, gamma));
}
BulletproofPlus bulletproof_plus_PROVE(uint64_t v, const rct::key &gamma)
{
return bulletproof_plus_PROVE(std::vector<uint64_t>(1, v), rct::keyV(1, gamma));
}
// Given a set of values v [0..2**N) and masks gamma, construct a range proof
BulletproofPlus bulletproof_plus_PROVE(const rct::keyV &sv, const rct::keyV &gamma)
{
// Sanity check on inputs
CHECK_AND_ASSERT_THROW_MES(sv.size() == gamma.size(), "Incompatible sizes of sv and gamma");
CHECK_AND_ASSERT_THROW_MES(!sv.empty(), "sv is empty");
for (const rct::key &sve: sv)
CHECK_AND_ASSERT_THROW_MES(is_reduced(sve), "Invalid sv input");
for (const rct::key &g: gamma)
CHECK_AND_ASSERT_THROW_MES(is_reduced(g), "Invalid gamma input");
init_exponents();
// Useful proof bounds
//
// N: number of bits in each range (here, 64)
// logN: base-2 logarithm
// M: first power of 2 greater than or equal to the number of range proofs to aggregate
// logM: base-2 logarithm
constexpr size_t logN = 6; // log2(64)
constexpr size_t N = 1<<logN;
size_t M, logM;
for (logM = 0; (M = 1<<logM) <= maxM && M < sv.size(); ++logM);
CHECK_AND_ASSERT_THROW_MES(M <= maxM, "sv/gamma are too large");
const size_t logMN = logM + logN;
const size_t MN = M * N;
rct::keyV V(sv.size());
rct::keyV aL(MN), aR(MN);
rct::keyV aL8(MN), aR8(MN);
rct::key temp;
rct::key temp2;
// Prepare output commitments and offset by a factor of 8**(-1)
//
// This offset is applied to other group elements as well;
// it allows us to apply a multiply-by-8 operation in the verifier efficiently
// to ensure that the resulting group elements are in the prime-order point subgroup
// and avoid much more constly multiply-by-group-order operations.
for (size_t i = 0; i < sv.size(); ++i)
{
rct::key gamma8, sv8;
sc_mul(gamma8.bytes, gamma[i].bytes, INV_EIGHT.bytes);
sc_mul(sv8.bytes, sv[i].bytes, INV_EIGHT.bytes);
rct::addKeys2(V[i], gamma8, sv8, rct::H);
}
// Decompose values
//
// Note that this effectively pads the set to a power of 2, which is required for the inner-product argument later.
for (size_t j = 0; j < M; ++j)
{
for (size_t i = N; i-- > 0; )
{
if (j < sv.size() && (sv[j][i/8] & (((uint64_t)1)<<(i%8))))
{
aL[j*N+i] = rct::identity();
aL8[j*N+i] = INV_EIGHT;
aR[j*N+i] = aR8[j*N+i] = rct::zero();
}
else
{
aL[j*N+i] = aL8[j*N+i] = rct::zero();
aR[j*N+i] = MINUS_ONE;
aR8[j*N+i] = MINUS_INV_EIGHT;
}
}
}
try_again:
// This is a Fiat-Shamir transcript
rct::key transcript = copy(initial_transcript);
transcript = transcript_update(transcript, rct::hash_to_scalar(V));
// A
rct::key alpha = rct::skGen();
rct::key pre_A = vector_exponent(aL8, aR8);
rct::key A;
sc_mul(temp.bytes, alpha.bytes, INV_EIGHT.bytes);
rct::addKeys(A, pre_A, rct::scalarmultBase(temp));
// Challenges
rct::key y = transcript_update(transcript, A);
if (y == rct::zero())
{
MINFO("y is 0, trying again");
goto try_again;
}
rct::key z = transcript = rct::hash_to_scalar(y);
if (z == rct::zero())
{
MINFO("z is 0, trying again");
goto try_again;
}
rct::key z_squared;
sc_mul(z_squared.bytes, z.bytes, z.bytes);
// Windowed vector
// d[j*N+i] = z**(2*(j+1)) * 2**i
//
// We compute this iteratively in order to reduce scalar operations.
rct::keyV d(MN, rct::zero());
d[0] = z_squared;
for (size_t i = 1; i < N; i++)
{
sc_mul(d[i].bytes, d[i-1].bytes, TWO.bytes);
}
for (size_t j = 1; j < M; j++)
{
for (size_t i = 0; i < N; i++)
{
sc_mul(d[j*N+i].bytes, d[(j-1)*N+i].bytes, z_squared.bytes);
}
}
rct::keyV y_powers = vector_of_scalar_powers(y, MN+2);
// Prepare inner product terms
rct::keyV aL1 = vector_subtract(aL, z);
rct::keyV aR1 = vector_add(aR, z);
rct::keyV d_y(MN);
for (size_t i = 0; i < MN; i++)
{
sc_mul(d_y[i].bytes, d[i].bytes, y_powers[MN-i].bytes);
}
aR1 = vector_add(aR1, d_y);
rct::key alpha1 = alpha;
temp = ONE;
for (size_t j = 0; j < sv.size(); j++)
{
sc_mul(temp.bytes, temp.bytes, z_squared.bytes);
sc_mul(temp2.bytes, y_powers[MN+1].bytes, temp.bytes);
sc_muladd(alpha1.bytes, temp2.bytes, gamma[j].bytes, alpha1.bytes);
}
// These are used in the inner product rounds
size_t nprime = MN;
std::vector<ge_p3> Gprime(MN);
std::vector<ge_p3> Hprime(MN);
rct::keyV aprime(MN);
rct::keyV bprime(MN);
const rct::key yinv = invert(y);
rct::keyV yinvpow(MN);
yinvpow[0] = ONE;
for (size_t i = 0; i < MN; ++i)
{
Gprime[i] = Gi_p3[i];
Hprime[i] = Hi_p3[i];
if (i > 0)
{
sc_mul(yinvpow[i].bytes, yinvpow[i-1].bytes, yinv.bytes);
}
aprime[i] = aL1[i];
bprime[i] = aR1[i];
}
rct::keyV L(logMN);
rct::keyV R(logMN);
int round = 0;
// Inner-product rounds
while (nprime > 1)
{
nprime /= 2;
rct::key cL = weighted_inner_product(slice(aprime, 0, nprime), slice(bprime, nprime, bprime.size()), y);
rct::key cR = weighted_inner_product(vector_scalar(slice(aprime, nprime, aprime.size()), y_powers[nprime]), slice(bprime, 0, nprime), y);
rct::key dL = rct::skGen();
rct::key dR = rct::skGen();
L[round] = compute_LR(nprime, yinvpow[nprime], Gprime, nprime, Hprime, 0, aprime, 0, bprime, nprime, cL, dL);
R[round] = compute_LR(nprime, y_powers[nprime], Gprime, 0, Hprime, nprime, aprime, nprime, bprime, 0, cR, dR);
const rct::key challenge = transcript_update(transcript, L[round], R[round]);
if (challenge == rct::zero())
{
MINFO("challenge is 0, trying again");
goto try_again;
}
const rct::key challenge_inv = invert(challenge);
sc_mul(temp.bytes, yinvpow[nprime].bytes, challenge.bytes);
hadamard_fold(Gprime, challenge_inv, temp);
hadamard_fold(Hprime, challenge, challenge_inv);
sc_mul(temp.bytes, challenge_inv.bytes, y_powers[nprime].bytes);
aprime = vector_add(vector_scalar(slice(aprime, 0, nprime), challenge), vector_scalar(slice(aprime, nprime, aprime.size()), temp));
bprime = vector_add(vector_scalar(slice(bprime, 0, nprime), challenge_inv), vector_scalar(slice(bprime, nprime, bprime.size()), challenge));
rct::key challenge_squared;
sc_mul(challenge_squared.bytes, challenge.bytes, challenge.bytes);
rct::key challenge_squared_inv;
sc_mul(challenge_squared_inv.bytes, challenge_inv.bytes, challenge_inv.bytes);
sc_muladd(alpha1.bytes, dL.bytes, challenge_squared.bytes, alpha1.bytes);
sc_muladd(alpha1.bytes, dR.bytes, challenge_squared_inv.bytes, alpha1.bytes);
++round;
}
// Final round computations
rct::key r = rct::skGen();
rct::key s = rct::skGen();
rct::key d_ = rct::skGen();
rct::key eta = rct::skGen();
std::vector<MultiexpData> A1_data;
A1_data.reserve(4);
A1_data.resize(4);
sc_mul(A1_data[0].scalar.bytes, r.bytes, INV_EIGHT.bytes);
A1_data[0].point = Gprime[0];
sc_mul(A1_data[1].scalar.bytes, s.bytes, INV_EIGHT.bytes);
A1_data[1].point = Hprime[0];
sc_mul(A1_data[2].scalar.bytes, d_.bytes, INV_EIGHT.bytes);
ge_p3 G_p3;
ge_frombytes_vartime(&G_p3, rct::G.bytes);
A1_data[2].point = G_p3;
sc_mul(temp.bytes, r.bytes, y.bytes);
sc_mul(temp.bytes, temp.bytes, bprime[0].bytes);
sc_mul(temp2.bytes, s.bytes, y.bytes);
sc_mul(temp2.bytes, temp2.bytes, aprime[0].bytes);
sc_add(temp.bytes, temp.bytes, temp2.bytes);
sc_mul(A1_data[3].scalar.bytes, temp.bytes, INV_EIGHT.bytes);
ge_p3 H_p3;
ge_frombytes_vartime(&H_p3, rct::H.bytes);
A1_data[3].point = H_p3;
rct::key A1 = multiexp(A1_data, 0);
sc_mul(temp.bytes, r.bytes, y.bytes);
sc_mul(temp.bytes, temp.bytes, s.bytes);
sc_mul(temp.bytes, temp.bytes, INV_EIGHT.bytes);
sc_mul(temp2.bytes, eta.bytes, INV_EIGHT.bytes);
rct::key B;
rct::addKeys2(B, temp2, temp, rct::H);
rct::key e = transcript_update(transcript, A1, B);
if (e == rct::zero())
{
MINFO("e is 0, trying again");
goto try_again;
}
rct::key e_squared;
sc_mul(e_squared.bytes, e.bytes, e.bytes);
rct::key r1;
sc_muladd(r1.bytes, aprime[0].bytes, e.bytes, r.bytes);
rct::key s1;
sc_muladd(s1.bytes, bprime[0].bytes, e.bytes, s.bytes);
rct::key d1;
sc_muladd(d1.bytes, d_.bytes, e.bytes, eta.bytes);
sc_muladd(d1.bytes, alpha1.bytes, e_squared.bytes, d1.bytes);
return BulletproofPlus(std::move(V), A, A1, B, r1, s1, d1, std::move(L), std::move(R));
}
BulletproofPlus bulletproof_plus_PROVE(const std::vector<uint64_t> &v, const rct::keyV &gamma)
{
CHECK_AND_ASSERT_THROW_MES(v.size() == gamma.size(), "Incompatible sizes of v and gamma");
// vG + gammaH
rct::keyV sv(v.size());
for (size_t i = 0; i < v.size(); ++i)
{
sv[i] = rct::d2h(v[i]);
}
return bulletproof_plus_PROVE(sv, gamma);
}
struct bp_plus_proof_data_t
{
rct::key y, z, e;
std::vector<rct::key> challenges;
size_t logM, inv_offset;
};
// Given a batch of range proofs, determine if they are all valid
bool bulletproof_plus_VERIFY(const std::vector<const BulletproofPlus*> &proofs)
{
init_exponents();
const size_t logN = 6;
const size_t N = 1 << logN;
// Set up
size_t max_length = 0; // size of each of the longest proof's inner-product vectors
size_t nV = 0; // number of output commitments across all proofs
size_t inv_offset = 0;
size_t max_logM = 0;
std::vector<bp_plus_proof_data_t> proof_data;
proof_data.reserve(proofs.size());
// We'll perform only a single batch inversion across all proofs in the batch,
// since batch inversion requires only one scalar inversion operation.
std::vector<rct::key> to_invert;
to_invert.reserve(11 * proofs.size()); // maximal size, given the aggregation limit
for (const BulletproofPlus *p: proofs)
{
const BulletproofPlus &proof = *p;
// Sanity checks
CHECK_AND_ASSERT_MES(is_reduced(proof.r1), false, "Input scalar not in range");
CHECK_AND_ASSERT_MES(is_reduced(proof.s1), false, "Input scalar not in range");
CHECK_AND_ASSERT_MES(is_reduced(proof.d1), false, "Input scalar not in range");
CHECK_AND_ASSERT_MES(proof.V.size() >= 1, false, "V does not have at least one element");
CHECK_AND_ASSERT_MES(proof.L.size() == proof.R.size(), false, "Mismatched L and R sizes");
CHECK_AND_ASSERT_MES(proof.L.size() > 0, false, "Empty proof");
max_length = std::max(max_length, proof.L.size());
nV += proof.V.size();
proof_data.push_back({});
bp_plus_proof_data_t &pd = proof_data.back();
// Reconstruct the challenges
rct::key transcript = copy(initial_transcript);
transcript = transcript_update(transcript, rct::hash_to_scalar(proof.V));
pd.y = transcript_update(transcript, proof.A);
CHECK_AND_ASSERT_MES(!(pd.y == rct::zero()), false, "y == 0");
pd.z = transcript = rct::hash_to_scalar(pd.y);
CHECK_AND_ASSERT_MES(!(pd.z == rct::zero()), false, "z == 0");
// Determine the number of inner-product rounds based on proof size
size_t M;
for (pd.logM = 0; (M = 1<<pd.logM) <= maxM && M < proof.V.size(); ++pd.logM);
CHECK_AND_ASSERT_MES(proof.L.size() == 6+pd.logM, false, "Proof is not the expected size");
max_logM = std::max(pd.logM, max_logM);
const size_t rounds = pd.logM+logN;
CHECK_AND_ASSERT_MES(rounds > 0, false, "Zero rounds");
// The inner-product challenges are computed per round
pd.challenges.resize(rounds);
for (size_t j = 0; j < rounds; ++j)
{
pd.challenges[j] = transcript_update(transcript, proof.L[j], proof.R[j]);
CHECK_AND_ASSERT_MES(!(pd.challenges[j] == rct::zero()), false, "challenges[j] == 0");
}
// Final challenge
pd.e = transcript_update(transcript,proof.A1,proof.B);
CHECK_AND_ASSERT_MES(!(pd.e == rct::zero()), false, "e == 0");
// Batch scalar inversions
pd.inv_offset = inv_offset;
for (size_t j = 0; j < rounds; ++j)
to_invert.push_back(pd.challenges[j]);
to_invert.push_back(pd.y);
inv_offset += rounds + 1;
}
CHECK_AND_ASSERT_MES(max_length < 32, false, "At least one proof is too large");
size_t maxMN = 1u << max_length;
rct::key temp;
rct::key temp2;
// Final batch proof data
std::vector<MultiexpData> multiexp_data;
multiexp_data.reserve(nV + (2 * (max_logM + logN) + 3) * proofs.size() + 2 * maxMN);
multiexp_data.resize(2 * maxMN);
const std::vector<rct::key> inverses = invert(std::move(to_invert));
to_invert.clear();
// Weights and aggregates
//
// The idea is to take the single multiscalar multiplication used in the verification
// of each proof in the batch and weight it using a random weighting factor, resulting
// in just one multiscalar multiplication check to zero for the entire batch.
// We can further simplify the verifier complexity by including common group elements
// only once in this single multiscalar multiplication.
// Common group elements' weighted scalar sums are tracked across proofs for this reason.
//
// To build a multiscalar multiplication for each proof, we use the method described in
// Section 6.1 of the preprint. Note that the result given there does not account for
// the construction of the inner-product inputs that are produced in the range proof
// verifier algorithm; we have done so here.
rct::key G_scalar = rct::zero();
rct::key H_scalar = rct::zero();
rct::keyV Gi_scalars(maxMN, rct::zero());
rct::keyV Hi_scalars(maxMN, rct::zero());
int proof_data_index = 0;
rct::keyV challenges_cache;
std::vector<ge_p3> proof8_V, proof8_L, proof8_R;
// Process each proof and add to the weighted batch
for (const BulletproofPlus *p: proofs)
{
const BulletproofPlus &proof = *p;
const bp_plus_proof_data_t &pd = proof_data[proof_data_index++];
CHECK_AND_ASSERT_MES(proof.L.size() == 6+pd.logM, false, "Proof is not the expected size");
const size_t M = 1 << pd.logM;
const size_t MN = M*N;
// Random weighting factor must be nonzero, which is exceptionally unlikely!
rct::key weight = ZERO;
while (weight == ZERO)
{
weight = rct::skGen();
}
// Rescale previously offset proof elements
//
// This ensures that all such group elements are in the prime-order subgroup.
proof8_V.resize(proof.V.size()); for (size_t i = 0; i < proof.V.size(); ++i) rct::scalarmult8(proof8_V[i], proof.V[i]);
proof8_L.resize(proof.L.size()); for (size_t i = 0; i < proof.L.size(); ++i) rct::scalarmult8(proof8_L[i], proof.L[i]);
proof8_R.resize(proof.R.size()); for (size_t i = 0; i < proof.R.size(); ++i) rct::scalarmult8(proof8_R[i], proof.R[i]);
ge_p3 proof8_A1;
ge_p3 proof8_B;
ge_p3 proof8_A;
rct::scalarmult8(proof8_A1, proof.A1);
rct::scalarmult8(proof8_B, proof.B);
rct::scalarmult8(proof8_A, proof.A);
// Compute necessary powers of the y-challenge
rct::key y_MN = copy(pd.y);
rct::key y_MN_1;
size_t temp_MN = MN;
while (temp_MN > 1)
{
sc_mul(y_MN.bytes, y_MN.bytes, y_MN.bytes);
temp_MN /= 2;
}
sc_mul(y_MN_1.bytes, y_MN.bytes, pd.y.bytes);
// V_j: -e**2 * z**(2*j+1) * y**(MN+1) * weight
rct::key e_squared;
sc_mul(e_squared.bytes, pd.e.bytes, pd.e.bytes);
rct::key z_squared;
sc_mul(z_squared.bytes, pd.z.bytes, pd.z.bytes);
sc_sub(temp.bytes, ZERO.bytes, e_squared.bytes);
sc_mul(temp.bytes, temp.bytes, y_MN_1.bytes);
sc_mul(temp.bytes, temp.bytes, weight.bytes);
for (size_t j = 0; j < proof8_V.size(); j++)
{
sc_mul(temp.bytes, temp.bytes, z_squared.bytes);
multiexp_data.emplace_back(temp, proof8_V[j]);
}
// B: -weight
sc_mul(temp.bytes, MINUS_ONE.bytes, weight.bytes);
multiexp_data.emplace_back(temp, proof8_B);
// A1: -weight*e
sc_mul(temp.bytes, temp.bytes, pd.e.bytes);
multiexp_data.emplace_back(temp, proof8_A1);
// A: -weight*e*e
rct::key minus_weight_e_squared;
sc_mul(minus_weight_e_squared.bytes, temp.bytes, pd.e.bytes);
multiexp_data.emplace_back(minus_weight_e_squared, proof8_A);
// G: weight*d1
sc_muladd(G_scalar.bytes, weight.bytes, proof.d1.bytes, G_scalar.bytes);
// Windowed vector
// d[j*N+i] = z**(2*(j+1)) * 2**i
rct::keyV d(MN, rct::zero());
d[0] = z_squared;
for (size_t i = 1; i < N; i++)
{
sc_add(d[i].bytes, d[i-1].bytes, d[i-1].bytes);
}
for (size_t j = 1; j < M; j++)
{
for (size_t i = 0; i < N; i++)
{
sc_mul(d[j*N+i].bytes, d[(j-1)*N+i].bytes, z_squared.bytes);
}
}
// More efficient computation of sum(d)
rct::key sum_d;
sc_mul(sum_d.bytes, TWO_SIXTY_FOUR_MINUS_ONE.bytes, sum_of_even_powers(pd.z, 2*M).bytes);
// H: weight*( r1*y*s1 + e**2*( y**(MN+1)*z*sum(d) + (z**2-z)*sum(y) ) )
rct::key sum_y = sum_of_scalar_powers(pd.y, MN);
sc_sub(temp.bytes, z_squared.bytes, pd.z.bytes);
sc_mul(temp.bytes, temp.bytes, sum_y.bytes);
sc_mul(temp2.bytes, y_MN_1.bytes, pd.z.bytes);
sc_mul(temp2.bytes, temp2.bytes, sum_d.bytes);
sc_add(temp.bytes, temp.bytes, temp2.bytes);
sc_mul(temp.bytes, temp.bytes, e_squared.bytes);
sc_mul(temp2.bytes, proof.r1.bytes, pd.y.bytes);
sc_mul(temp2.bytes, temp2.bytes, proof.s1.bytes);
sc_add(temp.bytes, temp.bytes, temp2.bytes);
sc_muladd(H_scalar.bytes, temp.bytes, weight.bytes, H_scalar.bytes);
// Compute the number of rounds for the inner-product argument
const size_t rounds = pd.logM+logN;
CHECK_AND_ASSERT_MES(rounds > 0, false, "Zero rounds");
const rct::key *challenges_inv = &inverses[pd.inv_offset];
const rct::key yinv = inverses[pd.inv_offset + rounds];
// Compute challenge products
challenges_cache.resize(1<<rounds);
challenges_cache[0] = challenges_inv[0];
challenges_cache[1] = pd.challenges[0];
for (size_t j = 1; j < rounds; ++j)
{
const size_t slots = 1<<(j+1);
for (size_t s = slots; s-- > 0; --s)
{
sc_mul(challenges_cache[s].bytes, challenges_cache[s/2].bytes, pd.challenges[j].bytes);
sc_mul(challenges_cache[s-1].bytes, challenges_cache[s/2].bytes, challenges_inv[j].bytes);
}
}
// Gi and Hi
rct::key e_r1_w_y;
sc_mul(e_r1_w_y.bytes, pd.e.bytes, proof.r1.bytes);
sc_mul(e_r1_w_y.bytes, e_r1_w_y.bytes, weight.bytes);
rct::key e_s1_w;
sc_mul(e_s1_w.bytes, pd.e.bytes, proof.s1.bytes);
sc_mul(e_s1_w.bytes, e_s1_w.bytes, weight.bytes);
rct::key e_squared_z_w;
sc_mul(e_squared_z_w.bytes, e_squared.bytes, pd.z.bytes);
sc_mul(e_squared_z_w.bytes, e_squared_z_w.bytes, weight.bytes);
rct::key minus_e_squared_z_w;
sc_sub(minus_e_squared_z_w.bytes, ZERO.bytes, e_squared_z_w.bytes);
rct::key minus_e_squared_w_y;
sc_sub(minus_e_squared_w_y.bytes, ZERO.bytes, e_squared.bytes);
sc_mul(minus_e_squared_w_y.bytes, minus_e_squared_w_y.bytes, weight.bytes);
sc_mul(minus_e_squared_w_y.bytes, minus_e_squared_w_y.bytes, y_MN.bytes);
for (size_t i = 0; i < MN; ++i)
{
rct::key g_scalar = copy(e_r1_w_y);
rct::key h_scalar;
// Use the binary decomposition of the index
sc_muladd(g_scalar.bytes, g_scalar.bytes, challenges_cache[i].bytes, e_squared_z_w.bytes);
sc_muladd(h_scalar.bytes, e_s1_w.bytes, challenges_cache[(~i) & (MN-1)].bytes, minus_e_squared_z_w.bytes);
// Complete the scalar derivation
sc_add(Gi_scalars[i].bytes, Gi_scalars[i].bytes, g_scalar.bytes);
sc_muladd(h_scalar.bytes, minus_e_squared_w_y.bytes, d[i].bytes, h_scalar.bytes);
sc_add(Hi_scalars[i].bytes, Hi_scalars[i].bytes, h_scalar.bytes);
// Update iterated values
sc_mul(e_r1_w_y.bytes, e_r1_w_y.bytes, yinv.bytes);
sc_mul(minus_e_squared_w_y.bytes, minus_e_squared_w_y.bytes, yinv.bytes);
}
// L_j: -weight*e*e*challenges[j]**2
// R_j: -weight*e*e*challenges[j]**(-2)
for (size_t j = 0; j < rounds; ++j)
{
sc_mul(temp.bytes, pd.challenges[j].bytes, pd.challenges[j].bytes);
sc_mul(temp.bytes, temp.bytes, minus_weight_e_squared.bytes);
multiexp_data.emplace_back(temp, proof8_L[j]);
sc_mul(temp.bytes, challenges_inv[j].bytes, challenges_inv[j].bytes);
sc_mul(temp.bytes, temp.bytes, minus_weight_e_squared.bytes);
multiexp_data.emplace_back(temp, proof8_R[j]);
}
}
// Verify all proofs in the weighted batch
multiexp_data.emplace_back(G_scalar, rct::G);
multiexp_data.emplace_back(H_scalar, rct::H);
for (size_t i = 0; i < maxMN; ++i)
{
multiexp_data[i * 2] = {Gi_scalars[i], Gi_p3[i]};
multiexp_data[i * 2 + 1] = {Hi_scalars[i], Hi_p3[i]};
}
if (!(multiexp(multiexp_data, 2 * maxMN) == rct::identity()))
{
MERROR("Verification failure");
return false;
}
return true;
}
bool bulletproof_plus_VERIFY(const std::vector<BulletproofPlus> &proofs)
{
std::vector<const BulletproofPlus*> proof_pointers;
proof_pointers.reserve(proofs.size());
for (const BulletproofPlus &proof: proofs)
proof_pointers.push_back(&proof);
return bulletproof_plus_VERIFY(proof_pointers);
}
bool bulletproof_plus_VERIFY(const BulletproofPlus &proof)
{
std::vector<const BulletproofPlus*> proofs;
proofs.push_back(&proof);
return bulletproof_plus_VERIFY(proofs);
}
}
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