text: // Copyright (c) 2017-2018, 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.
  //
  // Adapted from Java code by Sarang Noether
  
  #include <stdlib.h>
  #include <boost/thread/mutex.hpp>
  #include <boost/thread/lock_guard.hpp>
  #include "misc_log_ex.h"
  #include "span.h"
  #include "common/perf_timer.h"
  #include "cryptonote_config.h"
  extern "C"
  {
  #include "crypto/crypto-ops.h"
  }
  #include "rctOps.h"
  #include "multiexp.h"
  #include "bulletproofs.h"
  
  #undef MONERO_DEFAULT_LOG_CATEGORY
  #define MONERO_DEFAULT_LOG_CATEGORY "bulletproofs"
  
  //#define DEBUG_BP
  
  #if 1
  #define PERF_TIMER_START_BP(x) PERF_TIMER_START_UNIT(x, 1000000)
  #define PERF_TIMER_STOP_BP(x) PERF_TIMER_STOP(x)
  #else
  #define PERF_TIMER_START_BP(x) ((void*)0)
  #define PERF_TIMER_STOP_BP(x) ((void*)0)
  #endif
  
  #define STRAUS_SIZE_LIMIT 232
  #define PIPPENGER_SIZE_LIMIT 0
  
  namespace rct
  {
  
  static rct::key vector_exponent(const rct::keyV &a, const rct::keyV &b);
  static rct::keyV vector_powers(const rct::key &x, size_t n);
  static rct::keyV vector_dup(const rct::key &x, size_t n);
  static rct::key inner_product(const rct::keyV &a, const rct::keyV &b);
  
  static constexpr size_t maxN = 64;
  static constexpr size_t maxM = BULLETPROOF_MAX_OUTPUTS;
  static rct::key Hi[maxN*maxM], Gi[maxN*maxM];
  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;
  static const 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  } };
  static const 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 } };
  static const 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 } };
  static const rct::keyV oneN = vector_dup(rct::identity(), maxN);
  static const rct::keyV twoN = vector_powers(TWO, maxN);
  static const rct::key ip12 = inner_product(oneN, twoN);
  static boost::mutex init_mutex;
  
  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()));
  }
  
  static inline bool is_reduced(const rct::key &scalar)
  {
    return sc_check(scalar.bytes) == 0;
  }
  
  static rct::key get_exponent(const rct::key &base, size_t idx)
  {
    static const std::string salt("bulletproof");
    std::string hashed = std::string((const char*)base.bytes, sizeof(base)) + salt + tools::get_varint_data(idx);
    const rct::key e = rct::hashToPoint(rct::hash2rct(crypto::cn_fast_hash(hashed.data(), hashed.size())));
    CHECK_AND_ASSERT_THROW_MES(!(e == rct::identity()), "Exponent is point at infinity");
    return e;
  }
  
  static void init_exponents()
  {
    boost::lock_guard<boost::mutex> lock(init_mutex);
  
    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[i] = get_exponent(rct::H, i * 2);
      CHECK_AND_ASSERT_THROW_MES(ge_frombytes_vartime(&Hi_p3[i], Hi[i].bytes) == 0, "ge_frombytes_vartime failed");
      Gi[i] = get_exponent(rct::H, i * 2 + 1);
      CHECK_AND_ASSERT_THROW_MES(ge_frombytes_vartime(&Gi_p3[i], Gi[i].bytes) == 0, "ge_frombytes_vartime failed");
  
      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);
  
    MINFO("Hi/Gi cache size: " << (sizeof(Hi)+sizeof(Gi))/1024 << " kB");
    MINFO("Hi_p3/Gi_p3 cache size: " << (sizeof(Hi_p3)+sizeof(Gi_p3))/1024 << " kB");
    MINFO("Straus cache size: " << straus_get_cache_size(straus_HiGi_cache)/1024 << " kB");
    MINFO("Pippenger cache size: " << pippenger_get_cache_size(pippenger_HiGi_cache)/1024 << " kB");
    size_t cache_size = (sizeof(Hi)+sizeof(Hi_p3))*2 + straus_get_cache_size(straus_HiGi_cache) + pippenger_get_cache_size(pippenger_HiGi_cache);
    MINFO("Total cache size: " << cache_size/1024 << "kB");
    init_done = true;
  }
  
  /* Given two scalar arrays, construct a vector commitment */
  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());
  }
  
  /* Compute a custom vector-scalar commitment */
  static rct::key cross_vector_exponent8(size_t size, const std::vector<ge_p3> &A, size_t Ao, const std::vector<ge_p3> &B, size_t Bo, const rct::keyV &a, size_t ao, const rct::keyV &b, size_t bo, const rct::keyV *scale, const ge_p3 *extra_point, const rct::key *extra_scalar)
  {
    CHECK_AND_ASSERT_THROW_MES(size + Ao <= A.size(), "Incompatible size for A");
    CHECK_AND_ASSERT_THROW_MES(size + Bo <= B.size(), "Incompatible size for B");
    CHECK_AND_ASSERT_THROW_MES(size + ao <= a.size(), "Incompatible size for a");
    CHECK_AND_ASSERT_THROW_MES(size + bo <= b.size(), "Incompatible size for b");
    CHECK_AND_ASSERT_THROW_MES(size <= maxN*maxM, "size is too large");
    CHECK_AND_ASSERT_THROW_MES(!scale || size == scale->size() / 2, "Incompatible size for scale");
    CHECK_AND_ASSERT_THROW_MES(!!extra_point == !!extra_scalar, "only one of extra point/scalar present");
  
    std::vector<MultiexpData> multiexp_data;
    multiexp_data.resize(size*2 + (!!extra_point));
    for (size_t i = 0; i < size; ++i)
    {
      sc_mul(multiexp_data[i*2].scalar.bytes, a[ao+i].bytes, INV_EIGHT.bytes);;
      multiexp_data[i*2].point = A[Ao+i];
      sc_mul(multiexp_data[i*2+1].scalar.bytes, b[bo+i].bytes, INV_EIGHT.bytes);
      if (scale)
        sc_mul(multiexp_data[i*2+1].scalar.bytes, multiexp_data[i*2+1].scalar.bytes, (*scale)[Bo+i].bytes);
      multiexp_data[i*2+1].point = B[Bo+i];
    }
    if (extra_point)
    {
      sc_mul(multiexp_data.back().scalar.bytes, extra_scalar->bytes, INV_EIGHT.bytes);
      multiexp_data.back().point = *extra_point;
    }
    return multiexp(multiexp_data, 0);
  }
  
  /* Given a scalar, construct a vector of powers */
  static rct::keyV vector_powers(const rct::key &x, size_t n)
  {
    rct::keyV res(n);
    if (n == 0)
      return res;
    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, return the sum of its powers from 0 to n-1 */
  static rct::key vector_power_sum(const rct::key &x, size_t n)
  {
    if (n == 0)
      return rct::zero();
    rct::key res = rct::identity();
    if (n == 1)
      return res;
    rct::key prev = x;
    for (size_t i = 1; i < n; ++i)
    {
      if (i > 1)
        sc_mul(prev.bytes, prev.bytes, x.bytes);
      sc_add(res.bytes, res.bytes, prev.bytes);
    }
    return res;
  }
  
  /* Given two scalar arrays, construct the inner product */
  static rct::key inner_product(const epee::span<const rct::key> &a, const epee::span<const rct::key> &b)
  {
    CHECK_AND_ASSERT_THROW_MES(a.size() == b.size(), "Incompatible sizes of a and b");
    rct::key res = rct::zero();
    for (size_t i = 0; i < a.size(); ++i)
    {
      sc_muladd(res.bytes, a[i].bytes, b[i].bytes, res.bytes);
    }
    return res;
  }
  
  static rct::key inner_product(const rct::keyV &a, const rct::keyV &b)
  {
    return inner_product(epee::span<const rct::key>(a.data(), a.size()), epee::span<const rct::key>(b.data(), b.size()));
  }
  
  /* Given two scalar arrays, construct the Hadamard product */
  static rct::keyV hadamard(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_mul(res[i].bytes, a[i].bytes, b[i].bytes);
    }
    return res;
  }
  
  /* folds a curvepoint array using a two way scaled Hadamard product */
  static void hadamard_fold(std::vector<ge_p3> &v, const rct::keyV *scale, 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]);
      rct::key sa, sb;
      if (scale) sc_mul(sa.bytes, a.bytes, (*scale)[n].bytes); else sa = a;
      if (scale) sc_mul(sb.bytes, b.bytes, (*scale)[sz + n].bytes); else sb = b;
      ge_double_scalarmult_precomp_vartime2_p3(&v[n], sa.bytes, c[0], sb.bytes, c[1]);
    }
    v.resize(sz);
  }
  
  /* Add two vectors */
  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 and 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;
  }
  
  static rct::keyV vector_scalar(const rct::keyV &a, const rct::key &x)
  {
    return vector_scalar(epee::span<const rct::key>(a.data(), a.size()), x);
  }
  
  /* Create a vector from copies of a single value */
  static rct::keyV vector_dup(const rct::key &x, size_t N)
  {
    return rct::keyV(N, x);
  }
  
  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 scalar, the clever way */
  static rct::key invert(const rct::key &x)
  {
    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);
  
  #ifdef DEBUG_BP
    rct::key tmp;
    sc_mul(tmp.bytes, inv.bytes, x.bytes);
    CHECK_AND_ASSERT_THROW_MES(tmp == rct::identity(), "invert failed");
  #endif
    return inv;
  }
  
  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)
    {
      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);
  }
  
  static rct::key hash_cache_mash(rct::key &hash_cache, const rct::key &mash0, const rct::key &mash1)
  {
    rct::keyV data;
    data.reserve(3);
    data.push_back(hash_cache);
    data.push_back(mash0);
    data.push_back(mash1);
    return hash_cache = rct::hash_to_scalar(data);
  }
  
  static rct::key hash_cache_mash(rct::key &hash_cache, const rct::key &mash0, const rct::key &mash1, const rct::key &mash2)
  {
    rct::keyV data;
    data.reserve(4);
    data.push_back(hash_cache);
    data.push_back(mash0);
    data.push_back(mash1);
    data.push_back(mash2);
    return hash_cache = rct::hash_to_scalar(data);
  }
  
  static rct::key hash_cache_mash(rct::key &hash_cache, const rct::key &mash0, const rct::key &mash1, const rct::key &mash2, const rct::key &mash3)
  {
    rct::keyV data;
    data.reserve(5);
    data.push_back(hash_cache);
    data.push_back(mash0);
    data.push_back(mash1);
    data.push_back(mash2);
    data.push_back(mash3);
    return hash_cache = rct::hash_to_scalar(data);
  }
  
  /* Given a value v (0..2^N-1) and a mask gamma, construct a range proof */
  Bulletproof bulletproof_PROVE(const rct::key &sv, const rct::key &gamma)
  {
    return bulletproof_PROVE(rct::keyV(1, sv), rct::keyV(1, gamma));
  }
  
  Bulletproof bulletproof_PROVE(uint64_t v, const rct::key &gamma)
  {
    return bulletproof_PROVE(std::vector<uint64_t>(1, v), rct::keyV(1, gamma));
  }
  
  /* Given a set of values v (0..2^N-1) and masks gamma, construct a range proof */
  Bulletproof bulletproof_PROVE(const rct::keyV &sv, const rct::keyV &gamma)
  {
    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();
  
    PERF_TIMER_UNIT(PROVE, 1000000);
  
    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 tmp, tmp2;
  
    PERF_TIMER_START_BP(PROVE_v);
    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);
    }
    PERF_TIMER_STOP_BP(PROVE_v);
  
    PERF_TIMER_START_BP(PROVE_aLaR);
    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;
        }
      }
    }
    PERF_TIMER_STOP_BP(PROVE_aLaR);
  
    // DEBUG: Test to ensure this recovers the value
  #ifdef DEBUG_BP
    for (size_t j = 0; j < M; ++j)
    {
      uint64_t test_aL = 0, test_aR = 0;
      for (size_t i = 0; i < N; ++i)
      {
        if (aL[j*N+i] == rct::identity())
          test_aL += ((uint64_t)1)<<i;
        if (aR[j*N+i] == rct::zero())
          test_aR += ((uint64_t)1)<<i;
      }
      uint64_t v_test = 0;
      if (j < sv.size())
        for (int n = 0; n < 8; ++n) v_test |= (((uint64_t)sv[j][n]) << (8*n));
      CHECK_AND_ASSERT_THROW_MES(test_aL == v_test, "test_aL failed");
      CHECK_AND_ASSERT_THROW_MES(test_aR == v_test, "test_aR failed");
    }
  #endif
  
  try_again:
    rct::key hash_cache = rct::hash_to_scalar(V);
  
    PERF_TIMER_START_BP(PROVE_step1);
    // PAPER LINES 38-39
    rct::key alpha = rct::skGen();
    rct::key ve = vector_exponent(aL8, aR8);
    rct::key A;
    sc_mul(tmp.bytes, alpha.bytes, INV_EIGHT.bytes);
    rct::addKeys(A, ve, rct::scalarmultBase(tmp));
  
    // PAPER LINES 40-42
    rct::keyV sL = rct::skvGen(MN), sR = rct::skvGen(MN);
    rct::key rho = rct::skGen();
    ve = vector_exponent(sL, sR);
    rct::key S;
    rct::addKeys(S, ve, rct::scalarmultBase(rho));
    S = rct::scalarmultKey(S, INV_EIGHT);
  
    // PAPER LINES 43-45
    rct::key y = hash_cache_mash(hash_cache, A, S);
    if (y == rct::zero())
    {
      PERF_TIMER_STOP_BP(PROVE_step1);
      MINFO("y is 0, trying again");
      goto try_again;
    }
    rct::key z = hash_cache = rct::hash_to_scalar(y);
    if (z == rct::zero())
    {
      PERF_TIMER_STOP_BP(PROVE_step1);
      MINFO("z is 0, trying again");
      goto try_again;
    }
  
    // Polynomial construction by coefficients
    rct::keyV l0 = vector_subtract(aL, z);
    const rct::keyV &l1 = sL;
  
    // This computes the ugly sum/concatenation from PAPER LINE 65
    rct::keyV zero_twos(MN);
    const rct::keyV zpow = vector_powers(z, M+2);
    for (size_t i = 0; i < MN; ++i)
    {
      zero_twos[i] = rct::zero();
      for (size_t j = 1; j <= M; ++j)
      {
        if (i >= (j-1)*N && i < j*N)
        {
          CHECK_AND_ASSERT_THROW_MES(1+j < zpow.size(), "invalid zpow index");
          CHECK_AND_ASSERT_THROW_MES(i-(j-1)*N < twoN.size(), "invalid twoN index");
          sc_muladd(zero_twos[i].bytes, zpow[1+j].bytes, twoN[i-(j-1)*N].bytes, zero_twos[i].bytes);
        }
      }
    }
  
    rct::keyV r0 = vector_add(aR, z);
    const auto yMN = vector_powers(y, MN);
    r0 = hadamard(r0, yMN);
    r0 = vector_add(r0, zero_twos);
    rct::keyV r1 = hadamard(yMN, sR);
  
    // Polynomial construction before PAPER LINE 46
    rct::key t1_1 = inner_product(l0, r1);
    rct::key t1_2 = inner_product(l1, r0);
    rct::key t1;
    sc_add(t1.bytes, t1_1.bytes, t1_2.bytes);
    rct::key t2 = inner_product(l1, r1);
  
    PERF_TIMER_STOP_BP(PROVE_step1);
  
    PERF_TIMER_START_BP(PROVE_step2);
    // PAPER LINES 47-48
    rct::key tau1 = rct::skGen(), tau2 = rct::skGen();
  
    rct::key T1, T2;
    ge_p3 p3;
    sc_mul(tmp.bytes, t1.bytes, INV_EIGHT.bytes);
    sc_mul(tmp2.bytes, tau1.bytes, INV_EIGHT.bytes);
    ge_double_scalarmult_base_vartime_p3(&p3, tmp.bytes, &ge_p3_H, tmp2.bytes);
    ge_p3_tobytes(T1.bytes, &p3);
    sc_mul(tmp.bytes, t2.bytes, INV_EIGHT.bytes);
    sc_mul(tmp2.bytes, tau2.bytes, INV_EIGHT.bytes);
    ge_double_scalarmult_base_vartime_p3(&p3, tmp.bytes, &ge_p3_H, tmp2.bytes);
    ge_p3_tobytes(T2.bytes, &p3);
  
    // PAPER LINES 49-51
    rct::key x = hash_cache_mash(hash_cache, z, T1, T2);
    if (x == rct::zero())
    {
      PERF_TIMER_STOP_BP(PROVE_step2);
      MINFO("x is 0, trying again");
      goto try_again;
    }
  
    // PAPER LINES 52-53
    rct::key taux;
    sc_mul(taux.bytes, tau1.bytes, x.bytes);
    rct::key xsq;
    sc_mul(xsq.bytes, x.bytes, x.bytes);
    sc_muladd(taux.bytes, tau2.bytes, xsq.bytes, taux.bytes);
    for (size_t j = 1; j <= sv.size(); ++j)
    {
      CHECK_AND_ASSERT_THROW_MES(j+1 < zpow.size(), "invalid zpow index");
      sc_muladd(taux.bytes, zpow[j+1].bytes, gamma[j-1].bytes, taux.bytes);
    }
    rct::key mu;
    sc_muladd(mu.bytes, x.bytes, rho.bytes, alpha.bytes);
  
    // PAPER LINES 54-57
    rct::keyV l = l0;
    l = vector_add(l, vector_scalar(l1, x));
    rct::keyV r = r0;
    r = vector_add(r, vector_scalar(r1, x));
    PERF_TIMER_STOP_BP(PROVE_step2);
  
    PERF_TIMER_START_BP(PROVE_step3);
    rct::key t = inner_product(l, r);
  
    // DEBUG: Test if the l and r vectors match the polynomial forms
  #ifdef DEBUG_BP
    rct::key test_t;
    const rct::key t0 = inner_product(l0, r0);
    sc_muladd(test_t.bytes, t1.bytes, x.bytes, t0.bytes);
    sc_muladd(test_t.bytes, t2.bytes, xsq.bytes, test_t.bytes);
    CHECK_AND_ASSERT_THROW_MES(test_t == t, "test_t check failed");
  #endif
  
    // PAPER LINES 32-33
    rct::key x_ip = hash_cache_mash(hash_cache, x, taux, mu, t);
    if (x_ip == rct::zero())
    {
      PERF_TIMER_STOP_BP(PROVE_step3);
      MINFO("x_ip is 0, trying again");
      goto try_again;
    }
  
    // 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] = rct::identity();
    yinvpow[1] = yinv;
    for (size_t i = 0; i < MN; ++i)
    {
      Gprime[i] = Gi_p3[i];
      Hprime[i] = Hi_p3[i];
      if (i > 1)
        sc_mul(yinvpow[i].bytes, yinvpow[i-1].bytes, yinv.bytes);
      aprime[i] = l[i];
      bprime[i] = r[i];
    }
    rct::keyV L(logMN);
    rct::keyV R(logMN);
    int round = 0;
    rct::keyV w(logMN); // this is the challenge x in the inner product protocol
    PERF_TIMER_STOP_BP(PROVE_step3);
  
    PERF_TIMER_START_BP(PROVE_step4);
    // PAPER LINE 13
    const rct::keyV *scale = &yinvpow;
    while (nprime > 1)
    {
      // PAPER LINE 15
      nprime /= 2;
  
      // PAPER LINES 16-17
      PERF_TIMER_START_BP(PROVE_inner_product);
      rct::key cL = inner_product(slice(aprime, 0, nprime), slice(bprime, nprime, bprime.size()));
      rct::key cR = inner_product(slice(aprime, nprime, aprime.size()), slice(bprime, 0, nprime));
      PERF_TIMER_STOP_BP(PROVE_inner_product);
  
      // PAPER LINES 18-19
      PERF_TIMER_START_BP(PROVE_LR);
      sc_mul(tmp.bytes, cL.bytes, x_ip.bytes);
      L[round] = cross_vector_exponent8(nprime, Gprime, nprime, Hprime, 0, aprime, 0, bprime, nprime, scale, &ge_p3_H, &tmp);
      sc_mul(tmp.bytes, cR.bytes, x_ip.bytes);
      R[round] = cross_vector_exponent8(nprime, Gprime, 0, Hprime, nprime, aprime, nprime, bprime, 0, scale, &ge_p3_H, &tmp);
      PERF_TIMER_STOP_BP(PROVE_LR);
  
      // PAPER LINES 21-22
      w[round] = hash_cache_mash(hash_cache, L[round], R[round]);
      if (w[round] == rct::zero())
      {
        PERF_TIMER_STOP_BP(PROVE_step4);
        MINFO("w[round] is 0, trying again");
        goto try_again;
      }
  
      // PAPER LINES 24-25
      const rct::key winv = invert(w[round]);
      if (nprime > 1)
      {
        PERF_TIMER_START_BP(PROVE_hadamard2);
        hadamard_fold(Gprime, NULL, winv, w[round]);
        hadamard_fold(Hprime, scale, w[round], winv);
        PERF_TIMER_STOP_BP(PROVE_hadamard2);
      }
  
      // PAPER LINES 28-29
      PERF_TIMER_START_BP(PROVE_prime);
      aprime = vector_add(vector_scalar(slice(aprime, 0, nprime), w[round]), vector_scalar(slice(aprime, nprime, aprime.size()), winv));
      bprime = vector_add(vector_scalar(slice(bprime, 0, nprime), winv), vector_scalar(slice(bprime, nprime, bprime.size()), w[round]));
      PERF_TIMER_STOP_BP(PROVE_prime);
  
      scale = NULL;
      ++round;
    }
    PERF_TIMER_STOP_BP(PROVE_step4);
  
    // PAPER LINE 58 (with inclusions from PAPER LINE 8 and PAPER LINE 20)
    return Bulletproof(std::move(V), A, S, T1, T2, taux, mu, std::move(L), std::move(R), aprime[0], bprime[0], t);
  }
  
  Bulletproof bulletproof_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
    PERF_TIMER_START_BP(PROVE_v);
    rct::keyV sv(v.size());
    for (size_t i = 0; i < v.size(); ++i)
    {
      sv[i] = rct::zero();
      sv[i].bytes[0] = v[i] & 255;
      sv[i].bytes[1] = (v[i] >> 8) & 255;
      sv[i].bytes[2] = (v[i] >> 16) & 255;
      sv[i].bytes[3] = (v[i] >> 24) & 255;
      sv[i].bytes[4] = (v[i] >> 32) & 255;
      sv[i].bytes[5] = (v[i] >> 40) & 255;
      sv[i].bytes[6] = (v[i] >> 48) & 255;
      sv[i].bytes[7] = (v[i] >> 56) & 255;
    }
    PERF_TIMER_STOP_BP(PROVE_v);
    return bulletproof_PROVE(sv, gamma);
  }
  
  struct proof_data_t
  {
    rct::key x, y, z, x_ip;
    std::vector<rct::key> w;
    size_t logM, inv_offset;
  };
  
  /* Given a range proof, determine if it is valid */
  bool bulletproof_VERIFY(const std::vector<const Bulletproof*> &proofs)
  {
    init_exponents();
  
    PERF_TIMER_START_BP(VERIFY);
  
    const size_t logN = 6;
    const size_t N = 1 << logN;
  
    // sanity and figure out which proof is longest
    size_t max_length = 0;
    size_t nV = 0;
    std::vector<proof_data_t> proof_data;
    proof_data.reserve(proofs.size());
    size_t inv_offset = 0;
    std::vector<rct::key> to_invert;
    to_invert.reserve(11 * sizeof(proofs));
    for (const Bulletproof *p: proofs)
    {
      const Bulletproof &proof = *p;
  
      // check scalar range
      CHECK_AND_ASSERT_MES(is_reduced(proof.taux), false, "Input scalar not in range");
      CHECK_AND_ASSERT_MES(is_reduced(proof.mu), false, "Input scalar not in range");
      CHECK_AND_ASSERT_MES(is_reduced(proof.a), false, "Input scalar not in range");
      CHECK_AND_ASSERT_MES(is_reduced(proof.b), false, "Input scalar not in range");
      CHECK_AND_ASSERT_MES(is_reduced(proof.t), 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();
  
      // Reconstruct the challenges
      PERF_TIMER_START_BP(VERIFY_start);
      proof_data.resize(proof_data.size() + 1);
      proof_data_t &pd = proof_data.back();
      rct::key hash_cache = rct::hash_to_scalar(proof.V);
      pd.y = hash_cache_mash(hash_cache, proof.A, proof.S);
      CHECK_AND_ASSERT_MES(!(pd.y == rct::zero()), false, "y == 0");
      pd.z = hash_cache = rct::hash_to_scalar(pd.y);
      CHECK_AND_ASSERT_MES(!(pd.z == rct::zero()), false, "z == 0");
      pd.x = hash_cache_mash(hash_cache, pd.z, proof.T1, proof.T2);
      CHECK_AND_ASSERT_MES(!(pd.x == rct::zero()), false, "x == 0");
      pd.x_ip = hash_cache_mash(hash_cache, pd.x, proof.taux, proof.mu, proof.t);
      CHECK_AND_ASSERT_MES(!(pd.x_ip == rct::zero()), false, "x_ip == 0");
      PERF_TIMER_STOP_BP(VERIFY_start);
  
      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");
  
      const size_t rounds = pd.logM+logN;
      CHECK_AND_ASSERT_MES(rounds > 0, false, "Zero rounds");
  
      PERF_TIMER_START_BP(VERIFY_line_21_22);
      // PAPER LINES 21-22
      // The inner product challenges are computed per round
      pd.w.resize(rounds);
      for (size_t i = 0; i < rounds; ++i)
      {
        pd.w[i] = hash_cache_mash(hash_cache, proof.L[i], proof.R[i]);
        CHECK_AND_ASSERT_MES(!(pd.w[i] == rct::zero()), false, "w[i] == 0");
      }
      PERF_TIMER_STOP_BP(VERIFY_line_21_22);
  
      pd.inv_offset = inv_offset;
      for (size_t i = 0; i < rounds; ++i)
        to_invert.push_back(pd.w[i]);
      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 tmp;
  
    std::vector<MultiexpData> multiexp_data;
    multiexp_data.reserve(nV + (2 * (10/*logM*/ + logN) + 4) * proofs.size() + 2 * maxMN);
    multiexp_data.resize(2 * maxMN);
  
    PERF_TIMER_START_BP(VERIFY_line_24_25_invert);
    const std::vector<rct::key> inverses = invert(to_invert);
    PERF_TIMER_STOP_BP(VERIFY_line_24_25_invert);
  
    // setup weighted aggregates
    rct::key z1 = rct::zero();
    rct::key z3 = rct::zero();
    rct::keyV m_z4(maxMN, rct::zero()), m_z5(maxMN, rct::zero());
    rct::key m_y0 = rct::zero(), y1 = rct::zero();
    int proof_data_index = 0;
    for (const Bulletproof *p: proofs)
    {
      const Bulletproof &proof = *p;
      const 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;
      const rct::key weight_y = rct::skGen();
      const rct::key weight_z = rct::skGen();
  
      // pre-multiply some points by 8
      rct::keyV proof8_V = proof.V; for (rct::key &k: proof8_V) k = rct::scalarmult8(k);
      rct::keyV proof8_L = proof.L; for (rct::key &k: proof8_L) k = rct::scalarmult8(k);
      rct::keyV proof8_R = proof.R; for (rct::key &k: proof8_R) k = rct::scalarmult8(k);
      rct::key proof8_T1 = rct::scalarmult8(proof.T1);
      rct::key proof8_T2 = rct::scalarmult8(proof.T2);
      rct::key proof8_S = rct::scalarmult8(proof.S);
      rct::key proof8_A = rct::scalarmult8(proof.A);
  
      PERF_TIMER_START_BP(VERIFY_line_61);
      // PAPER LINE 61
      sc_mulsub(m_y0.bytes, proof.taux.bytes, weight_y.bytes, m_y0.bytes);
  
      const rct::keyV zpow = vector_powers(pd.z, M+3);
  
      rct::key k;
      const rct::key ip1y = vector_power_sum(pd.y, MN);
      sc_mulsub(k.bytes, zpow[2].bytes, ip1y.bytes, rct::zero().bytes);
      for (size_t j = 1; j <= M; ++j)
      {
        CHECK_AND_ASSERT_MES(j+2 < zpow.size(), false, "invalid zpow index");
        sc_mulsub(k.bytes, zpow[j+2].bytes, ip12.bytes, k.bytes);
      }
      PERF_TIMER_STOP_BP(VERIFY_line_61);
  
      PERF_TIMER_START_BP(VERIFY_line_61rl_new);
      sc_muladd(tmp.bytes, pd.z.bytes, ip1y.bytes, k.bytes);
      sc_sub(tmp.bytes, proof.t.bytes, tmp.bytes);
      sc_muladd(y1.bytes, tmp.bytes, weight_y.bytes, y1.bytes);
      for (size_t j = 0; j < proof8_V.size(); j++)
      {
        sc_mul(tmp.bytes, zpow[j+2].bytes, weight_y.bytes);
        multiexp_data.emplace_back(tmp, proof8_V[j]);
      }
      sc_mul(tmp.bytes, pd.x.bytes, weight_y.bytes);
      multiexp_data.emplace_back(tmp, proof8_T1);
      rct::key xsq;
      sc_mul(xsq.bytes, pd.x.bytes, pd.x.bytes);
      sc_mul(tmp.bytes, xsq.bytes, weight_y.bytes);
      multiexp_data.emplace_back(tmp, proof8_T2);
      PERF_TIMER_STOP_BP(VERIFY_line_61rl_new);
  
      PERF_TIMER_START_BP(VERIFY_line_62);
      // PAPER LINE 62
      multiexp_data.emplace_back(weight_z, proof8_A);
      sc_mul(tmp.bytes, pd.x.bytes, weight_z.bytes);
      multiexp_data.emplace_back(tmp, proof8_S);
      PERF_TIMER_STOP_BP(VERIFY_line_62);
  
      // Compute the number of rounds for the inner product
      const size_t rounds = pd.logM+logN;
      CHECK_AND_ASSERT_MES(rounds > 0, false, "Zero rounds");
  
      PERF_TIMER_START_BP(VERIFY_line_24_25);
      // Basically PAPER LINES 24-25
      // Compute the curvepoints from G[i] and H[i]
      rct::key yinvpow = rct::identity();
      rct::key ypow = rct::identity();
  
      const rct::key *winv = &inverses[pd.inv_offset];
      const rct::key yinv = inverses[pd.inv_offset + rounds];
  
      // precalc
      PERF_TIMER_START_BP(VERIFY_line_24_25_precalc);
      rct::keyV w_cache(1<<rounds);
      w_cache[0] = winv[0];
      w_cache[1] = pd.w[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(w_cache[s].bytes, w_cache[s/2].bytes, pd.w[j].bytes);
          sc_mul(w_cache[s-1].bytes, w_cache[s/2].bytes, winv[j].bytes);
        }
      }
      PERF_TIMER_STOP_BP(VERIFY_line_24_25_precalc);
  
      for (size_t i = 0; i < MN; ++i)
      {
        rct::key g_scalar = proof.a;
        rct::key h_scalar;
        if (i == 0)
          h_scalar = proof.b;
        else
          sc_mul(h_scalar.bytes, proof.b.bytes, yinvpow.bytes);
  
        // Convert the index to binary IN REVERSE and construct the scalar exponent
        sc_mul(g_scalar.bytes, g_scalar.bytes, w_cache[i].bytes);
        sc_mul(h_scalar.bytes, h_scalar.bytes, w_cache[(~i) & (MN-1)].bytes);
  
        // Adjust the scalars using the exponents from PAPER LINE 62
        sc_add(g_scalar.bytes, g_scalar.bytes, pd.z.bytes);
        CHECK_AND_ASSERT_MES(2+i/N < zpow.size(), false, "invalid zpow index");
        CHECK_AND_ASSERT_MES(i%N < twoN.size(), false, "invalid twoN index");
        sc_mul(tmp.bytes, zpow[2+i/N].bytes, twoN[i%N].bytes);
        if (i == 0)
        {
          sc_add(tmp.bytes, tmp.bytes, pd.z.bytes);
          sc_sub(h_scalar.bytes, h_scalar.bytes, tmp.bytes);
        }
        else
        {
          sc_muladd(tmp.bytes, pd.z.bytes, ypow.bytes, tmp.bytes);
          sc_mulsub(h_scalar.bytes, tmp.bytes, yinvpow.bytes, h_scalar.bytes);
        }
  
        sc_mulsub(m_z4[i].bytes, g_scalar.bytes, weight_z.bytes, m_z4[i].bytes);
        sc_mulsub(m_z5[i].bytes, h_scalar.bytes, weight_z.bytes, m_z5[i].bytes);
  
        if (i == 0)
        {
          yinvpow = yinv;
          ypow = pd.y;
        }
        else if (i != MN-1)
        {
          sc_mul(yinvpow.bytes, yinvpow.bytes, yinv.bytes);
          sc_mul(ypow.bytes, ypow.bytes, pd.y.bytes);
        }
      }
  
      PERF_TIMER_STOP_BP(VERIFY_line_24_25);
  
      // PAPER LINE 26
      PERF_TIMER_START_BP(VERIFY_line_26_new);
      sc_muladd(z1.bytes, proof.mu.bytes, weight_z.bytes, z1.bytes);
      for (size_t i = 0; i < rounds; ++i)
      {
        sc_mul(tmp.bytes, pd.w[i].bytes, pd.w[i].bytes);
        sc_mul(tmp.bytes, tmp.bytes, weight_z.bytes);
        multiexp_data.emplace_back(tmp, proof8_L[i]);
        sc_mul(tmp.bytes, winv[i].bytes, winv[i].bytes);
        sc_mul(tmp.bytes, tmp.bytes, weight_z.bytes);
        multiexp_data.emplace_back(tmp, proof8_R[i]);
      }
      sc_mulsub(tmp.bytes, proof.a.bytes, proof.b.bytes, proof.t.bytes);
      sc_mul(tmp.bytes, tmp.bytes, pd.x_ip.bytes);
      sc_muladd(z3.bytes, tmp.bytes, weight_z.bytes, z3.bytes);
      PERF_TIMER_STOP_BP(VERIFY_line_26_new);
    }
  
    // now check all proofs at once
    PERF_TIMER_START_BP(VERIFY_step2_check);
    sc_sub(tmp.bytes, m_y0.bytes, z1.bytes);
    multiexp_data.emplace_back(tmp, rct::G);
    sc_sub(tmp.bytes, z3.bytes, y1.bytes);
    multiexp_data.emplace_back(tmp, rct::H);
    for (size_t i = 0; i < maxMN; ++i)
    {
      multiexp_data[i * 2] = {m_z4[i], Gi_p3[i]};
      multiexp_data[i * 2 + 1] = {m_z5[i], Hi_p3[i]};
    }
    if (!(multiexp(multiexp_data, 2 * maxMN) == rct::identity()))
    {
      PERF_TIMER_STOP_BP(VERIFY_step2_check);
      MERROR("Verification failure");
      return false;
    }
    PERF_TIMER_STOP_BP(VERIFY_step2_check);
  
    PERF_TIMER_STOP_BP(VERIFY);
    return true;
  }
  
  bool bulletproof_VERIFY(const std::vector<Bulletproof> &proofs)
  {
    std::vector<const Bulletproof*> proof_pointers;
    proof_pointers.reserve(proofs.size());
    for (const Bulletproof &proof: proofs)
      proof_pointers.push_back(&proof);
    return bulletproof_VERIFY(proof_pointers);
  }
  
  bool bulletproof_VERIFY(const Bulletproof &proof)
  {
    std::vector<const Bulletproof*> proofs;
    proofs.push_back(&proof);
    return bulletproof_VERIFY(proofs);
  }
  
  }
  