add bulletproofs to the build, with basic unit tests

Based on Java code from Sarang Noether

1 files changed, 760 insertions, 0 deletions

diff --git a/src/ringct/bulletproofs.cc b/src/ringct/bulletproofs.cc
new file mode 100644
index 000000000..7b4ac12a7
--- /dev/null
+++ b/src/ringct/bulletproofs.cc
@@ -0,0 +1,760 @@
+// Copyright (c) 2017, 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 <openssl/ssl.h>
+#include <boost/thread/mutex.hpp>
+#include "misc_log_ex.h"
+#include "common/perf_timer.h"
+extern "C"
+{
+#include "crypto/crypto-ops.h"
+}
+#include "rctOps.h"
+#include "bulletproofs.h"
+
+#undef MONERO_DEFAULT_LOG_CATEGORY
+#define MONERO_DEFAULT_LOG_CATEGORY "bulletproofs"
+
+//#define DEBUG_BP
+
+#define PERF_TIMER_START_BP(x) PERF_TIMER_START_UNIT(x, 1000000)
+
+namespace rct
+{
+
+static rct::key vector_exponent(const rct::keyV &a, const rct::keyV &b);
+static rct::keyV vector_powers(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 rct::key Hi[maxN], Gi[maxN];
+static ge_dsmp Gprecomp[64], Hprecomp[64];
+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::keyV oneN = vector_powers(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 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);
+  return rct::hashToPoint(rct::hash2rct(crypto::cn_fast_hash(hashed.data(), hashed.size())));
+}
+
+static void init_exponents()
+{
+  boost::lock_guard<boost::mutex> lock(init_mutex);
+
+  static bool init_done = false;
+  if (init_done)
+    return;
+  for (size_t i = 0; i < maxN; ++i)
+  {
+    Hi[i] = get_exponent(rct::H, i * 2);
+    rct::precomp(Hprecomp[i], Hi[i]);
+    Gi[i] = get_exponent(rct::H, i * 2 + 1);
+    rct::precomp(Gprecomp[i], Gi[i]);
+  }
+  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, "Incompatible sizes of a and maxN");
+  rct::key res = rct::identity();
+  for (size_t i = 0; i < a.size(); ++i)
+  {
+    rct::key term;
+    rct::addKeys3(term, a[i], Gprecomp[i], b[i], Hprecomp[i]);
+    rct::addKeys(res, res, term);
+  }
+  return res;
+}
+
+/* Compute a custom vector-scalar commitment */
+static rct::key vector_exponent_custom(const rct::keyV &A, const rct::keyV &B, 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() == b.size(), "Incompatible sizes of a and b");
+  CHECK_AND_ASSERT_THROW_MES(a.size() == A.size(), "Incompatible sizes of a and A");
+  CHECK_AND_ASSERT_THROW_MES(a.size() <= maxN, "Incompatible sizes of a and maxN");
+  rct::key res = rct::identity();
+  for (size_t i = 0; i < a.size(); ++i)
+  {
+    rct::key term;
+#if 0
+    // we happen to know where A and B might fall, so don't bother checking the rest
+    ge_dsmp *Acache = NULL, *Bcache = NULL;
+    ge_dsmp Acache_custom[1], Bcache_custom[1];
+    if (Gi[i] == A[i])
+      Acache = Gprecomp + i;
+    else if (i<32 && Gi[i+32] == A[i])
+      Acache = Gprecomp + i + 32;
+    else
+    {
+      rct::precomp(Acache_custom[0], A[i]);
+      Acache = Acache_custom;
+    }
+    if (i == 0 && B[i] == Hi[0])
+      Bcache = Hprecomp;
+    else
+    {
+      rct::precomp(Bcache_custom[0], B[i]);
+      Bcache = Bcache_custom;
+    }
+    rct::addKeys3(term, a[i], *Acache, b[i], *Bcache);
+#else
+    ge_dsmp Acache, Bcache;
+    rct::precomp(Bcache, B[i]);
+    rct::addKeys3(term, a[i], A[i], b[i], Bcache);
+#endif
+    rct::addKeys(res, res, term);
+  }
+  return res;
+}
+
+/* Given a scalar, construct a vector of powers */
+static rct::keyV vector_powers(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 two scalar arrays, construct the inner product */
+static rct::key inner_product(const rct::keyV &a, const rct::keyV &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;
+}
+
+/* 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;
+}
+
+/* Given two curvepoint arrays, construct the Hadamard product */
+static rct::keyV hadamard2(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)
+  {
+    rct::addKeys(res[i], a[i], b[i]);
+  }
+  return res;
+}
+
+/* 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;
+}
+
+/* Subtract two vectors */
+static rct::keyV vector_subtract(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_sub(res[i].bytes, a[i].bytes, b[i].bytes);
+  }
+  return res;
+}
+
+/* Multiply a scalar and a vector */
+static rct::keyV vector_scalar(const rct::keyV &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;
+}
+
+/* Exponentiate a curve vector by a scalar */
+static rct::keyV vector_scalar2(const rct::keyV &a, const rct::key &x)
+{
+  rct::keyV res(a.size());
+  for (size_t i = 0; i < a.size(); ++i)
+  {
+    rct::scalarmultKey(res[i], a[i], x);
+  }
+  return res;
+}
+
+static rct::key switch_endianness(rct::key k)
+{
+  std::reverse(k.bytes, k.bytes + sizeof(k));
+  return k;
+}
+
+/* Compute the inverse of a scalar, the stupid way */
+static rct::key invert(const rct::key &x)
+{
+  rct::key inv;
+
+  BN_CTX *ctx = BN_CTX_new();
+  BIGNUM *X = BN_new();
+  BIGNUM *L = BN_new();
+  BIGNUM *I = BN_new();
+
+  BN_bin2bn(switch_endianness(x).bytes, sizeof(rct::key), X);
+  BN_bin2bn(switch_endianness(rct::curveOrder()).bytes, sizeof(rct::key), L);
+
+  CHECK_AND_ASSERT_THROW_MES(BN_mod_inverse(I, X, L, ctx), "Failed to invert");
+
+  const int len = BN_num_bytes(I);
+  CHECK_AND_ASSERT_THROW_MES((size_t)len <= sizeof(rct::key), "Invalid number length");
+  inv = rct::zero();
+  BN_bn2bin(I, inv.bytes);
+  std::reverse(inv.bytes, inv.bytes + len);
+
+  BN_free(I);
+  BN_free(L);
+  BN_free(X);
+  BN_CTX_free(ctx);
+
+#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;
+}
+
+/* Compute the slice of a vector */
+static rct::keyV 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");
+  rct::keyV res(stop - start);
+  for (size_t i = start; i < stop; ++i)
+  {
+    res[i - start] = a[i];
+  }
+  return res;
+}
+
+/* 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)
+{
+  init_exponents();
+
+  PERF_TIMER_UNIT(PROVE, 1000000);
+
+  constexpr size_t logN = 6; // log2(64)
+  constexpr size_t N = 1<<logN;
+
+  rct::key V;
+  rct::keyV aL(N), aR(N);
+
+  PERF_TIMER_START_BP(PROVE_v);
+  rct::addKeys2(V, sv, gamma, rct::H);
+  PERF_TIMER_STOP(PROVE_v);
+
+  PERF_TIMER_START_BP(PROVE_aLaR);
+  for (size_t i = N; i-- > 0; )
+  {
+    if (sv[i/8] & (((uint64_t)1)<<(i%8)))
+    {
+      aL[i] = rct::identity();
+    }
+    else
+    {
+      aL[i] = rct::zero();
+    }
+    sc_sub(aR[i].bytes, aL[i].bytes, rct::identity().bytes);
+  }
+  PERF_TIMER_STOP(PROVE_aLaR);
+
+
+  // DEBUG: Test to ensure this recovers the value
+#ifdef DEBUG_BP
+  uint64_t test_aL = 0, test_aR = 0;
+  for (size_t i = 0; i < N; ++i)
+  {
+    if (aL[i] == rct::identity())
+      test_aL += ((uint64_t)1)<<i;
+    if (aR[i] == rct::zero())
+      test_aR += ((uint64_t)1)<<i;
+  }
+  uint64_t v_test = 0;
+  for (int n = 0; n < 8; ++n) v_test |= (((uint64_t)sv[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
+
+  PERF_TIMER_START_BP(PROVE_step1);
+  // PAPER LINES 38-39
+  rct::key alpha = rct::skGen();
+  rct::key ve = vector_exponent(aL, aR);
+  rct::key A;
+  rct::addKeys(A, ve, rct::scalarmultKey(rct::H, alpha));
+
+  // PAPER LINES 40-42
+  rct::keyV sL = rct::skvGen(N), sR = rct::skvGen(N);
+  rct::key rho = rct::skGen();
+  ve = vector_exponent(sL, sR);
+  rct::key S;
+  rct::addKeys(S, ve, rct::scalarmultKey(rct::H, rho));
+
+  // PAPER LINES 43-45
+  rct::keyV hashed;
+  hashed.push_back(A);
+  hashed.push_back(S);
+  rct::key y = rct::hash_to_scalar(hashed);
+  rct::key z = rct::hash_to_scalar(y);
+
+  // Polynomial construction before PAPER LINE 46
+  rct::key t0 = rct::zero();
+  rct::key t1 = rct::zero();
+  rct::key t2 = rct::zero();
+
+  const auto yN = vector_powers(y, N);
+
+  rct::key ip1y = inner_product(oneN, yN);
+  rct::key tmp;
+  sc_muladd(t0.bytes, z.bytes, ip1y.bytes, t0.bytes);
+
+  rct::key zsq;
+  sc_mul(zsq.bytes, z.bytes, z.bytes);
+  sc_muladd(t0.bytes, zsq.bytes, sv.bytes, t0.bytes);
+
+  rct::key k = rct::zero();
+  sc_mulsub(k.bytes, zsq.bytes, ip1y.bytes, k.bytes);
+
+  rct::key zcu;
+  sc_mul(zcu.bytes, zsq.bytes, z.bytes);
+  sc_mulsub(k.bytes, zcu.bytes, ip12.bytes, k.bytes);
+  sc_add(t0.bytes, t0.bytes, k.bytes);
+
+  // DEBUG: Test the value of t0 has the correct form
+#ifdef DEBUG_BP
+  rct::key test_t0 = rct::zero();
+  rct::key iph = inner_product(aL, hadamard(aR, yN));
+  sc_add(test_t0.bytes, test_t0.bytes, iph.bytes);
+  rct::key ips = inner_product(vector_subtract(aL, aR), yN);
+  sc_muladd(test_t0.bytes, z.bytes, ips.bytes, test_t0.bytes);
+  rct::key ipt = inner_product(twoN, aL);
+  sc_muladd(test_t0.bytes, zsq.bytes, ipt.bytes, test_t0.bytes);
+  sc_add(test_t0.bytes, test_t0.bytes, k.bytes);
+  CHECK_AND_ASSERT_THROW_MES(t0 == test_t0, "t0 check failed");
+#endif
+  PERF_TIMER_STOP(PROVE_step1);
+
+  PERF_TIMER_START_BP(PROVE_step2);
+  const auto HyNsR = hadamard(yN, sR);
+  const auto vpIz = vector_scalar(oneN, z);
+  const auto vp2zsq = vector_scalar(twoN, zsq);
+  const auto aL_vpIz = vector_subtract(aL, vpIz);
+  const auto aR_vpIz = vector_add(aR, vpIz);
+
+  rct::key ip1 = inner_product(aL_vpIz, HyNsR);
+  sc_add(t1.bytes, t1.bytes, ip1.bytes);
+
+  rct::key ip2 = inner_product(sL, vector_add(hadamard(yN, aR_vpIz), vp2zsq));
+  sc_add(t1.bytes, t1.bytes, ip2.bytes);
+
+  rct::key ip3 = inner_product(sL, HyNsR);
+  sc_add(t2.bytes, t2.bytes, ip3.bytes);
+
+  // PAPER LINES 47-48
+  rct::key tau1 = rct::skGen(), tau2 = rct::skGen();
+
+  rct::key T1 = rct::addKeys(rct::scalarmultBase(t1), rct::scalarmultKey(rct::H, tau1));
+  rct::key T2 = rct::addKeys(rct::scalarmultBase(t2), rct::scalarmultKey(rct::H, tau2));
+
+  // PAPER LINES 49-51
+  hashed.clear();
+  hashed.push_back(z);
+  hashed.push_back(T1);
+  hashed.push_back(T2);
+  rct::key x = rct::hash_to_scalar(hashed);
+
+  // PAPER LINES 52-53
+  rct::key taux = rct::zero();
+  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);
+  sc_muladd(taux.bytes, gamma.bytes, zsq.bytes, taux.bytes);
+  rct::key mu;
+  sc_muladd(mu.bytes, x.bytes, rho.bytes, alpha.bytes);
+
+  // PAPER LINES 54-57
+  rct::keyV l = vector_add(aL_vpIz, vector_scalar(sL, x));
+  rct::keyV r = vector_add(hadamard(yN, vector_add(aR_vpIz, vector_scalar(sR, x))), vp2zsq);
+  PERF_TIMER_STOP(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;
+  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
+  hashed.clear();
+  hashed.push_back(x);
+  hashed.push_back(taux);
+  hashed.push_back(mu);
+  hashed.push_back(t);
+  rct::key x_ip = rct::hash_to_scalar(hashed);
+
+  // These are used in the inner product rounds
+  size_t nprime = N;
+  rct::keyV Gprime(N);
+  rct::keyV Hprime(N);
+  rct::keyV aprime(N);
+  rct::keyV bprime(N);
+  const rct::key yinv = invert(y);
+  rct::key yinvpow = rct::identity();
+  for (size_t i = 0; i < N; ++i)
+  {
+    Gprime[i] = Gi[i];
+    Hprime[i] = scalarmultKey(Hi[i], yinvpow);
+    sc_mul(yinvpow.bytes, yinvpow.bytes, yinv.bytes);
+    aprime[i] = l[i];
+    bprime[i] = r[i];
+  }
+  rct::keyV L(logN);
+  rct::keyV R(logN);
+  int round = 0;
+  rct::keyV w(logN); // this is the challenge x in the inner product protocol
+  PERF_TIMER_STOP(PROVE_step3);
+
+  PERF_TIMER_START_BP(PROVE_step4);
+  // PAPER LINE 13
+  while (nprime > 1)
+  {
+    // PAPER LINE 15
+    nprime /= 2;
+
+    // PAPER LINES 16-17
+    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));
+
+    // PAPER LINES 18-19
+    L[round] = vector_exponent_custom(slice(Gprime, nprime, Gprime.size()), slice(Hprime, 0, nprime), slice(aprime, 0, nprime), slice(bprime, nprime, bprime.size()));
+    sc_mul(tmp.bytes, cL.bytes, x_ip.bytes);
+    rct::addKeys(L[round], L[round], rct::scalarmultBase(tmp));
+    R[round] = vector_exponent_custom(slice(Gprime, 0, nprime), slice(Hprime, nprime, Hprime.size()), slice(aprime, nprime, aprime.size()), slice(bprime, 0, nprime));
+    sc_mul(tmp.bytes, cR.bytes, x_ip.bytes);
+    rct::addKeys(R[round], R[round], rct::scalarmultBase(tmp));
+
+    // PAPER LINES 21-22
+    hashed.clear();
+    if (round == 0)
+    {
+      hashed.push_back(L[0]);
+      hashed.push_back(R[0]);
+      w[0] = rct::hash_to_scalar(hashed);
+    }
+    else
+    {
+      hashed.push_back(w[round - 1]);
+      hashed.push_back(L[round]);
+      hashed.push_back(R[round]);
+      w[round] = rct::hash_to_scalar(hashed);
+    }
+
+    // PAPER LINES 24-25
+    const rct::key winv = invert(w[round]);
+    Gprime = hadamard2(vector_scalar2(slice(Gprime, 0, nprime), winv), vector_scalar2(slice(Gprime, nprime, Gprime.size()), w[round]));
+    Hprime = hadamard2(vector_scalar2(slice(Hprime, 0, nprime), w[round]), vector_scalar2(slice(Hprime, nprime, Hprime.size()), winv));
+
+    // PAPER LINES 28-29
+    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]));
+
+    ++round;
+  }
+  PERF_TIMER_STOP(PROVE_step4);
+
+  // PAPER LINE 58 (with inclusions from PAPER LINE 8 and PAPER LINE 20)
+  return Bulletproof(V, A, S, T1, T2, taux, mu, L, R, aprime[0], bprime[0], t);
+}
+
+Bulletproof bulletproof_PROVE(uint64_t v, const rct::key &gamma)
+{
+  // vG + gammaH
+  PERF_TIMER_START_BP(PROVE_v);
+  rct::key sv = rct::zero();
+  sv.bytes[0] = v & 255;
+  sv.bytes[1] = (v >> 8) & 255;
+  sv.bytes[2] = (v >> 16) & 255;
+  sv.bytes[3] = (v >> 24) & 255;
+  sv.bytes[4] = (v >> 32) & 255;
+  sv.bytes[5] = (v >> 40) & 255;
+  sv.bytes[6] = (v >> 48) & 255;
+  sv.bytes[7] = (v >> 56) & 255;
+  PERF_TIMER_STOP(PROVE_v);
+  return bulletproof_PROVE(sv, gamma);
+}
+
+/* Given a range proof, determine if it is valid */
+bool bulletproof_VERIFY(const Bulletproof &proof)
+{
+  init_exponents();
+
+  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");
+  CHECK_AND_ASSERT_MES(proof.L.size() == 6, false, "Proof is not for 64 bits");
+
+  const size_t logN = proof.L.size();
+  const size_t N = 1 << logN;
+
+  // Reconstruct the challenges
+  PERF_TIMER_START_BP(VERIFY);
+  PERF_TIMER_START_BP(VERIFY_start);
+  rct::keyV hashed;
+  hashed.push_back(proof.A);
+  hashed.push_back(proof.S);
+  rct::key y = rct::hash_to_scalar(hashed);
+  rct::key z = rct::hash_to_scalar(y);
+  hashed.clear();
+  hashed.push_back(z);
+  hashed.push_back(proof.T1);
+  hashed.push_back(proof.T2);
+  rct::key x = rct::hash_to_scalar(hashed);
+  PERF_TIMER_STOP(VERIFY_start);
+
+  PERF_TIMER_START_BP(VERIFY_line_60);
+  // Reconstruct the challenges
+  hashed.clear();
+  hashed.push_back(x);
+  hashed.push_back(proof.taux);
+  hashed.push_back(proof.mu);
+  hashed.push_back(proof.t);
+  rct::key x_ip = hash_to_scalar(hashed);
+  PERF_TIMER_STOP(VERIFY_line_60);
+
+  PERF_TIMER_START_BP(VERIFY_line_61);
+  // PAPER LINE 61
+  rct::key L61Left = rct::addKeys(rct::scalarmultKey(rct::H, proof.taux), rct::scalarmultBase(proof.t));
+
+  rct::key k = rct::zero();
+  const auto yN = vector_powers(y, N);
+  rct::key ip1y = inner_product(oneN, yN);
+  rct::key zsq;
+  sc_mul(zsq.bytes, z.bytes, z.bytes);
+  rct::key tmp, tmp2;
+  sc_mulsub(k.bytes, zsq.bytes, ip1y.bytes, k.bytes);
+  rct::key zcu;
+  sc_mul(zcu.bytes, zsq.bytes, z.bytes);
+  sc_mulsub(k.bytes, zcu.bytes, ip12.bytes, k.bytes);
+  PERF_TIMER_STOP(VERIFY_line_61);
+
+  PERF_TIMER_START_BP(VERIFY_line_61rl);
+  sc_muladd(tmp.bytes, z.bytes, ip1y.bytes, k.bytes);
+  rct::key L61Right = rct::scalarmultBase(tmp);
+
+  tmp = rct::scalarmultKey(proof.V, zsq);
+  rct::addKeys(L61Right, L61Right, tmp);
+
+  tmp = rct::scalarmultKey(proof.T1, x);
+  rct::addKeys(L61Right, L61Right, tmp);
+
+  rct::key xsq;
+  sc_mul(xsq.bytes, x.bytes, x.bytes);
+  tmp = rct::scalarmultKey(proof.T2, xsq);
+  rct::addKeys(L61Right, L61Right, tmp);
+  PERF_TIMER_STOP(VERIFY_line_61rl);
+
+  if (!(L61Right == L61Left))
+  {
+    MERROR("Verification failure at step 1");
+    return false;
+  }
+
+  PERF_TIMER_START_BP(VERIFY_line_62);
+  // PAPER LINE 62
+  rct::key P = rct::addKeys(proof.A, rct::scalarmultKey(proof.S, x));
+  PERF_TIMER_STOP(VERIFY_line_62);
+
+  // Compute the number of rounds for the inner product
+  const size_t rounds = proof.L.size();
+  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
+  rct::keyV w(rounds);
+  hashed.clear();
+  hashed.push_back(proof.L[0]);
+  hashed.push_back(proof.R[0]);
+  w[0] = rct::hash_to_scalar(hashed);
+  for (size_t i = 1; i < rounds; ++i)
+  {
+    hashed.clear();
+    hashed.push_back(w[i-1]);
+    hashed.push_back(proof.L[i]);
+    hashed.push_back(proof.R[i]);
+    w[i] = rct::hash_to_scalar(hashed);
+  }
+  PERF_TIMER_STOP(VERIFY_line_21_22);
+
+  PERF_TIMER_START_BP(VERIFY_line_24_25);
+  // Basically PAPER LINES 24-25
+  // Compute the curvepoints from G[i] and H[i]
+  rct::key inner_prod = rct::identity();
+  rct::key yinvpow = rct::identity();
+  rct::key ypow = rct::identity();
+
+  PERF_TIMER_START_BP(VERIFY_line_24_25_invert);
+  const rct::key yinv = invert(y);
+  rct::keyV winv(rounds);
+  for (size_t i = 0; i < rounds; ++i)
+    winv[i] = invert(w[i]);
+  PERF_TIMER_STOP(VERIFY_line_24_25_invert);
+
+  for (size_t i = 0; i < N; ++i)
+  {
+    // Convert the index to binary IN REVERSE and construct the scalar exponent
+    rct::key g_scalar = proof.a;
+    rct::key h_scalar;
+    sc_mul(h_scalar.bytes, proof.b.bytes, yinvpow.bytes);
+
+    for (size_t j = rounds; j-- > 0; )
+    {
+      size_t J = w.size() - j - 1;
+
+      if ((i & (((size_t)1)<<j)) == 0)
+      {
+        sc_mul(g_scalar.bytes, g_scalar.bytes, winv[J].bytes);
+        sc_mul(h_scalar.bytes, h_scalar.bytes, w[J].bytes);
+      }
+      else
+      {
+        sc_mul(g_scalar.bytes, g_scalar.bytes, w[J].bytes);
+        sc_mul(h_scalar.bytes, h_scalar.bytes, winv[J].bytes);
+      }
+    }
+
+    // Adjust the scalars using the exponents from PAPER LINE 62
+    sc_add(g_scalar.bytes, g_scalar.bytes, z.bytes);
+    sc_mul(tmp.bytes, zsq.bytes, twoN[i].bytes);
+    sc_muladd(tmp.bytes, z.bytes, ypow.bytes, tmp.bytes);
+    sc_mulsub(h_scalar.bytes, tmp.bytes, yinvpow.bytes, h_scalar.bytes);
+
+    // Now compute the basepoint's scalar multiplication
+    // Each of these could be written as a multiexp operation instead
+    rct::addKeys3(tmp, g_scalar, Gprecomp[i], h_scalar, Hprecomp[i]);
+    rct::addKeys(inner_prod, inner_prod, tmp);
+
+    if (i != N-1)
+    {
+      sc_mul(yinvpow.bytes, yinvpow.bytes, yinv.bytes);
+      sc_mul(ypow.bytes, ypow.bytes, y.bytes);
+    }
+  }
+  PERF_TIMER_STOP(VERIFY_line_24_25);
+
+  PERF_TIMER_START_BP(VERIFY_line_26);
+  // PAPER LINE 26
+  rct::key pprime;
+  sc_sub(tmp.bytes, rct::zero().bytes, proof.mu.bytes);
+  rct::addKeys(pprime, P, rct::scalarmultKey(rct::H, tmp));
+
+  for (size_t i = 0; i < rounds; ++i)
+  {
+    sc_mul(tmp.bytes, w[i].bytes, w[i].bytes);
+    sc_mul(tmp2.bytes, winv[i].bytes, winv[i].bytes);
+#if 1
+    ge_dsmp cacheL, cacheR;
+    rct::precomp(cacheL, proof.L[i]);
+    rct::precomp(cacheR, proof.R[i]);
+    rct::addKeys3(tmp, tmp, cacheL, tmp2, cacheR);
+    rct::addKeys(pprime, pprime, tmp);
+#else
+    rct::addKeys(pprime, pprime, rct::scalarmultKey(proof.L[i], tmp));
+    rct::addKeys(pprime, pprime, rct::scalarmultKey(proof.R[i], tmp2));
+#endif
+  }
+  sc_mul(tmp.bytes, proof.t.bytes, x_ip.bytes);
+  rct::addKeys(pprime, pprime, rct::scalarmultBase(tmp));
+  PERF_TIMER_STOP(VERIFY_line_26);
+
+  PERF_TIMER_START_BP(VERIFY_step2_check);
+  sc_mul(tmp.bytes, proof.a.bytes, proof.b.bytes);
+  sc_mul(tmp.bytes, tmp.bytes, x_ip.bytes);
+  tmp = rct::scalarmultBase(tmp);
+  rct::addKeys(tmp, tmp, inner_prod);
+  PERF_TIMER_STOP(VERIFY_step2_check);
+  if (!(pprime == tmp))
+  {
+    MERROR("Verification failure at step 2");
+    return false;
+  }
+
+  PERF_TIMER_STOP(VERIFY);
+  return true;
+}
+
+}