crypto.cpp

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// Copyrights(c) 2017-2021, The Electroneum Project
// Copyrights(c) 2014-2019, The Monero Project
// 
// All rights reserved.
// 
// Redistribution and use in source and binary forms, with or without modification, are
// permitted provided that the following conditions are met:
// 
// 1. Redistributions of source code must retain the above copyright notice, this list of
//    conditions and the following disclaimer.
// 
// 2. Redistributions in binary form must reproduce the above copyright notice, this list
//    of conditions and the following disclaimer in the documentation and/or other
//    materials provided with the distribution.
// 
// 3. Neither the name of the copyright holder nor the names of its contributors may be
//    used to endorse or promote products derived from this software without specific
//    prior written permission.
// 
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
// MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL
// THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
// STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF
// THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
// 
// Parts of this file are originally copyright (c) 2012-2013 The Cryptonote developers
 
#include <unistd.h>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <cstdlib>
#include <cstring>
#include <boost/thread/mutex.hpp>
#include <boost/thread/lock_guard.hpp>
#include <boost/shared_ptr.hpp>
 
#include "common/varint.h"
#include "crypto.h"
#include "hash.h"
 
namespace {
  static void local_abort(const char *msg)
  {
    fprintf(stderr, "%s\n", msg);
#ifdef NDEBUG
    _exit(1);
#else
    abort();
#endif
  }
}
 
namespace crypto {
 
  using std::abort;
  using std::int32_t;
  using std::int64_t;
  using std::size_t;
  using std::uint32_t;
  using std::uint64_t;
 
  extern "C" {
#include "crypto-ops.h"
#include "random.h"
  }
 
  const crypto::public_key null_pkey = crypto::public_key{};
  const crypto::secret_key null_skey = crypto::secret_key{};
 
    static inline unsigned char *operator &(ec_point &point) {
        return &reinterpret_cast<unsigned char &>(point);
    }
 
    static inline const unsigned char *operator &(const ec_point &point) {
        return &reinterpret_cast<const unsigned char &>(point);
    }
 
    static inline unsigned char *operator &(ec_scalar &scalar) {
        return &reinterpret_cast<unsigned char &>(scalar);
    }
 
    static inline const unsigned char *operator &(const ec_scalar &scalar) {
        return &reinterpret_cast<const unsigned char &>(scalar);
    }
 
  void generate_random_bytes_thread_safe(size_t N, uint8_t *bytes)
  {
    static boost::mutex random_lock;
    boost::lock_guard<boost::mutex> lock(random_lock);
    generate_random_bytes_not_thread_safe(N, bytes);
  }
 
  static inline bool less32(const unsigned char *k0, const unsigned char *k1)
  {
    for (int n = 31; n >= 0; --n)
    {
      if (k0[n] < k1[n])
        return true;
      if (k0[n] > k1[n])
        return false;
    }
    return false;
  }
 
  void random32_unbiased(unsigned char *bytes)
  {
    // l = 2^252 + 27742317777372353535851937790883648493.
    // it fits 15 in 32 bytes
    static const unsigned char limit[32] = { 0xe3, 0x6a, 0x67, 0x72, 0x8b, 0xce, 0x13, 0x29, 0x8f, 0x30, 0x82, 0x8c, 0x0b, 0xa4, 0x10, 0x39, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0xf0 };
    do
    {
      generate_random_bytes_thread_safe(32, bytes);
    } while (!sc_isnonzero(bytes) && !less32(bytes, limit)); // should be good about 15/16 of the time
    sc_reduce32(bytes);
  }
  /* generate a random 32-byte (256-bit) integer and copy it to res */
  void random_scalar(ec_scalar &res) {
    random32_unbiased((unsigned char*)res.data);
  }
 
  void hash_to_scalar(const void *data, size_t length, ec_scalar &res) {
    cn_fast_hash(data, length, reinterpret_cast<hash &>(res));
    sc_reduce32(&res);
  }
 
  /* 
   * generate public and secret keys from a random 256-bit integer
   * TODO: allow specifying random value (for wallet recovery)
   * 
   */
  secret_key crypto_ops::generate_keys(public_key &pub, secret_key &sec, const secret_key& recovery_key, bool recover) {
    ge_p3 point;
 
    secret_key rng;
 
    if (recover)
    {
      rng = recovery_key;
    }
    else
    {
      random_scalar(rng);
    }
    sec = rng;
    sc_reduce32(&unwrap(sec));  // reduce in case second round of keys (sendkeys)
 
    ge_scalarmult_base(&point, &unwrap(sec));
    ge_p3_tobytes(&pub, &point);
 
    return rng;
  }
 
  bool crypto_ops::check_key(const public_key &key) {
    ge_p3 point;
    return ge_frombytes_vartime(&point, &key) == 0;
  }
 
  bool crypto_ops::secret_key_to_public_key(const secret_key &sec, public_key &pub) {
    ge_p3 point;
    if (sc_check(&unwrap(sec)) != 0) {
      return false;
    }
    ge_scalarmult_base(&point, &unwrap(sec));
    ge_p3_tobytes(&pub, &point);
    return true;
  }
 
  bool crypto_ops::generate_key_derivation(const public_key &key1, const secret_key &key2, key_derivation &derivation) {
    ge_p3 point;
    ge_p2 point2;
    ge_p1p1 point3;
    assert(sc_check(&key2) == 0);
    if (ge_frombytes_vartime(&point, &key1) != 0) {
      return false;
    }
    ge_scalarmult(&point2, &unwrap(key2), &point);
    ge_mul8(&point3, &point2);
    ge_p1p1_to_p2(&point2, &point3);
    ge_tobytes(&derivation, &point2);
    return true;
  }
 
  void crypto_ops::derivation_to_scalar(const key_derivation &derivation, size_t output_index, ec_scalar &res) {
    struct {
      key_derivation derivation;
      char output_index[(sizeof(size_t) * 8 + 6) / 7];
    } buf;
    char *end = buf.output_index;
    buf.derivation = derivation;
    tools::write_varint(end, output_index);
    assert(end <= buf.output_index + sizeof buf.output_index);
    hash_to_scalar(&buf, end - reinterpret_cast<char *>(&buf), res);
  }
 
  bool crypto_ops::derive_public_key(const key_derivation &derivation, size_t output_index,
    const public_key &base, public_key &derived_key) {
    ec_scalar scalar;
    ge_p3 point1;
    ge_p3 point2;
    ge_cached point3;
    ge_p1p1 point4;
    ge_p2 point5;
    if (ge_frombytes_vartime(&point1, &base) != 0) {
      return false;
    }
    derivation_to_scalar(derivation, output_index, scalar);
    ge_scalarmult_base(&point2, &scalar);
    ge_p3_to_cached(&point3, &point2);
    ge_add(&point4, &point1, &point3);
    ge_p1p1_to_p2(&point5, &point4);
    ge_tobytes(&derived_key, &point5);
    return true;
  }
 
  void crypto_ops::derive_secret_key(const key_derivation &derivation, size_t output_index,
    const secret_key &base, secret_key &derived_key) {
    ec_scalar scalar;
    assert(sc_check(&base) == 0);
    derivation_to_scalar(derivation, output_index, scalar);
    sc_add(&unwrap(derived_key), &unwrap(base), &scalar);
  }
 
  bool crypto_ops::derive_subaddress_public_key(const public_key &out_key, const key_derivation &derivation, std::size_t output_index, public_key &derived_key) {
    ec_scalar scalar;
    ge_p3 point1;
    ge_p3 point2;
    ge_cached point3;
    ge_p1p1 point4;
    ge_p2 point5;
    if (ge_frombytes_vartime(&point1, &out_key) != 0) {
      return false;
    }
    derivation_to_scalar(derivation, output_index, scalar);
    ge_scalarmult_base(&point2, &scalar);
    ge_p3_to_cached(&point3, &point2);
    ge_sub(&point4, &point1, &point3);
    ge_p1p1_to_p2(&point5, &point4);
    ge_tobytes(&derived_key, &point5);
    return true;
  }
 
  struct s_comm {
    hash h;
    ec_point key;
    ec_point comm;
  };
 
  struct s_comm_2 {
    hash msg;
    ec_point D;
    ec_point X;
    ec_point Y;
  };
 
  struct s_comm_3 {
      hash h;
      uint32_t i;
      ec_point key;
      ec_point comm;
  };
 
  void crypto_ops::generate_signature(const hash &prefix_hash, const public_key &pub, const secret_key &sec, signature &sig) {
    ge_p3 tmp3;
    ec_scalar k;
    s_comm buf;
#if !defined(NDEBUG)
    {
      ge_p3 t;
      public_key t2;
      assert(sc_check(&sec) == 0);
      ge_scalarmult_base(&t, &sec);
      ge_p3_tobytes(&t2, &t);
      assert(pub == t2);
    }
#endif
    buf.h = prefix_hash;
    buf.key = pub;
  try_again:
    random_scalar(k);
    if (((const uint32_t*)(&k))[7] == 0) // we don't want tiny numbers here
      goto try_again;
    ge_scalarmult_base(&tmp3, &k);
    ge_p3_tobytes(&buf.comm, &tmp3);
    hash_to_scalar(&buf, sizeof(s_comm), sig.c);
    if (!sc_isnonzero((const unsigned char*)sig.c.data))
      goto try_again;
    sc_mulsub(&sig.r, &sig.c, &unwrap(sec), &k);
    if (!sc_isnonzero((const unsigned char*)sig.r.data))
      goto try_again;
  }
 
  bool crypto_ops::check_signature(const hash &prefix_hash, const public_key &pub, const signature &sig) {
    ge_p2 tmp2;
    ge_p3 tmp3;
    ec_scalar c;
    s_comm buf;
    assert(check_key(pub));
    buf.h = prefix_hash;
    buf.key = pub;
    if (ge_frombytes_vartime(&tmp3, &pub) != 0) {
      return false;
    }
    if (sc_check(&sig.c) != 0 || sc_check(&sig.r) != 0 || !sc_isnonzero(&sig.c)) {
      return false;
    }
    ge_double_scalarmult_base_vartime(&tmp2, &sig.c, &tmp3, &sig.r);
    ge_tobytes(&buf.comm, &tmp2);
    static const ec_point infinity = {{ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}};
    if (memcmp(&buf.comm, &infinity, 32) == 0)
      return false;
    hash_to_scalar(&buf, sizeof(s_comm), c);
    sc_sub(&c, &c, &sig.c);
    return sc_isnonzero(&c) == 0;
  }
 
  void crypto_ops::generate_tx_proof(const hash &prefix_hash, const public_key &R, const public_key &A, const boost::optional<public_key> &B, const public_key &D, const secret_key &r, signature &sig) {
    // sanity check
    ge_p3 R_p3;
    ge_p3 A_p3;
    ge_p3 B_p3;
    ge_p3 D_p3;
    if (ge_frombytes_vartime(&R_p3, &R) != 0) throw std::runtime_error("tx pubkey is invalid");
    if (ge_frombytes_vartime(&A_p3, &A) != 0) throw std::runtime_error("recipient view pubkey is invalid");
    if (B && ge_frombytes_vartime(&B_p3, &*B) != 0) throw std::runtime_error("recipient spend pubkey is invalid");
    if (ge_frombytes_vartime(&D_p3, &D) != 0) throw std::runtime_error("key derivation is invalid");
#if !defined(NDEBUG)
    {
      assert(sc_check(&r) == 0);
      // check R == r*G or R == r*B
      public_key dbg_R;
      if (B)
      {
        ge_p2 dbg_R_p2;
        ge_scalarmult(&dbg_R_p2, &r, &B_p3);
        ge_tobytes(&dbg_R, &dbg_R_p2);
      }
      else
      {
        ge_p3 dbg_R_p3;
        ge_scalarmult_base(&dbg_R_p3, &r);
        ge_p3_tobytes(&dbg_R, &dbg_R_p3);
      }
      assert(R == dbg_R);
      // check D == r*A
      ge_p2 dbg_D_p2;
      ge_scalarmult(&dbg_D_p2, &r, &A_p3);
      public_key dbg_D;
      ge_tobytes(&dbg_D, &dbg_D_p2);
      assert(D == dbg_D);
    }
#endif
 
    // pick random k
    ec_scalar k;
    random_scalar(k);
    
    s_comm_2 buf;
    buf.msg = prefix_hash;
    buf.D = D;
 
    if (B)
    {
      // compute X = k*B
      ge_p2 X_p2;
      ge_scalarmult(&X_p2, &k, &B_p3);
      ge_tobytes(&buf.X, &X_p2);
    }
    else
    {
      // compute X = k*G
      ge_p3 X_p3;
      ge_scalarmult_base(&X_p3, &k);
      ge_p3_tobytes(&buf.X, &X_p3);
    }
    
    // compute Y = k*A
    ge_p2 Y_p2;
    ge_scalarmult(&Y_p2, &k, &A_p3);
    ge_tobytes(&buf.Y, &Y_p2);
 
    // sig.c = Hs(Msg || D || X || Y)
    hash_to_scalar(&buf, sizeof(buf), sig.c);
 
    // sig.r = k - sig.c*r
    sc_mulsub(&sig.r, &sig.c, &unwrap(r), &k);
  }
 
  bool crypto_ops::check_tx_proof(const hash &prefix_hash, const public_key &R, const public_key &A, const boost::optional<public_key> &B, const public_key &D, const signature &sig) {
    // sanity check
    ge_p3 R_p3;
    ge_p3 A_p3;
    ge_p3 B_p3;
    ge_p3 D_p3;
    if (ge_frombytes_vartime(&R_p3, &R) != 0) return false;
    if (ge_frombytes_vartime(&A_p3, &A) != 0) return false;
    if (B && ge_frombytes_vartime(&B_p3, &*B) != 0) return false;
    if (ge_frombytes_vartime(&D_p3, &D) != 0) return false;
    if (sc_check(&sig.c) != 0 || sc_check(&sig.r) != 0) return false;
 
    // compute sig.c*R
    ge_p3 cR_p3;
    {
      ge_p2 cR_p2;
      ge_scalarmult(&cR_p2, &sig.c, &R_p3);
      public_key cR;
      ge_tobytes(&cR, &cR_p2);
      if (ge_frombytes_vartime(&cR_p3, &cR) != 0) return false;
    }
 
    ge_p1p1 X_p1p1;
    if (B)
    {
      // compute X = sig.c*R + sig.r*B
      ge_p2 rB_p2;
      ge_scalarmult(&rB_p2, &sig.r, &B_p3);
      public_key rB;
      ge_tobytes(&rB, &rB_p2);
      ge_p3 rB_p3;
      if (ge_frombytes_vartime(&rB_p3, &rB) != 0) return false;
      ge_cached rB_cached;
      ge_p3_to_cached(&rB_cached, &rB_p3);
      ge_add(&X_p1p1, &cR_p3, &rB_cached);
    }
    else
    {
      // compute X = sig.c*R + sig.r*G
      ge_p3 rG_p3;
      ge_scalarmult_base(&rG_p3, &sig.r);
      ge_cached rG_cached;
      ge_p3_to_cached(&rG_cached, &rG_p3);
      ge_add(&X_p1p1, &cR_p3, &rG_cached);
    }
    ge_p2 X_p2;
    ge_p1p1_to_p2(&X_p2, &X_p1p1);
 
    // compute sig.c*D
    ge_p2 cD_p2;
    ge_scalarmult(&cD_p2, &sig.c, &D_p3);
 
    // compute sig.r*A
    ge_p2 rA_p2;
    ge_scalarmult(&rA_p2, &sig.r, &A_p3);
 
    // compute Y = sig.c*D + sig.r*A
    public_key cD;
    public_key rA;
    ge_tobytes(&cD, &cD_p2);
    ge_tobytes(&rA, &rA_p2);
    ge_p3 cD_p3;
    ge_p3 rA_p3;
    if (ge_frombytes_vartime(&cD_p3, &cD) != 0) return false;
    if (ge_frombytes_vartime(&rA_p3, &rA) != 0) return false;
    ge_cached rA_cached;
    ge_p3_to_cached(&rA_cached, &rA_p3);
    ge_p1p1 Y_p1p1;
    ge_add(&Y_p1p1, &cD_p3, &rA_cached);
    ge_p2 Y_p2;
    ge_p1p1_to_p2(&Y_p2, &Y_p1p1);
 
    // compute c2 = Hs(Msg || D || X || Y)
    s_comm_2 buf;
    buf.msg = prefix_hash;
    buf.D = D;
    ge_tobytes(&buf.X, &X_p2);
    ge_tobytes(&buf.Y, &Y_p2);
    ec_scalar c2;
    hash_to_scalar(&buf, sizeof(s_comm_2), c2);
 
    // test if c2 == sig.c
    sc_sub(&c2, &c2, &sig.c);
    return sc_isnonzero(&c2) == 0;
  }
 
  void hash_to_ec(const public_key &key, ge_p3 &res) {
    hash h;
    ge_p2 point;
    ge_p1p1 point2;
    cn_fast_hash(std::addressof(key), sizeof(public_key), h);
    ge_fromfe_frombytes_vartime(&point, reinterpret_cast<const unsigned char *>(&h));
    ge_mul8(&point2, &point);
    ge_p1p1_to_p3(&res, &point2);
  }
  void hash_to_ec(const hash& h, ge_p3 &res) {
    ge_p2 point;
    ge_p1p1 point2;
    ge_fromfe_frombytes_vartime(&point, reinterpret_cast<const unsigned char *>(&h));
    ge_mul8(&point2, &point);
    ge_p1p1_to_p3(&res, &point2);
  }
 
  void crypto_ops::generate_key_image(const public_key &pub, const secret_key &sec, key_image &image) {
    ge_p3 point;
    ge_p2 point2;
    assert(sc_check(&sec) == 0);
    hash_to_ec(pub, point);
    ge_scalarmult(&point2, &unwrap(sec), &point);
    ge_tobytes(&image, &point2);
  }
 
  size_t rs_comm_size(size_t pubs_count) {
    return sizeof(rs_comm) + pubs_count * sizeof(ec_point_pair);
  }
 
  void crypto_ops::generate_input_signature(const hash prefix_hash, const uint32_t input_index, const secret_key sec_view, const secret_key sec_spend, signature &sig){
 
    // add the two secret keys and reduce modulo l to get a new valid secret key for signing.
    secret_key sec = addSecretKeys(sec_view, sec_spend);
 
    //corresponding public key for the combined secret key
    public_key pub;
    ge_p3 t;
    assert(sc_check(&sec) == 0);
    ge_scalarmult_base(&t, &sec);
    ge_p3_tobytes(&pub, &t);
 
    //generate signature
    ge_p3 tmp3;
    ec_scalar k;
    s_comm_3 buf;
    buf.h = prefix_hash;
    buf.i = input_index;
    buf.key = pub;
    try_again:
    random_scalar(k);
    if (((const uint32_t*)(&k))[7] == 0) // we don't want tiny numbers here
      goto try_again;
    ge_scalarmult_base(&tmp3, &k);
    ge_p3_tobytes(&buf.comm, &tmp3);
    hash_to_scalar(&buf, sizeof(s_comm_3), sig.c);
    if (!sc_isnonzero((const unsigned char*)sig.c.data))
      goto try_again;
    sc_mulsub(&sig.r, &sig.c, &unwrap(sec), &k);
    if (!sc_isnonzero((const unsigned char*)sig.r.data))
      goto try_again;
  }
 
  bool crypto_ops::verify_input_signature(const hash &prefix_hash,const uint32_t input_index, const public_key pub_view, const public_key pub_spend, signature sig)
  {
    public_key pub = addKeys(pub_view, pub_spend);
 
    ge_p2 tmp2;
    ge_p3 tmp3;
    ec_scalar c;
    s_comm_3 buf;
    assert(check_key(pub));
    buf.h = prefix_hash;
    buf.i = input_index;
    buf.key = pub;
    if (ge_frombytes_vartime(&tmp3, &pub) != 0) {
      return false;
    }
    if (sc_check(&sig.c) != 0 || sc_check(&sig.r) != 0 || !sc_isnonzero(&sig.c)) {
      return false;
    }
    ge_double_scalarmult_base_vartime(&tmp2, &sig.c, &tmp3, &sig.r);
    ge_tobytes(&buf.comm, &tmp2);
    static const ec_point infinity = {{ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}};
    if (memcmp(&buf.comm, &infinity, 32) == 0)
      return false;
    hash_to_scalar(&buf, sizeof(s_comm_3), c);
    sc_sub(&c, &c, &sig.c);
    return sc_isnonzero(&c) == 0;
  }
 
  //for curve points: AB = A + B
  public_key crypto_ops::addKeys(const public_key &A, const public_key &B) {
    public_key AB;
    ge_p3 B2, A2;
    assert(ge_frombytes_vartime(&B2, &B) == 0 && ge_frombytes_vartime(&A2, &A) == 0);
    ge_cached tmp2;
    ge_p3_to_cached(&tmp2, &B2);
    ge_p1p1 tmp3;
    ge_add(&tmp3, &A2, &tmp2);
    ge_p1p1_to_p3(&A2, &tmp3);
    ge_p3_tobytes(&AB, &A2);
 
    return AB;
  }
 
  secret_key crypto_ops::addSecretKeys(const secret_key& a, const secret_key& b){
      assert(sc_check(&a) == 0);
      assert(sc_check(&b) == 0);
      secret_key d;
      sc_add(&unwrap(d), &unwrap(a), &unwrap(b));
      sc_reduce32(&d);
      return d;
  }
 
  void crypto_ops::generate_ring_signature(const hash &prefix_hash, const key_image &image,
    const public_key *const *pubs, size_t pubs_count,
    const secret_key &sec, size_t sec_index,
    signature *sig) {
    size_t i;
    ge_p3 image_unp;
    ge_dsmp image_pre;
    ec_scalar sum, k, h;
    boost::shared_ptr<rs_comm> buf(reinterpret_cast<rs_comm *>(malloc(rs_comm_size(pubs_count))), free);
    if (!buf)
      local_abort("malloc failure");
    assert(sec_index < pubs_count);
#if !defined(NDEBUG)
    {
      ge_p3 t;
      public_key t2;
      key_image t3;
      assert(sc_check(&sec) == 0);
      ge_scalarmult_base(&t, &sec);
      ge_p3_tobytes(&t2, &t);
      assert(*pubs[sec_index] == t2);
      generate_key_image(*pubs[sec_index], sec, t3);
      assert(image == t3);
      for (i = 0; i < pubs_count; i++) {
        assert(check_key(*pubs[i]));
      }
    }
#endif
    if (ge_frombytes_vartime(&image_unp, &image) != 0) {
      local_abort("invalid key image");
    }
    ge_dsm_precomp(image_pre, &image_unp);
    sc_0(&sum);
    buf->h = prefix_hash;
    for (i = 0; i < pubs_count; i++) {
      ge_p2 tmp2;
      ge_p3 tmp3;
      // Comments below use the same notation as the Cryptonote whitepaper.
      if (i == sec_index) {
        // Generate a random q_s
        random_scalar(k);
        // L_s = q_s*G
        ge_scalarmult_base(&tmp3, &k);
        // L_s in byte form
        ge_p3_tobytes(&buf->ab[i].a, &tmp3);
        // tmp3 now becomes H_p(P_s)
        hash_to_ec(*pubs[i], tmp3);
        // R_s = q_s * H_p(P_s)
        ge_scalarmult(&tmp2, &k, &tmp3);
        // R_s in byte form
        ge_tobytes(&buf->ab[i].b, &tmp2);
      } else {
        // Generate our random scalars w_i & q_i
        random_scalar(sig[i].c);
        random_scalar(sig[i].r);
        // Takes the byte form of P_i and converts to an ed25519 point.
        if (ge_frombytes_vartime(&tmp3, &*pubs[i]) != 0) {
          local_abort("invalid pubkey");
        }
        // Returns L_i = (q_i*G) + (w_i*P_i)
        ge_double_scalarmult_base_vartime(&tmp2, &sig[i].c, &tmp3, &sig[i].r);
        // L_i in byte form
        ge_tobytes(&buf->ab[i].a, &tmp2);
        hash_to_ec(*pubs[i], tmp3);
        // R_i = (q_i * H_p(P_i)) + w_i*I
        ge_double_scalarmult_precomp_vartime(&tmp2, &sig[i].r, &tmp3, &sig[i].c, image_pre);
        // R_i in byte form
        ge_tobytes(&buf->ab[i].b, &tmp2);
        // Add c_i to the running sum of c_i's
        sc_add(&sum, &sum, &sig[i].c);
      }
    }
    // Keccak1600 Hash (H_s) of the buffer of all L&R, which is then converted to a 32 byte integer modulo l.
    hash_to_scalar(buf.get(), rs_comm_size(pubs_count), h);
    // c_s = c-sum(c_i)  where i != s
    sc_sub(&sig[sec_index].c, &h, &sum);
    // Close the loop: r_s = q_s - c_s*x
    // where x is the real output private key. This is the same x used to generate the key image
    sc_mulsub(&sig[sec_index].r, &sig[sec_index].c, &unwrap(sec), &k);
  }
 
  bool crypto_ops::check_ring_signature(const hash &prefix_hash, const key_image &image,
    const public_key *const *pubs, size_t pubs_count,
    const signature *sig) {
    size_t i;
    ge_p3 image_unp;
    ge_dsmp image_pre;
    ec_scalar sum, h;
    boost::shared_ptr<rs_comm> buf(reinterpret_cast<rs_comm *>(malloc(rs_comm_size(pubs_count))), free);
    if (!buf)
      return false;
 
#if !defined(NDEBUG)
    for (i = 0; i < pubs_count; i++) {
      assert(check_key(*pubs[i]));
    }
#endif
    if (ge_frombytes_vartime(&image_unp, &image) != 0) {
      return false;
    }
    ge_dsm_precomp(image_pre, &image_unp);
    sc_0(&sum);
    buf->h = prefix_hash;
    for (i = 0; i < pubs_count; i++) {
      ge_p2 tmp2;
      ge_p3 tmp3;
      if (sc_check(&sig[i].c) != 0 || sc_check(&sig[i].r) != 0) {
        return false;
      }
      if (ge_frombytes_vartime(&tmp3, &*pubs[i]) != 0) {
        return false;
      }
      ge_double_scalarmult_base_vartime(&tmp2, &sig[i].c, &tmp3, &sig[i].r);
      ge_tobytes(&buf->ab[i].a, &tmp2);
      hash_to_ec(*pubs[i], tmp3);
      ge_double_scalarmult_precomp_vartime(&tmp2, &sig[i].r, &tmp3, &sig[i].c, image_pre);
      ge_tobytes(&buf->ab[i].b, &tmp2);
      sc_add(&sum, &sum, &sig[i].c);
    }
    hash_to_scalar(buf.get(), rs_comm_size(pubs_count), h);
    sc_sub(&h, &h, &sum);
    return sc_isnonzero(&h) == 0;
  }
 
  std::string crypto_ops::sign_message(const std::string &message, const std::string &privateKey) {
 
    if(privateKey.size() != 32) {
      return std::string("");
    }
 
    ed25519_public_key pKey;
    ed25519_publickey((unsigned char*)privateKey.data(), pKey);
 
    ed25519_signature signature;
    ed25519_sign((unsigned char*)message.data(), message.size(), (unsigned char*)privateKey.data(), pKey, signature);
 
    return std::string(reinterpret_cast<char const*>(signature), 64);
  }
 
  bool crypto_ops::verify_signature(const std::string &message, const std::string &publicKey, const std::string &signature) {
 
    if(publicKey.size() != 32 || signature.size() !=64) {
      return false;
    }
    return ed25519_sign_open((unsigned char*)message.data(), message.size(), (unsigned char*)publicKey.data(), (unsigned char*)signature.data()) == 0;
  }
 
  bool crypto_ops::verify_signature(const std::string &message, const std::vector<std::string> publicKey, const std::string &signature) {
 
    if(publicKey.empty() || signature.size() != 64) {
      return false;
    }
 
    bool result = false;
    for(auto key : publicKey) {
      if(ed25519_sign_open((unsigned char*)message.data(), message.size(), (unsigned char*)key.data(), (unsigned char*)signature.data()) == 0) {
        result = true;
        break;
      }
    }
 
    return result;
  }
 
  std::vector<std::string> crypto_ops::create_ed25519_keypair() {
    ed25519_secret_key sk, pk;
    ed25519_randombytes_unsafe(sk, sizeof(ed25519_secret_key));
 
    ed25519_publickey(sk, pk);
 
    std::vector<std::string> result = {
            boost::algorithm::hex(std::string(reinterpret_cast<char const*>(sk), 32)),
            boost::algorithm::hex(std::string(reinterpret_cast<char const*>(pk), 32))
    };
 
    return result;
  }
}