ec_group.cpp

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/*
* ECC Domain Parameters
*
* (C) 2007 Falko Strenzke, FlexSecure GmbH
* (C) 2008,2018 Jack Lloyd
* (C) 2018 Tobias Niemann
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
 
#include <botan/ec_group.h>
#include <botan/internal/point_mul.h>
#include <botan/internal/primality.h>
#include <botan/ber_dec.h>
#include <botan/der_enc.h>
#include <botan/pem.h>
#include <botan/reducer.h>
#include <botan/mutex.h>
#include <botan/rng.h>
#include <vector>
 
#if defined(BOTAN_HAS_EC_HASH_TO_CURVE)
  #include <botan/internal/ec_h2c.h>
#endif
 
namespace Botan {
 
class EC_Group_Data final
   {
   public:
 
      EC_Group_Data(const BigInt& p,
                    const BigInt& a,
                    const BigInt& b,
                    const BigInt& g_x,
                    const BigInt& g_y,
                    const BigInt& order,
                    const BigInt& cofactor,
                    const OID& oid,
                    EC_Group_Source source) :
         m_curve(p, a, b),
         m_base_point(m_curve, g_x, g_y),
         m_g_x(g_x),
         m_g_y(g_y),
         m_order(order),
         m_cofactor(cofactor),
         m_mod_order(order),
         m_base_mult(m_base_point, m_mod_order),
         m_oid(oid),
         m_p_bits(p.bits()),
         m_order_bits(order.bits()),
         m_a_is_minus_3(a == p - 3),
         m_a_is_zero(a.is_zero()),
         m_source(source)
         {
         }
 
      bool params_match(const BigInt& p, const BigInt& a, const BigInt& b,
                        const BigInt& g_x, const BigInt& g_y,
                        const BigInt& order, const BigInt& cofactor) const
         {
         return (this->p() == p &&
                 this->a() == a &&
                 this->b() == b &&
                 this->order() == order &&
                 this->cofactor() == cofactor &&
                 this->g_x() == g_x &&
                 this->g_y() == g_y);
         }
 
      bool params_match(const EC_Group_Data& other) const
         {
         return params_match(other.p(), other.a(), other.b(),
                             other.g_x(), other.g_y(),
                             other.order(), other.cofactor());
         }
 
      void set_oid(const OID& oid)
         {
         BOTAN_STATE_CHECK(m_oid.empty());
         m_oid = oid;
         }
 
      const OID& oid() const { return m_oid; }
      const BigInt& p() const { return m_curve.get_p(); }
      const BigInt& a() const { return m_curve.get_a(); }
      const BigInt& b() const { return m_curve.get_b(); }
      const BigInt& order() const { return m_order; }
      const BigInt& cofactor() const { return m_cofactor; }
      const BigInt& g_x() const { return m_g_x; }
      const BigInt& g_y() const { return m_g_y; }
 
      size_t p_bits() const { return m_p_bits; }
      size_t p_bytes() const { return (m_p_bits + 7) / 8; }
 
      size_t order_bits() const { return m_order_bits; }
      size_t order_bytes() const { return (m_order_bits + 7) / 8; }
 
      const CurveGFp& curve() const { return m_curve; }
      const PointGFp& base_point() const { return m_base_point; }
 
      bool a_is_minus_3() const { return m_a_is_minus_3; }
      bool a_is_zero() const { return m_a_is_zero; }
 
      BigInt mod_order(const BigInt& x) const { return m_mod_order.reduce(x); }
 
      BigInt square_mod_order(const BigInt& x) const
         {
         return m_mod_order.square(x);
         }
 
      BigInt multiply_mod_order(const BigInt& x, const BigInt& y) const
         {
         return m_mod_order.multiply(x, y);
         }
 
      BigInt multiply_mod_order(const BigInt& x, const BigInt& y, const BigInt& z) const
         {
         return m_mod_order.multiply(m_mod_order.multiply(x, y), z);
         }
 
      BigInt inverse_mod_order(const BigInt& x) const
         {
         return inverse_mod(x, m_order);
         }
 
      PointGFp blinded_base_point_multiply(const BigInt& k,
                                           RandomNumberGenerator& rng,
                                           std::vector<BigInt>& ws) const
         {
         return m_base_mult.mul(k, rng, m_order, ws);
         }
 
      EC_Group_Source source() const { return m_source; }
 
   private:
      CurveGFp m_curve;
      PointGFp m_base_point;
 
      BigInt m_g_x;
      BigInt m_g_y;
      BigInt m_order;
      BigInt m_cofactor;
      Modular_Reducer m_mod_order;
      PointGFp_Base_Point_Precompute m_base_mult;
      OID m_oid;
      size_t m_p_bits;
      size_t m_order_bits;
      bool m_a_is_minus_3;
      bool m_a_is_zero;
      EC_Group_Source m_source;
   };
 
class EC_Group_Data_Map final
   {
   public:
      EC_Group_Data_Map() {}
 
      size_t clear()
         {
         lock_guard_type<mutex_type> lock(m_mutex);
         size_t count = m_registered_curves.size();
         m_registered_curves.clear();
         return count;
         }
 
      std::shared_ptr<EC_Group_Data> lookup(const OID& oid)
         {
         lock_guard_type<mutex_type> lock(m_mutex);
 
         for(auto i : m_registered_curves)
            {
            if(i->oid() == oid)
               return i;
            }
 
         // Not found, check hardcoded data
         std::shared_ptr<EC_Group_Data> data = EC_Group::EC_group_info(oid);
 
         if(data)
            {
            for(auto curve : m_registered_curves)
               {
               if(curve->oid().empty() == true && curve->params_match(*data))
                  {
                  curve->set_oid(oid);
                  return curve;
                  }
               }
 
            m_registered_curves.push_back(data);
            return data;
            }
 
         // Nope, unknown curve
         return std::shared_ptr<EC_Group_Data>();
         }
 
      std::shared_ptr<EC_Group_Data> lookup_or_create(const BigInt& p,
                                                      const BigInt& a,
                                                      const BigInt& b,
                                                      const BigInt& g_x,
                                                      const BigInt& g_y,
                                                      const BigInt& order,
                                                      const BigInt& cofactor,
                                                      const OID& oid,
                                                      EC_Group_Source source)
         {
         lock_guard_type<mutex_type> lock(m_mutex);
 
         for(auto i : m_registered_curves)
            {
            /*
            * The params may be the same but you are trying to register under a
            * different OID than the one we are using, so using a different
            * group, since EC_Group's model assumes a single OID per group.
            */
            if(!oid.empty() && !i->oid().empty() && i->oid() != oid)
               {
               continue;
               }
 
            const bool same_oid = !oid.empty() && i->oid() == oid;
            const bool same_params = i->params_match(p, a, b, g_x, g_y, order, cofactor);
 
            /*
            * If the params and OID are the same then we are done, just return
            * the already registered curve obj.
            */
            if(same_params && same_oid)
               {
               return i;
               }
 
            /*
            * If same params and the new OID is empty, then that's ok too
            */
            if(same_params && oid.empty())
               {
               return i;
               }
 
            /*
            * Check for someone trying to reuse an already in-use OID
            */
            if(same_oid && !same_params)
               {
               throw Invalid_Argument("Attempting to register a curve using OID " + oid.to_string() +
                                      " but a distinct curve is already registered using that OID");
               }
 
            /*
            * If the same curve was previously created without an OID but is now
            * being registered again using an OID, save that OID.
            */
            if(same_params && i->oid().empty() && !oid.empty())
               {
               i->set_oid(oid);
               return i;
               }
            }
 
         /*
         Not found in current list, so we need to create a new entry
 
         If an OID is set, try to look up relative our static tables to detect a duplicate
         registration under an OID
         */
 
         std::shared_ptr<EC_Group_Data> new_group =
            std::make_shared<EC_Group_Data>(p, a, b, g_x, g_y, order, cofactor, oid, source);
 
         if(oid.has_value())
            {
            std::shared_ptr<EC_Group_Data> data = EC_Group::EC_group_info(oid);
            if(data != nullptr && !new_group->params_match(*data))
               throw Invalid_Argument("Attempting to register an EC group under OID of hardcoded group");
            }
         else
            {
            // Here try to use the order as a hint to look up the group id, to identify common groups
            const OID oid_from_store = EC_Group::EC_group_identity_from_order(order);
            if(oid_from_store.has_value())
               {
               std::shared_ptr<EC_Group_Data> data = EC_Group::EC_group_info(oid_from_store);
 
               /*
               If EC_group_identity_from_order returned an OID then looking up that OID
               must always return a result.
               */
               BOTAN_ASSERT_NOMSG(data != nullptr);
 
               /*
               It is possible (if unlikely) that someone is registering another group
               that happens to have an order equal to that of a well known group -
               so verify all values before assigning the OID.
               */
               if(new_group->params_match(*data))
                  {
                  new_group->set_oid(oid_from_store);
                  }
               }
            }
 
         m_registered_curves.push_back(new_group);
         return new_group;
         }
 
   private:
      mutex_type m_mutex;
      std::vector<std::shared_ptr<EC_Group_Data>> m_registered_curves;
   };
 
//static
EC_Group_Data_Map& EC_Group::ec_group_data()
   {
   /*
   * This exists purely to ensure the allocator is constructed before g_ec_data,
   * which ensures that its destructor runs after ~g_ec_data is complete.
   */
 
   static Allocator_Initializer g_init_allocator;
   static EC_Group_Data_Map g_ec_data;
   return g_ec_data;
   }
 
//static
size_t EC_Group::clear_registered_curve_data()
   {
   return ec_group_data().clear();
   }
 
//static
std::shared_ptr<EC_Group_Data>
EC_Group::load_EC_group_info(const char* p_str,
                             const char* a_str,
                             const char* b_str,
                             const char* g_x_str,
                             const char* g_y_str,
                             const char* order_str,
                             const OID& oid)
   {
   const BigInt p(p_str);
   const BigInt a(a_str);
   const BigInt b(b_str);
   const BigInt g_x(g_x_str);
   const BigInt g_y(g_y_str);
   const BigInt order(order_str);
   const BigInt cofactor(1); // implicit
 
   return std::make_shared<EC_Group_Data>(p, a, b, g_x, g_y, order, cofactor, oid, EC_Group_Source::Builtin);
   }
 
//static
std::shared_ptr<EC_Group_Data> EC_Group::BER_decode_EC_group(const uint8_t bits[], size_t len,
                                                             EC_Group_Source source)
   {
   BER_Decoder ber(bits, len);
   BER_Object obj = ber.get_next_object();
 
   if(obj.type() == ASN1_Type::Null)
      {
      throw Decoding_Error("Cannot handle ImplicitCA ECC parameters");
      }
   else if(obj.type() == ASN1_Type::ObjectId)
      {
      OID dom_par_oid;
      BER_Decoder(bits, len).decode(dom_par_oid);
      return ec_group_data().lookup(dom_par_oid);
      }
   else if(obj.type() == ASN1_Type::Sequence)
      {
      BigInt p, a, b, order, cofactor;
      std::vector<uint8_t> base_pt;
      std::vector<uint8_t> seed;
 
      BER_Decoder(bits, len)
         .start_sequence()
           .decode_and_check<size_t>(1, "Unknown ECC param version code")
           .start_sequence()
            .decode_and_check(OID("1.2.840.10045.1.1"),
                              "Only prime ECC fields supported")
             .decode(p)
           .end_cons()
           .start_sequence()
             .decode_octet_string_bigint(a)
             .decode_octet_string_bigint(b)
             .decode_optional_string(seed, ASN1_Type::BitString, ASN1_Type::BitString)
           .end_cons()
           .decode(base_pt, ASN1_Type::OctetString)
           .decode(order)
           .decode(cofactor)
         .end_cons()
         .verify_end();
 
      if(p.bits() < 64 || p.is_negative() || !is_bailie_psw_probable_prime(p))
         throw Decoding_Error("Invalid ECC p parameter");
 
      if(a.is_negative() || a >= p)
         throw Decoding_Error("Invalid ECC a parameter");
 
      if(b <= 0 || b >= p)
         throw Decoding_Error("Invalid ECC b parameter");
 
      if(order <= 0 || !is_bailie_psw_probable_prime(order))
         throw Decoding_Error("Invalid ECC order parameter");
 
      if(cofactor <= 0 || cofactor >= 16)
         throw Decoding_Error("Invalid ECC cofactor parameter");
 
      std::pair<BigInt, BigInt> base_xy = Botan::OS2ECP(base_pt.data(), base_pt.size(), p, a, b);
 
      return ec_group_data().lookup_or_create(p, a, b, base_xy.first, base_xy.second,
                                              order, cofactor, OID(), source);
      }
   else
      {
      throw Decoding_Error("Unexpected tag while decoding ECC domain params");
      }
   }
 
EC_Group::EC_Group()
   {
   }
 
EC_Group::~EC_Group()
   {
   // shared_ptr possibly freed here
   }
 
EC_Group::EC_Group(const OID& domain_oid)
   {
   this->m_data = ec_group_data().lookup(domain_oid);
   if(!this->m_data)
      throw Invalid_Argument("Unknown EC_Group " + domain_oid.to_string());
   }
 
EC_Group::EC_Group(const std::string& str)
   {
   if(str.empty())
      return; // no initialization / uninitialized
 
   try
      {
      const OID oid = OID::from_string(str);
      if(oid.has_value())
         m_data = ec_group_data().lookup(oid);
      }
   catch(...)
      {
      }
 
   if(m_data == nullptr)
      {
      if(str.size() > 30 && str.substr(0, 29) == "-----BEGIN EC PARAMETERS-----")
         {
         // OK try it as PEM ...
         secure_vector<uint8_t> ber = PEM_Code::decode_check_label(str, "EC PARAMETERS");
         this->m_data = BER_decode_EC_group(ber.data(), ber.size(), EC_Group_Source::ExternalSource);
         }
      }
 
   if(m_data == nullptr)
      throw Invalid_Argument("Unknown ECC group '" + str + "'");
   }
 
//static
EC_Group EC_Group::EC_Group_from_PEM(const std::string& pem)
   {
   const auto ber = PEM_Code::decode_check_label(pem, "EC PARAMETERS");
   return EC_Group(ber.data(), ber.size());
   }
 
EC_Group::EC_Group(const BigInt& p,
                   const BigInt& a,
                   const BigInt& b,
                   const BigInt& base_x,
                   const BigInt& base_y,
                   const BigInt& order,
                   const BigInt& cofactor,
                   const OID& oid)
   {
   m_data = ec_group_data().lookup_or_create(p, a, b, base_x, base_y, order, cofactor, oid,
                                             EC_Group_Source::ExternalSource);
   }
 
EC_Group::EC_Group(const uint8_t ber[], size_t ber_len)
   {
   m_data = BER_decode_EC_group(ber, ber_len, EC_Group_Source::ExternalSource);
   }
 
const EC_Group_Data& EC_Group::data() const
   {
   if(m_data == nullptr)
      throw Invalid_State("EC_Group uninitialized");
   return *m_data;
   }
 
bool EC_Group::a_is_minus_3() const
   {
   return data().a_is_minus_3();
   }
 
bool EC_Group::a_is_zero() const
   {
   return data().a_is_zero();
   }
 
size_t EC_Group::get_p_bits() const
   {
   return data().p_bits();
   }
 
size_t EC_Group::get_p_bytes() const
   {
   return data().p_bytes();
   }
 
size_t EC_Group::get_order_bits() const
   {
   return data().order_bits();
   }
 
size_t EC_Group::get_order_bytes() const
   {
   return data().order_bytes();
   }
 
const BigInt& EC_Group::get_p() const
   {
   return data().p();
   }
 
const BigInt& EC_Group::get_a() const
   {
   return data().a();
   }
 
const BigInt& EC_Group::get_b() const
   {
   return data().b();
   }
 
const PointGFp& EC_Group::get_base_point() const
   {
   return data().base_point();
   }
 
const BigInt& EC_Group::get_order() const
   {
   return data().order();
   }
 
const BigInt& EC_Group::get_g_x() const
   {
   return data().g_x();
   }
 
const BigInt& EC_Group::get_g_y() const
   {
   return data().g_y();
   }
 
const BigInt& EC_Group::get_cofactor() const
   {
   return data().cofactor();
   }
 
BigInt EC_Group::mod_order(const BigInt& k) const
   {
   return data().mod_order(k);
   }
 
BigInt EC_Group::square_mod_order(const BigInt& x) const
   {
   return data().square_mod_order(x);
   }
 
BigInt EC_Group::multiply_mod_order(const BigInt& x, const BigInt& y) const
   {
   return data().multiply_mod_order(x, y);
   }
 
BigInt EC_Group::multiply_mod_order(const BigInt& x, const BigInt& y, const BigInt& z) const
   {
   return data().multiply_mod_order(x, y, z);
   }
 
BigInt EC_Group::inverse_mod_order(const BigInt& x) const
   {
   return data().inverse_mod_order(x);
   }
 
const OID& EC_Group::get_curve_oid() const
   {
   return data().oid();
   }
 
EC_Group_Source EC_Group::source() const
   {
   return data().source();
   }
 
size_t EC_Group::point_size(PointGFp::Compression_Type format) const
   {
   // Hybrid and standard format are (x,y), compressed is y, +1 format byte
   if(format == PointGFp::COMPRESSED)
      return (1 + get_p_bytes());
   else
      return (1 + 2*get_p_bytes());
   }
 
PointGFp EC_Group::OS2ECP(const uint8_t bits[], size_t len) const
   {
   return Botan::OS2ECP(bits, len, data().curve());
   }
 
PointGFp EC_Group::point(const BigInt& x, const BigInt& y) const
   {
   #if 0
   BigInt l = (x*x*x + x*get_a() + get_b()) % get_p();
   BigInt r = (y*y) % get_p();
 
   if(l != r)
      {
      printf("invalid point in EC_Group::point\n");
      }
#endif
   // TODO: randomize the representation?
   return PointGFp(data().curve(), x, y);
   }
 
PointGFp EC_Group::point_multiply(const BigInt& x, const PointGFp& pt, const BigInt& y) const
   {
   PointGFp_Multi_Point_Precompute xy_mul(get_base_point(), pt);
   return xy_mul.multi_exp(x, y);
   }
 
PointGFp EC_Group::blinded_base_point_multiply(const BigInt& k,
                                               RandomNumberGenerator& rng,
                                               std::vector<BigInt>& ws) const
   {
   return data().blinded_base_point_multiply(k, rng, ws);
   }
 
BigInt EC_Group::blinded_base_point_multiply_x(const BigInt& k,
                                               RandomNumberGenerator& rng,
                                               std::vector<BigInt>& ws) const
   {
   const PointGFp pt = data().blinded_base_point_multiply(k, rng, ws);
 
   if(pt.is_zero())
      return BigInt::zero();
   return pt.get_affine_x();
   }
 
BigInt EC_Group::random_scalar(RandomNumberGenerator& rng) const
   {
   return BigInt::random_integer(rng, BigInt::one(), get_order());
   }
 
PointGFp EC_Group::blinded_var_point_multiply(const PointGFp& point,
                                              const BigInt& k,
                                              RandomNumberGenerator& rng,
                                              std::vector<BigInt>& ws) const
   {
   PointGFp_Var_Point_Precompute mul(point, rng, ws);
   return mul.mul(k, rng, get_order(), ws);
   }
 
PointGFp EC_Group::zero_point() const
   {
   return PointGFp(data().curve());
   }
 
PointGFp EC_Group::hash_to_curve(const std::string& hash_fn,
                                 const uint8_t input[],
                                 size_t input_len,
                                 const std::string& domain,
                                 bool random_oracle) const
   {
   return this->hash_to_curve(hash_fn,
                              input,
                              input_len,
                              reinterpret_cast<const uint8_t*>(domain.c_str()),
                              domain.size(),
                              random_oracle);
   }
 
PointGFp EC_Group::hash_to_curve(const std::string& hash_fn,
                                 const uint8_t input[],
                                 size_t input_len,
                                 const uint8_t domain_sep[],
                                 size_t domain_sep_len,
                                 bool random_oracle) const
   {
#if defined(BOTAN_HAS_EC_HASH_TO_CURVE)
 
   // Only have SSWU currently
   if(get_a().is_zero() || get_b().is_zero() || get_p() % 4 == 1)
      {
      throw Not_Implemented("EC_Group::hash_to_curve not available for this curve type");
      }
 
   return hash_to_curve_sswu(*this, hash_fn,
                             input, input_len,
                             domain_sep, domain_sep_len,
                             random_oracle);
 
#else
   BOTAN_UNUSED(hash_fn, random_oracle, input, input_len, domain_sep, domain_sep_len);
   throw Not_Implemented("EC_Group::hash_to_curve functionality not available in this configuration");
#endif
   }
 
std::vector<uint8_t>
EC_Group::DER_encode(EC_Group_Encoding form) const
   {
   std::vector<uint8_t> output;
 
   DER_Encoder der(output);
 
   if(form == EC_Group_Encoding::Explicit)
      {
      const size_t ecpVers1 = 1;
      const OID curve_type("1.2.840.10045.1.1"); // prime field
 
      const size_t p_bytes = get_p_bytes();
 
      der.start_sequence()
            .encode(ecpVers1)
            .start_sequence()
               .encode(curve_type)
               .encode(get_p())
            .end_cons()
            .start_sequence()
               .encode(BigInt::encode_1363(get_a(), p_bytes),
                       ASN1_Type::OctetString)
               .encode(BigInt::encode_1363(get_b(), p_bytes),
                       ASN1_Type::OctetString)
            .end_cons()
              .encode(get_base_point().encode(PointGFp::UNCOMPRESSED), ASN1_Type::OctetString)
            .encode(get_order())
            .encode(get_cofactor())
         .end_cons();
      }
   else if(form == EC_Group_Encoding::NamedCurve)
      {
      const OID oid = get_curve_oid();
      if(oid.empty())
         {
         throw Encoding_Error("Cannot encode EC_Group as OID because OID not set");
         }
      der.encode(oid);
      }
   else if(form == EC_Group_Encoding::ImplicitCA)
      {
      der.encode_null();
      }
   else
      {
      throw Internal_Error("EC_Group::DER_encode: Unknown encoding");
      }
 
   return output;
   }
 
std::string EC_Group::PEM_encode() const
   {
   const std::vector<uint8_t> der = DER_encode(EC_Group_Encoding::Explicit);
   return PEM_Code::encode(der, "EC PARAMETERS");
   }
 
bool EC_Group::operator==(const EC_Group& other) const
   {
   if(m_data == other.m_data)
      return true; // same shared rep
 
   return (get_p() == other.get_p() &&
           get_a() == other.get_a() &&
           get_b() == other.get_b() &&
           get_g_x() == other.get_g_x() &&
           get_g_y() == other.get_g_y() &&
           get_order() == other.get_order() &&
           get_cofactor() == other.get_cofactor());
   }
 
bool EC_Group::verify_public_element(const PointGFp& point) const
   {
   //check that public point is not at infinity
   if(point.is_zero())
      return false;
 
   //check that public point is on the curve
   if(point.on_the_curve() == false)
      return false;
 
   //check that public point has order q
   if((point * get_order()).is_zero() == false)
      return false;
 
   if(get_cofactor() > 1)
      {
      if((point * get_cofactor()).is_zero())
         return false;
      }
 
   return true;
   }
 
bool EC_Group::verify_group(RandomNumberGenerator& rng,
                            bool strong) const
   {
   const bool is_builtin = source() == EC_Group_Source::Builtin;
 
   if(is_builtin && !strong)
      return true;
 
   const BigInt& p = get_p();
   const BigInt& a = get_a();
   const BigInt& b = get_b();
   const BigInt& order = get_order();
   const PointGFp& base_point = get_base_point();
 
   if(p <= 3 || order <= 0)
      return false;
   if(a < 0 || a >= p)
      return false;
   if(b <= 0 || b >= p)
      return false;
 
   const size_t test_prob = 128;
   const bool is_randomly_generated = is_builtin;
 
   //check if field modulus is prime
   if(!is_prime(p, rng, test_prob, is_randomly_generated))
      {
      return false;
      }
 
   //check if order is prime
   if(!is_prime(order, rng, test_prob, is_randomly_generated))
      {
      return false;
      }
 
   //compute the discriminant: 4*a^3 + 27*b^2 which must be nonzero
   const Modular_Reducer mod_p(p);
 
   const BigInt discriminant = mod_p.reduce(
      mod_p.multiply(4, mod_p.cube(a)) +
      mod_p.multiply(27, mod_p.square(b)));
 
   if(discriminant == 0)
      {
      return false;
      }
 
   //check for valid cofactor
   if(get_cofactor() < 1)
      {
      return false;
      }
 
   //check if the base point is on the curve
   if(!base_point.on_the_curve())
      {
      return false;
      }
   if((base_point * get_cofactor()).is_zero())
      {
      return false;
      }
   //check if order of the base point is correct
   if(!(base_point * order).is_zero())
      {
      return false;
      }
 
   return true;
   }
 
}