Elliptic Curve Operations
============================

In addition to high level operations for signatures, key agreement,
and message encryption using elliptic curve cryptography, the library
contains lower level interfaces for performing operations such as
elliptic curve point multiplication.

Only curves over prime fields are supported.

Many of these functions take a workspace, either a vector of words or
a vector of BigInts. These are used to minimize memory allocations
during common operations.

.. warning::
   You should only use these interfaces if you know what you are doing.

.. cpp:class:: EC_Group

      .. cpp:function:: EC_Group(const OID& oid)

         Initialize an ``EC_Group`` using an OID referencing the curve
         parameters.

      .. cpp:function:: EC_Group(const std::string& name)

         Initialize an ``EC_Group`` using a name or OID (for example
         "secp256r1", or "1.2.840.10045.3.1.7")

      .. cpp:function:: 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 = OID())

          Initialize an elliptic curve group from the relevant parameters. This
          is used for example to create custom (application-specific) curves.

      .. cpp:function:: EC_Group(const std::vector<uint8_t>& ber_encoding)

         Initialize an ``EC_Group`` by decoding a DER encoded parameter block.

      .. cpp:function:: std::vector<uint8_t> DER_encode(EC_Group_Encoding form) const

         Return the DER encoding of this group.

      .. cpp:function:: std::string PEM_encode() const

         Return the PEM encoding of this group (base64 of DER encoding plus
         header/trailer).

      .. cpp:function:: bool a_is_minus_3() const

         Return true if the ``a`` parameter is congruent to -3 mod p.

      .. cpp:function:: bool a_is_zero() const

         Return true if the ``a`` parameter is congruent to 0 mod p.

      .. cpp:function:: size_t get_p_bits() const

         Return size of the prime in bits.

      .. cpp:function:: size_t get_p_bytes() const

         Return size of the prime in bytes.

      .. cpp:function:: size_t get_order_bits() const

         Return size of the group order in bits.

      .. cpp:function:: size_t get_order_bytes() const

         Return size of the group order in bytes.

      .. cpp:function:: const BigInt& get_p() const

         Return the prime modulus.

      .. cpp:function:: const BigInt& get_a() const

         Return the ``a`` parameter of the elliptic curve equation.

      .. cpp:function:: const BigInt& get_b() const

         Return the ``b`` parameter of the elliptic curve equation.

      .. cpp:function:: const PointGFp& get_base_point() const

         Return the groups base point element.

      .. cpp:function:: const BigInt& get_g_x() const

         Return the x coordinate of the base point element.

      .. cpp:function:: const BigInt& get_g_y() const

         Return the y coordinate of the base point element.

      .. cpp:function:: const BigInt& get_order() const

         Return the order of the group generated by the base point.

      .. cpp:function:: const BigInt& get_cofactor() const

         Return the cofactor of the curve. In most cases this will be 1.

      .. cpp:function:: BigInt mod_order(const BigInt& x) const

         Reduce argument ``x`` modulo the curve order.

      .. cpp:function:: BigInt inverse_mod_order(const BigInt& x) const

         Return inverse of argument ``x`` modulo the curve order.

      .. cpp:function:: BigInt multiply_mod_order(const BigInt& x, const BigInt& y) const

         Multiply ``x`` and ``y`` and reduce the result modulo the curve order.

      .. cpp:function:: bool verify_public_element(const PointGFp& y) const

         Return true if ``y`` seems to be a valid group element.

      .. cpp:function:: const OID& get_curve_oid() const

         Return the OID used to identify the curve. May be empty.

      .. cpp:function:: PointGFp point(const BigInt& x, const BigInt& y) const

         Create and return a point with affine elements ``x`` and ``y``. Note
         this function *does not* verify that ``x`` and ``y`` satisfy the curve
         equation.

      .. cpp:function:: PointGFp point_multiply(const BigInt& x, const PointGFp& pt, const BigInt& y) const

         Multi-exponentation. Returns base_point*x + pt*y. Not constant time.
         (Ordinarily used for signature verification.)

      .. cpp:function:: PointGFp blinded_base_point_multiply(const BigInt& k, \
                                            RandomNumberGenerator& rng, \
                                            std::vector<BigInt>& ws) const

         Return ``base_point*k`` in a way that attempts to resist side channels.

      .. cpp:function:: BigInt blinded_base_point_multiply_x(const BigInt& k, \
                                           RandomNumberGenerator& rng, \
                                           std::vector<BigInt>& ws) const

         Like `blinded_base_point_multiply` but returns only the x coordinate.

      .. cpp:function:: PointGFp blinded_var_point_multiply(const PointGFp& point, \
                                          const BigInt& k, \
                                          RandomNumberGenerator& rng, \
                                          std::vector<BigInt>& ws) const

         Return ``point*k`` in a way that attempts to resist side channels.

      .. cpp:function:: BigInt random_scalar(RandomNumberGenerator& rng) const

         Return a random scalar (ie an integer between 1 and the group order).

      .. cpp:function:: PointGFp zero_point() const

         Return the zero point (aka the point at infinity).

      .. cpp:function:: PointGFp OS2ECP(const uint8_t bits[], size_t len) const

         Decode a point from the binary encoding. This function verifies that
         the decoded point is a valid element on the curve.

      .. cpp:function:: bool verify_group(RandomNumberGenerator& rng, bool strong = false) const

         Attempt to verify the group seems valid.

      .. cpp:function:: static const std::set<std::string>& known_named_groups()

         Return a list of known groups, ie groups for which ``EC_Group(name)``
         will succeed.

.. cpp:class:: PointGFp

   Stores elliptic curve points in Jacobian representation.

   .. cpp:function:: std::vector<uint8_t> encode(PointGFp::Compression_Type format) const

      Encode a point in a way that can later be decoded with `EC_Group::OS2ECP`.

   .. cpp:function:: PointGFp& operator+=(const PointGFp& rhs)

      Point addition.

   .. cpp:function:: PointGFp& operator-=(const PointGFp& rhs)

      Point subtraction.

   .. cpp:function:: PointGFp& operator*=(const BigInt& scalar)

      Point multiplication using Montgomery ladder.

      .. warning::
         Prefer the blinded functions in ``EC_Group``

   .. cpp:function:: PointGFp& negate()

      Negate this point.

   .. cpp:function:: BigInt get_affine_x() const

      Return the affine ``x`` coordinate of the point.

   .. cpp:function:: BigInt get_affine_y() const

      Return the affine ``y`` coordinate of the point.

   .. cpp:function:: void force_affine()

      Convert the point to its equivalent affine coordinates. Throws
      if this is the point at infinity.

   .. cpp:function:: static void force_all_affine(std::vector<PointGFp>& points, \
                                                  secure_vector<word>& ws)

      Force several points to be affine at once. Uses Montgomery's
      trick to reduce number of inversions required, so this is much
      faster than calling ``force_affine`` on each point in sequence.

   .. cpp:function:: bool is_affine() const

      Return true if this point is in affine coordinates.

   .. cpp:function:: bool is_zero() const

      Return true if this point is zero (aka point at infinity).

   .. cpp:function:: bool on_the_curve() const

      Return true if this point is on the curve.

   .. cpp:function:: void randomize_repr(RandomNumberGenerator& rng)

      Randomize the point representation.

   .. cpp:function:: bool operator==(const PointGFp& other) const

      Point equality. This compares the affine representations.

   .. cpp:function:: void add(const PointGFp& other, std::vector<BigInt>& workspace)

      Point addition, taking a workspace.

   .. cpp:function:: void add_affine(const PointGFp& other, std::vector<BigInt>& workspace)

      Mixed (Jacobian+affine) addition, taking a workspace.

      .. warning::

         This function assumes that ``other`` is affine, if this is
         not correct the result will be invalid.

   .. cpp:function:: void mult2(std::vector<BigInt>& workspace)

      Point doubling.

   .. cpp:function:: void mult2i(size_t i, std::vector<BigInt>& workspace)

      Repeated point doubling.

   .. cpp:function:: PointGFp plus(const PointGFp& other, std::vector<BigInt>& workspace) const

      Point addition, returning the result.

   .. cpp:function:: PointGFp double_of(std::vector<BigInt>& workspace) const

      Point doubling, returning the result.

   .. cpp:function:: PointGFp zero() const

      Return the point at infinity



