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gerrit-public.fairphone.software / platform / external / python / cryptography / 00734a5e50038d900cc3712dfc2631af1c0ed35f / . / docs / hazmat / primitives / asymmetric / ec.rst
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.. hazmat::
Elliptic curve cryptography
===========================
.. module:: cryptography.hazmat.primitives.asymmetric.ec
.. function:: generate_private_key(curve, backend)
.. versionadded:: 0.5
Generate a new private key on ``curve`` for use with ``backend``.
:param curve: An instance of :class:`EllipticCurve`.
:param backend: An instance of
:class:`~cryptography.hazmat.backends.interfaces.EllipticCurveBackend`.
:returns: A new instance of :class:`EllipticCurvePrivateKey`.
.. function:: derive_private_key(private_value, curve, backend)
.. versionadded:: 1.6
Derive a private key from ``private_value`` on ``curve`` for use with
``backend``.
:param int private_value: The secret scalar value.
:param curve: An instance of :class:`EllipticCurve`.
:param backend: An instance of
:class:`~cryptography.hazmat.backends.interfaces.EllipticCurveBackend`.
:returns: A new instance of :class:`EllipticCurvePrivateKey`.
Elliptic Curve Signature Algorithms
-----------------------------------
.. class:: ECDSA(algorithm)
.. versionadded:: 0.5
The ECDSA signature algorithm first standardized in NIST publication
`FIPS 186-3`_, and later in `FIPS 186-4`_.
:param algorithm: An instance of
:class:`~cryptography.hazmat.primitives.hashes.HashAlgorithm`.
.. doctest::
>>> from cryptography.hazmat.backends import default_backend
>>> from cryptography.hazmat.primitives import hashes
>>> from cryptography.hazmat.primitives.asymmetric import ec
>>> private_key = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... )
>>> data = b"this is some data I'd like to sign"
>>> signature = private_key.sign(
... data,
... ec.ECDSA(hashes.SHA256())
... )
The ``signature`` is a ``bytes`` object, whose contents is DER encoded as
described in :rfc:`3279`. This can be decoded using
:func:`~cryptography.hazmat.primitives.asymmetric.utils.decode_dss_signature`.
If your data is too large to be passed in a single call, you can hash it
separately and pass that value using
:class:`~cryptography.hazmat.primitives.asymmetric.utils.Prehashed`.
.. doctest::
>>> from cryptography.hazmat.primitives.asymmetric import utils
>>> chosen_hash = hashes.SHA256()
>>> hasher = hashes.Hash(chosen_hash, default_backend())
>>> hasher.update(b"data & ")
>>> hasher.update(b"more data")
>>> digest = hasher.finalize()
>>> sig = private_key.sign(
... digest,
... ec.ECDSA(utils.Prehashed(chosen_hash))
... )
Verification requires the public key, the signature itself, the signed
data, and knowledge of the hashing algorithm that was used when producing
the signature:
>>> public_key = private_key.public_key()
>>> public_key.verify(signature, data, ec.ECDSA(hashes.SHA256()))
If the signature is not valid, an
:class:`~cryptography.exceptions.InvalidSignature` exception will be raised.
If your data is too large to be passed in a single call, you can hash it
separately and pass that value using
:class:`~cryptography.hazmat.primitives.asymmetric.utils.Prehashed`.
.. doctest::
>>> chosen_hash = hashes.SHA256()
>>> hasher = hashes.Hash(chosen_hash, default_backend())
>>> hasher.update(b"data & ")
>>> hasher.update(b"more data")
>>> digest = hasher.finalize()
>>> public_key.verify(
... sig,
... digest,
... ec.ECDSA(utils.Prehashed(chosen_hash))
... )
.. note::
Although in this case the public key was derived from the private one,
in a typical setting you will not possess the private key. The
`Key loading`_ section explains how to load the public key from other
sources.
.. class:: EllipticCurvePrivateNumbers(private_value, public_numbers)
.. versionadded:: 0.5
The collection of integers that make up an EC private key.
.. attribute:: public_numbers
:type: :class:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurvePublicNumbers`
The :class:`EllipticCurvePublicNumbers` which makes up the EC public
key associated with this EC private key.
.. attribute:: private_value
:type: int
The private value.
.. method:: private_key(backend)
Convert a collection of numbers into a private key suitable for doing
actual cryptographic operations.
:param backend: An instance of
:class:`~cryptography.hazmat.backends.interfaces.EllipticCurveBackend`.
:returns: A new instance of :class:`EllipticCurvePrivateKey`.
.. class:: EllipticCurvePublicNumbers(x, y, curve)
.. warning::
The point represented by this object is not validated in any way until
:meth:`EllipticCurvePublicNumbers.public_key` is called and may not
represent a valid point on the curve. You should not attempt to perform
any computations using the values from this class until you have either
validated it yourself or called ``public_key()`` successfully.
.. versionadded:: 0.5
The collection of integers that make up an EC public key.
.. attribute:: curve
:type: :class:`EllipticCurve`
The elliptic curve for this key.
.. attribute:: x
:type: int
The affine x component of the public point used for verifying.
.. attribute:: y
:type: int
The affine y component of the public point used for verifying.
.. method:: public_key(backend)
Convert a collection of numbers into a public key suitable for doing
actual cryptographic operations.
:param backend: An instance of
:class:`~cryptography.hazmat.backends.interfaces.EllipticCurveBackend`.
:raises ValueError: Raised if the point is invalid for the curve.
:returns: A new instance of :class:`EllipticCurvePublicKey`.
.. method:: encode_point()
.. warning::
This method is deprecated as of version 2.5. Callers should migrate
to using
:meth:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurvePublicKey.public_bytes`.
.. versionadded:: 1.1
Encodes an elliptic curve point to a byte string as described in
`SEC 1 v2.0`_ section 2.3.3. This method only supports uncompressed
points.
:return bytes: The encoded point.
.. classmethod:: from_encoded_point(curve, data)
.. versionadded:: 1.1
.. note::
This has been deprecated in favor of
:meth:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurvePublicKey.from_encoded_point`
Decodes a byte string as described in `SEC 1 v2.0`_ section 2.3.3 and
returns an :class:`EllipticCurvePublicNumbers`. This method only
supports uncompressed points.
:param curve: An
:class:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurve`
instance.
:param bytes data: The serialized point byte string.
:returns: An :class:`EllipticCurvePublicNumbers` instance.
:raises ValueError: Raised on invalid point type or data length.
:raises TypeError: Raised when curve is not an
:class:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurve`.
Elliptic Curve Key Exchange algorithm
-------------------------------------
.. class:: ECDH()
.. versionadded:: 1.1
The Elliptic Curve Diffie-Hellman Key Exchange algorithm first standardized
in NIST publication `800-56A`_, and later in `800-56Ar2`_.
For most applications the ``shared_key`` should be passed to a key
derivation function. This allows mixing of additional information into the
key, derivation of multiple keys, and destroys any structure that may be
present.
.. warning::
This example does not give `forward secrecy`_ and is only provided as a
demonstration of the basic Diffie-Hellman construction. For real world
applications always use the ephemeral form described after this example.
.. doctest::
>>> from cryptography.hazmat.backends import default_backend
>>> from cryptography.hazmat.primitives import hashes
>>> from cryptography.hazmat.primitives.asymmetric import ec
>>> from cryptography.hazmat.primitives.kdf.hkdf import HKDF
>>> # Generate a private key for use in the exchange.
>>> server_private_key = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... )
>>> # In a real handshake the peer is a remote client. For this
>>> # example we'll generate another local private key though.
>>> peer_private_key = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... )
>>> shared_key = server_private_key.exchange(
... ec.ECDH(), peer_private_key.public_key())
>>> # Perform key derivation.
>>> derived_key = HKDF(
... algorithm=hashes.SHA256(),
... length=32,
... salt=None,
... info=b'handshake data',
... backend=default_backend()
... ).derive(shared_key)
>>> # And now we can demonstrate that the handshake performed in the
>>> # opposite direction gives the same final value
>>> same_shared_key = peer_private_key.exchange(
... ec.ECDH(), server_private_key.public_key())
>>> # Perform key derivation.
>>> same_derived_key = HKDF(
... algorithm=hashes.SHA256(),
... length=32,
... salt=None,
... info=b'handshake data',
... backend=default_backend()
... ).derive(same_shared_key)
>>> derived_key == same_derived_key
True
ECDHE (or EECDH), the ephemeral form of this exchange, is **strongly
preferred** over simple ECDH and provides `forward secrecy`_ when used.
You must generate a new private key using :func:`generate_private_key` for
each :meth:`~EllipticCurvePrivateKey.exchange` when performing an ECDHE key
exchange. An example of the ephemeral form:
.. doctest::
>>> from cryptography.hazmat.backends import default_backend
>>> from cryptography.hazmat.primitives import hashes
>>> from cryptography.hazmat.primitives.asymmetric import ec
>>> from cryptography.hazmat.primitives.kdf.hkdf import HKDF
>>> # Generate a private key for use in the exchange.
>>> private_key = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... )
>>> # In a real handshake the peer_public_key will be received from the
>>> # other party. For this example we'll generate another private key
>>> # and get a public key from that.
>>> peer_public_key = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... ).public_key()
>>> shared_key = private_key.exchange(ec.ECDH(), peer_public_key)
>>> # Perform key derivation.
>>> derived_key = HKDF(
... algorithm=hashes.SHA256(),
... length=32,
... salt=None,
... info=b'handshake data',
... backend=default_backend()
... ).derive(shared_key)
>>> # For the next handshake we MUST generate another private key.
>>> private_key_2 = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... )
>>> peer_public_key_2 = ec.generate_private_key(
... ec.SECP384R1(), default_backend()
... ).public_key()
>>> shared_key_2 = private_key_2.exchange(ec.ECDH(), peer_public_key_2)
>>> derived_key_2 = HKDF(
... algorithm=hashes.SHA256(),
... length=32,
... salt=None,
... info=b'handshake data',
... backend=default_backend()
... ).derive(shared_key_2)
Elliptic Curves
---------------
Elliptic curves provide equivalent security at much smaller key sizes than
other asymmetric cryptography systems such as RSA or DSA. For many operations
elliptic curves are also significantly faster; `elliptic curve diffie-hellman
is faster than diffie-hellman`_.
.. note::
Curves with a size of `less than 224 bits`_ should not be used. You should
strongly consider using curves of at least 224 :term:`bits`.
Generally the NIST prime field ("P") curves are significantly faster than the
other types suggested by NIST at both signing and verifying with ECDSA.
Prime fields also `minimize the number of security concerns for elliptic-curve
cryptography`_. However, there is `some concern`_ that both the prime field and
binary field ("B") NIST curves may have been weakened during their generation.
Currently `cryptography` only supports NIST curves, none of which are
considered "safe" by the `SafeCurves`_ project run by Daniel J. Bernstein and
Tanja Lange.
All named curves are instances of :class:`EllipticCurve`.
.. class:: SECP256R1
.. versionadded:: 0.5
SECG curve ``secp256r1``. Also called NIST P-256.
.. class:: SECP384R1
.. versionadded:: 0.5
SECG curve ``secp384r1``. Also called NIST P-384.
.. class:: SECP521R1
.. versionadded:: 0.5
SECG curve ``secp521r1``. Also called NIST P-521.
.. class:: SECP224R1
.. versionadded:: 0.5
SECG curve ``secp224r1``. Also called NIST P-224.
.. class:: SECP192R1
.. versionadded:: 0.5
SECG curve ``secp192r1``. Also called NIST P-192.
.. class:: SECP256K1
.. versionadded:: 0.9
SECG curve ``secp256k1``.
.. class:: BrainpoolP256R1
.. versionadded:: 2.2
Brainpool curve specified in :rfc:`5639`. These curves are discouraged
for new systems.
.. class:: BrainpoolP384R1
.. versionadded:: 2.2
Brainpool curve specified in :rfc:`5639`. These curves are discouraged
for new systems.
.. class:: BrainpoolP512R1
.. versionadded:: 2.2
Brainpool curve specified in :rfc:`5639`. These curves are discouraged
for new systems.
.. class:: SECT571K1
.. versionadded:: 0.5
SECG curve ``sect571k1``. Also called NIST K-571. These binary curves are
discouraged for new systems.
.. class:: SECT409K1
.. versionadded:: 0.5
SECG curve ``sect409k1``. Also called NIST K-409. These binary curves are
discouraged for new systems.
.. class:: SECT283K1
.. versionadded:: 0.5
SECG curve ``sect283k1``. Also called NIST K-283. These binary curves are
discouraged for new systems.
.. class:: SECT233K1
.. versionadded:: 0.5
SECG curve ``sect233k1``. Also called NIST K-233. These binary curves are
discouraged for new systems.
.. class:: SECT163K1
.. versionadded:: 0.5
SECG curve ``sect163k1``. Also called NIST K-163. These binary curves are
discouraged for new systems.
.. class:: SECT571R1
.. versionadded:: 0.5
SECG curve ``sect571r1``. Also called NIST B-571. These binary curves are
discouraged for new systems.
.. class:: SECT409R1
.. versionadded:: 0.5
SECG curve ``sect409r1``. Also called NIST B-409. These binary curves are
discouraged for new systems.
.. class:: SECT283R1
.. versionadded:: 0.5
SECG curve ``sect283r1``. Also called NIST B-283. These binary curves are
discouraged for new systems.
.. class:: SECT233R1
.. versionadded:: 0.5
SECG curve ``sect233r1``. Also called NIST B-233. These binary curves are
discouraged for new systems.
.. class:: SECT163R2
.. versionadded:: 0.5
SECG curve ``sect163r2``. Also called NIST B-163. These binary curves are
discouraged for new systems.
Key Interfaces
~~~~~~~~~~~~~~
.. class:: EllipticCurve
.. versionadded:: 0.5
A named elliptic curve.
.. attribute:: name
:type: str
The name of the curve. Usually the name used for the ASN.1 OID such as
``secp256k1``.
.. attribute:: key_size
:type: int
Size (in :term:`bits`) of a secret scalar for the curve (as generated
by :func:`generate_private_key`).
.. class:: EllipticCurveSignatureAlgorithm
.. versionadded:: 0.5
.. versionchanged:: 1.6
:class:`~cryptography.hazmat.primitives.asymmetric.utils.Prehashed`
can now be used as an ``algorithm``.
A signature algorithm for use with elliptic curve keys.
.. attribute:: algorithm
:type: :class:`~cryptography.hazmat.primitives.hashes.HashAlgorithm` or
:class:`~cryptography.hazmat.primitives.asymmetric.utils.Prehashed`
The digest algorithm to be used with the signature scheme.
.. class:: EllipticCurvePrivateKey
.. versionadded:: 0.5
An elliptic curve private key for use with an algorithm such as `ECDSA`_ or
`EdDSA`_. An elliptic curve private key that is not an
:term:`opaque key` also implements
:class:`EllipticCurvePrivateKeyWithSerialization` to provide serialization
methods.
.. method:: exchange(algorithm, peer_public_key)
.. versionadded:: 1.1
Performs a key exchange operation using the provided algorithm with
the peer's public key.
For most applications the ``shared_key`` should be passed to a key
derivation function. This allows mixing of additional information into the
key, derivation of multiple keys, and destroys any structure that may be
present.
:param algorithm: The key exchange algorithm, currently only
:class:`~cryptography.hazmat.primitives.asymmetric.ec.ECDH` is
supported.
:param EllipticCurvePublicKey peer_public_key: The public key for the
peer.
:returns bytes: A shared key.
.. method:: public_key()
:return: :class:`EllipticCurvePublicKey`
The EllipticCurvePublicKey object for this private key.
.. method:: sign(data, signature_algorithm)
.. versionadded:: 1.5
Sign one block of data which can be verified later by others using the
public key.
:param bytes data: The message string to sign.
:param signature_algorithm: An instance of
:class:`EllipticCurveSignatureAlgorithm`, such as :class:`ECDSA`.
:return bytes: Signature.
.. attribute:: key_size
.. versionadded:: 1.9
:type: int
Size (in :term:`bits`) of a secret scalar for the curve (as generated
by :func:`generate_private_key`).
.. class:: EllipticCurvePrivateKeyWithSerialization
.. versionadded:: 0.8
This interface contains additional methods relating to serialization.
Any object with this interface also has all the methods from
:class:`EllipticCurvePrivateKey`.
.. method:: private_numbers()
Create a :class:`EllipticCurvePrivateNumbers` object.
:returns: An :class:`EllipticCurvePrivateNumbers` instance.
.. method:: private_bytes(encoding, format, encryption_algorithm)
Allows serialization of the key to bytes. Encoding (
:attr:`~cryptography.hazmat.primitives.serialization.Encoding.PEM` or
:attr:`~cryptography.hazmat.primitives.serialization.Encoding.DER`),
format (
:attr:`~cryptography.hazmat.primitives.serialization.PrivateFormat.TraditionalOpenSSL`
or
:attr:`~cryptography.hazmat.primitives.serialization.PrivateFormat.PKCS8`)
and encryption algorithm (such as
:class:`~cryptography.hazmat.primitives.serialization.BestAvailableEncryption`
or :class:`~cryptography.hazmat.primitives.serialization.NoEncryption`)
are chosen to define the exact serialization.
:param encoding: A value from the
:class:`~cryptography.hazmat.primitives.serialization.Encoding` enum.
:param format: A value from the
:class:`~cryptography.hazmat.primitives.serialization.PrivateFormat` enum.
:param encryption_algorithm: An instance of an object conforming to the
:class:`~cryptography.hazmat.primitives.serialization.KeySerializationEncryption`
interface.
:return bytes: Serialized key.
.. class:: EllipticCurvePublicKey
.. versionadded:: 0.5
An elliptic curve public key.
.. attribute:: curve
:type: :class:`EllipticCurve`
The elliptic curve for this key.
.. method:: public_numbers()
Create a :class:`EllipticCurvePublicNumbers` object.
:returns: An :class:`EllipticCurvePublicNumbers` instance.
.. method:: public_bytes(encoding, format)
Allows serialization of the key data to bytes. When encoding the public
key the encodings (
:attr:`~cryptography.hazmat.primitives.serialization.Encoding.PEM`,
:attr:`~cryptography.hazmat.primitives.serialization.Encoding.DER`) and
format (
:attr:`~cryptography.hazmat.primitives.serialization.PublicFormat.SubjectPublicKeyInfo`)
are chosen to define the exact serialization. When encoding the point
the encoding
:attr:`~cryptography.hazmat.primitives.serialization.Encoding.X962`
should be used with the formats (
:attr:`~cryptography.hazmat.primitives.serialization.PublicFormat.UncompressedPoint`
or
:attr:`~cryptography.hazmat.primitives.serialization.PublicFormat.CompressedPoint`
).
:param encoding: A value from the
:class:`~cryptography.hazmat.primitives.serialization.Encoding` enum.
:param format: A value from the
:class:`~cryptography.hazmat.primitives.serialization.PublicFormat` enum.
:return bytes: Serialized data.
.. method:: verify(signature, data, signature_algorithm)
.. versionadded:: 1.5
Verify one block of data was signed by the private key associated
with this public key.
:param bytes signature: The signature to verify.
:param bytes data: The message string that was signed.
:param signature_algorithm: An instance of
:class:`EllipticCurveSignatureAlgorithm`.
:raises cryptography.exceptions.InvalidSignature: If the signature does
not validate.
.. attribute:: key_size
.. versionadded:: 1.9
:type: int
Size (in :term:`bits`) of a secret scalar for the curve (as generated
by :func:`generate_private_key`).
.. classmethod:: from_encoded_point(curve, data)
.. versionadded:: 2.5
Decodes a byte string as described in `SEC 1 v2.0`_ section 2.3.3 and
returns an :class:`EllipticCurvePublicKey`. This class method supports
compressed points.
:param curve: An
:class:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurve`
instance.
:param bytes data: The serialized point byte string.
:returns: An :class:`EllipticCurvePublicKey` instance.
:raises ValueError: Raised when an invalid point is supplied.
:raises TypeError: Raised when curve is not an
:class:`~cryptography.hazmat.primitives.asymmetric.ec.EllipticCurve`.
.. class:: EllipticCurvePublicKeyWithSerialization
.. versionadded:: 0.6
Alias for :class:`EllipticCurvePublicKey`.
Serialization
~~~~~~~~~~~~~
This sample demonstrates how to generate a private key and serialize it.
.. doctest::
>>> from cryptography.hazmat.backends import default_backend
>>> from cryptography.hazmat.primitives import hashes
>>> from cryptography.hazmat.primitives.asymmetric import ec
>>> from cryptography.hazmat.primitives import serialization
>>> private_key = ec.generate_private_key(ec.SECP384R1(), default_backend())
>>> serialized_private = private_key.private_bytes(
... encoding=serialization.Encoding.PEM,
... format=serialization.PrivateFormat.PKCS8,
... encryption_algorithm=serialization.BestAvailableEncryption(b'testpassword')
... )
>>> serialized_private.splitlines()[0]
b'-----BEGIN ENCRYPTED PRIVATE KEY-----'
You can also serialize the key without a password, by relying on
:class:`~cryptography.hazmat.primitives.serialization.NoEncryption`.
The public key is serialized as follows:
.. doctest::
>>> public_key = private_key.public_key()
>>> serialized_public = public_key.public_bytes(
... encoding=serialization.Encoding.PEM,
... format=serialization.PublicFormat.SubjectPublicKeyInfo
... )
>>> serialized_public.splitlines()[0]
b'-----BEGIN PUBLIC KEY-----'
This is the part that you would normally share with the rest of the world.
Key loading
~~~~~~~~~~~
This extends the sample in the previous section, assuming that the variables
``serialized_private`` and ``serialized_public`` contain the respective keys
in PEM format.
.. doctest::
>>> loaded_public_key = serialization.load_pem_public_key(
... serialized_public,
... backend=default_backend()
... )
>>> loaded_private_key = serialization.load_pem_private_key(
... serialized_private,
... # or password=None, if in plain text
... password=b'testpassword',
... backend=default_backend()
... )
Elliptic Curve Object Identifiers
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. class:: EllipticCurveOID
.. versionadded:: 2.4
.. attribute:: SECP192R1
Corresponds to the dotted string ``"1.2.840.10045.3.1.1"``.
.. attribute:: SECP224R1
Corresponds to the dotted string ``"1.3.132.0.33"``.
.. attribute:: SECP256K1
Corresponds to the dotted string ``"1.3.132.0.10"``.
.. attribute:: SECP256R1
Corresponds to the dotted string ``"1.2.840.10045.3.1.7"``.
.. attribute:: SECP384R1
Corresponds to the dotted string ``"1.3.132.0.34"``.
.. attribute:: SECP521R1
Corresponds to the dotted string ``"1.3.132.0.35"``.
.. attribute:: BRAINPOOLP256R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.36.3.3.2.8.1.1.7"``.
.. attribute:: BRAINPOOLP384R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.36.3.3.2.8.1.1.11"``.
.. attribute:: BRAINPOOLP512R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.36.3.3.2.8.1.1.13"``.
.. attribute:: SECT163K1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.1"``.
.. attribute:: SECT163R2
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.15"``.
.. attribute:: SECT233K1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.26"``.
.. attribute:: SECT233R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.27"``.
.. attribute:: SECT283K1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.16"``.
.. attribute:: SECT283R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.17"``.
.. attribute:: SECT409K1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.36"``.
.. attribute:: SECT409R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.37"``.
.. attribute:: SECT571K1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.38"``.
.. attribute:: SECT571R1
.. versionadded:: 2.5
Corresponds to the dotted string ``"1.3.132.0.39"``.
.. _`FIPS 186-3`: https://csrc.nist.gov/csrc/media/publications/fips/186/3/archive/2009-06-25/documents/fips_186-3.pdf
.. _`FIPS 186-4`: https://csrc.nist.gov/publications/detail/fips/186/4/final
.. _`800-56A`: https://csrc.nist.gov/publications/detail/sp/800-56a/revised/archive/2007-03-14
.. _`800-56Ar2`: https://csrc.nist.gov/publications/detail/sp/800-56a/rev-2/final
.. _`some concern`: https://crypto.stackexchange.com/questions/10263/should-we-trust-the-nist-recommended-ecc-parameters
.. _`less than 224 bits`: https://www.cosic.esat.kuleuven.be/ecrypt/ecrypt2/documents/D.SPA.20.pdf
.. _`elliptic curve diffie-hellman is faster than diffie-hellman`: https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1100&context=cseconfwork
.. _`minimize the number of security concerns for elliptic-curve cryptography`: https://cr.yp.to/ecdh/curve25519-20060209.pdf
.. _`SafeCurves`: https://safecurves.cr.yp.to/
.. _`ECDSA`: https://en.wikipedia.org/wiki/ECDSA
.. _`EdDSA`: https://en.wikipedia.org/wiki/EdDSA
.. _`forward secrecy`: https://en.wikipedia.org/wiki/Forward_secrecy
.. _`SEC 1 v2.0`: http://www.secg.org/sec1-v2.pdf