.. 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