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Python library for fast elliptic curve crypto

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.. image:: :target: :alt: PyPI

.. image:: :target: :alt: Travis CI

.. image:: :target: :alt: Documentation Status

.. contents::


This is a python package for doing fast elliptic curve cryptography, specifically digital signatures.


There is no nonce reuse, no branching on secret material, and all points are validated before any operations are performed on them. Timing side challenges are mitigated via Montgomery point multiplication. Nonces are generated per RFC6979_. The default curve used throughout the package is P256 which provides 128 bits of security. If you require a higher level of security you can specify the curve parameter in a method to use a curve over a bigger field e.g. P384. All that being said, crypto is tricky and I'm not beyond making mistakes. Please use a more established and reviewed library for security critical applications. Open an issue or email me if you see any security issue or risk with this library.

Python Versions Supported

The initial release of this package was targeted at python2.7. Earlier versions may work but have no guarantee of correctness or stability. As of release 1.2.1+ python3 is supported as well. Due to python2's EOL on January 1st 2020 release 2.x of this package only supports python3.5+.

Operating Systems Supported

This package is targeted at the Linux and MacOS operating systems. Due to the the dependency on the GMP C library building this package on Windows is difficult and no official support or distributions are provided for Windows OSes. See issue11_ for what users have done to get things building.

Supported Primitives

Curves over Prime Fields ~~~~~~~~~~~~~~~~~~~~~~~~

+---------------------------+-----------------------------------------+-------------+ | Name | Class | Proposed By | +===========================+=========================================+=============+ | P192 / secp192r1 | :code:

| NIST / NSA | +---------------------------+-----------------------------------------+-------------+ | P224 / secp224r1 | :code:
| NIST / NSA | +---------------------------+-----------------------------------------+-------------+ | P256 / secp256r1 | :code:
| NIST / NSA | +---------------------------+-----------------------------------------+-------------+ | P384 / secp384r1 | :code:
| NIST / NSA | +---------------------------+-----------------------------------------+-------------+ | P521 / secp521r1 | :code:
| NIST / NSA | +---------------------------+-----------------------------------------+-------------+ | secp192k1 | :code:
| Certicom | +---------------------------+-----------------------------------------+-------------+ | secp224k1 | :code:
| Certicom | +---------------------------+-----------------------------------------+-------------+ | secp256k1 (bitcoin curve) | :code:
| Certicom | +---------------------------+-----------------------------------------+-------------+ | brainpoolP160r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+ | brainpoolP192r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+ | brainpoolP224r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+ | brainpoolP256r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+ | brainpoolP320r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+ | brainpoolP384r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+ | brainpoolP512r1 | :code:
| BSI | +---------------------------+-----------------------------------------+-------------+

Arbitrary Curves ~~~~~~~~~~~~~~~~ As of version 1.5.1 construction of arbitrary curves in Weierstrass form (:code:

y^2 = x^3 + ax + b (mod p)
) is supported. I advise against using custom curves for any security critical applications. It's up to you to make sure that the parameters you pass here are correct, no validation of the base point is done, and in general no sanity checks are done. Use at your own risk.

.. code:: python

from fastecdsa.curve import Curve
curve = Curve(
    name,  # (str): The name of the curve
    p,  # (long): The value of p in the curve equation.
    a,  # (long): The value of a in the curve equation.
    b,  # (long): The value of b in the curve equation.
    q,  # (long): The order of the base point of the curve.
    gx,  # (long): The x coordinate of the base point of the curve.
    gy,  # (long): The y coordinate of the base point of the curve.
    oid  # (str): The object identifier of the curve (optional).

Hash Functions ~~~~~~~~~~~~~~ Any hash function in the :code:

module (:code:
md5, sha1, sha224, sha256, sha384, sha512
) will work, as will any hash function that implements the same interface / core functionality as the those in :code:
. For instance, if you wish to use SHA3 as the hash function the :code:
package will work with this library as long as it is at version >=1.0b1 (as previous versions didn't work with the :code:
module which is used in nonce generation). Note that :code:
sha3_224, sha3_256, sha3_384, sha3_512
are all in :code:
as of python3.6.


Curves over Prime Fields ~~~~~~~~~~~~~~~~~~~~~~~~ Currently it does elliptic curve arithmetic significantly faster than the :code:

package. You can see the times for 1,000 signature and verification operations over various curves below. These were run on an early 2014 MacBook Air with a 1.4 GHz Intel Core i5.

+-----------+------------------------+--------------------+---------+ | Curve | :code:

time | :code:
time | Speedup | +-----------+------------------------+--------------------+---------+ | P192 | 3.62s | 1m35.49s | ~26x | +-----------+------------------------+--------------------+---------+ | P224 | 4.50s | 2m13.42s | ~29x | +-----------+------------------------+--------------------+---------+ | P256 | 6.15s | 2m52.43s | ~28x | +-----------+------------------------+--------------------+---------+ | P384 | 12.11s | 6m21.01s | ~31x | +-----------+------------------------+--------------------+---------+ | P521 | 22.21s | 11m39.53s | ~31x | +-----------+------------------------+--------------------+---------+ | secp256k1 | 5.92s | 2m57.19s | ~30x | +-----------+------------------------+--------------------+---------+

Benchmarking ~~~~~~~~~~~~ If you'd like to benchmark performance on your machine you can do so using the command:

.. code:: bash

$ python benchmark

This will use the :code:

module to benchmark 1000 signature and verification operations for each curve supported by this package. Alternatively, if you have not cloned the repo but have installed the package via e.g. :code:
you can use the following command:

.. code:: bash

$ python -m fastecdsa.benchmark


You can use pip: :code:

$ pip install fastecdsa
or clone the repo and use :code:
$ python install
. Note that you need to have a C compiler. You also need to have GMP_ on your system as the underlying C code in this package includes the :code:
header (and links against gmp via the :code:
flag). You can install all dependencies as follows:

apt ~~~

.. code:: bash

$ sudo apt-get install python-dev libgmp3-dev

yum ~~~

.. code:: bash

$ sudo yum install python-devel gmp-devel


Generating Keys ~~~~~~~~~~~~~~~ You can use this package to generate keys if you like. Recall that private keys on elliptic curves are integers, and public keys are points i.e. integer pairs.

.. code:: python

from fastecdsa import keys, curve

"""The reason there are two ways to generate a keypair is that generating the public key requires a point multiplication, which can be expensive. That means sometimes you may want to delay generating the public key until it is actually needed."""

generate a keypair (i.e. both keys) for curve P256

priv_key, pub_key = keys.gen_keypair(curve.P256)

generate a private key for curve P256

priv_key = keys.gen_private_key(curve.P256)

get the public key corresponding to the private key we just generated

pub_key = keys.get_public_key(priv_key, curve.P256)

Signing and Verifying ~~~~~~~~~~~~~~~~~~~~~ Some basic usage is shown below:

.. code:: python

from fastecdsa import curve, ecdsa, keys
from hashlib import sha384

m = "a message to sign via ECDSA" # some message

''' use default curve and hash function (P256 and SHA2) ''' private_key = keys.gen_private_key(curve.P256) public_key = keys.get_public_key(private_key, curve.P256)

standard signature, returns two integers

r, s = ecdsa.sign(m, private_key)

should return True as the signature we just generated is valid.

valid = ecdsa.verify((r, s), m, public_key)

''' specify a different hash function to use with ECDSA ''' r, s = ecdsa.sign(m, private_key, hashfunc=sha384) valid = ecdsa.verify((r, s), m, public_key, hashfunc=sha384)

''' specify a different curve to use with ECDSA ''' private_key = keys.gen_private_key(curve.P224) public_key = keys.get_public_key(private_key, curve.P224) r, s = ecdsa.sign(m, private_key, curve=curve.P224) valid = ecdsa.verify((r, s), m, public_key, curve=curve.P224)

''' using SHA3 via pysha3>=1.0b1 package ''' import sha3 # pip install [--user] pysha3==1.0b1 from hashlib import sha3_256 private_key, public_key = keys.gen_keypair(curve.P256) r, s = ecdsa.sign(m, private_key, hashfunc=sha3_256) valid = ecdsa.verify((r, s), m, public_key, hashfunc=sha3_256)

Arbitrary Elliptic Curve Arithmetic ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The :code:

class allows arbitrary arithmetic to be performed over curves. The two main operations are point addition and point multiplication (by a scalar) which can be done via the standard python operators (:code:
and :code:

.. code:: python

# example taken from the document below (section 4.3.2):

from fastecdsa.curve import P256 from fastecdsa.point import Point

xs = 0xde2444bebc8d36e682edd27e0f271508617519b3221a8fa0b77cab3989da97c9 ys = 0xc093ae7ff36e5380fc01a5aad1e66659702de80f53cec576b6350b243042a256 S = Point(xs, ys, curve=P256)

xt = 0x55a8b00f8da1d44e62f6b3b25316212e39540dc861c89575bb8cf92e35e0986b yt = 0x5421c3209c2d6c704835d82ac4c3dd90f61a8a52598b9e7ab656e9d8c8b24316 T = Point(xt, yt, curve=P256)

Point Addition

R = S + T

Point Subtraction: (xs, ys) - (xt, yt) = (xs, ys) + (xt, -yt)

R = S - T

Point Doubling

R = S + S # produces the same value as the operation below R = 2 * S # S * 2 works fine too i.e. order doesn't matter

d = 0xc51e4753afdec1e6b6c6a5b992f43f8dd0c7a8933072708b6522468b2ffb06fd

Scalar Multiplication

R = d * S # S * d works fine too i.e. order doesn't matter

e = 0xd37f628ece72a462f0145cbefe3f0b355ee8332d37acdd83a358016aea029db7

Joint Scalar Multiplication

R = d * S + e * T

Importing and Exporting Keys ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can also export keys as files, ASN.1 encoded and formatted per RFC5480_ and RFC5915_. Both private keys and public keys can be exported as follows:

.. code:: python

from fastecdsa.curve import P256
from fastecdsa.keys import export_key, gen_keypair

d, Q = gen_keypair(P256)

save the private key to disk

export_key(d, curve=P256, filepath='/path/to/exported/p256.key')

save the public key to disk

export_key(Q, curve=P256, filepath='/path/to/exported/')

Keys stored in this format can also be imported. The import function will figure out if the key is a public or private key and parse it accordingly:

.. code:: python

from fastecdsa.keys import import_key

if the file is a private key then parsed_d is a long and parsed_Q is a Point object

if the file is a public key then parsed_d will be None

parsed_d, parsed_Q = import_key('/path/to/file.key')

Other encoding formats can also be specified, such as SEC1_ for public keys. This is done using classes found in the :code:

package, and passing them as keyword args to the key functions:

.. code:: python

from fastecdsa.curve import P256
from fastecdsa.encoding.sec1 import SEC1Encoder
from fastecdsa.keys import export_key, gen_keypair, import_key

_, Q = gen_keypair(P256) export_key(Q, curve=P256, filepath='/path/to/p256.key', encoder=SEC1Encoder) parsed_Q = import_key('/path/to/p256.key', curve=P256, public=True, decoder=SEC1Encoder)

Encoding Signatures ~~~~~~~~~~~~~~~~~~~ DER encoding of ECDSA signatures as defined in RFC2459_ is also supported. The :code:

provides the :code:
class which encodes signatures:

.. code:: python

from fastecdsa.encoding.der import DEREncoder

r, s = 0xdeadc0de, 0xbadc0de encoded = DEREncoder.encode_signature(r, s) decoded_r, decoded_s = DEREncoder.decode_signature(encoded)


Thanks to those below for contributing improvements:

  • boneyard93501
  • clouds56
  • m-kus
  • sirk390
  • targon
  • NotStatilko
  • bbbrumley
  • luinxz
  • JJChiDguez
  • J08nY

.. _issue11: .. _GMP: .. _RFC2459: .. _RFC5480: .. _RFC5915: .. _RFC6979: .. _SEC1:

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