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fastecdsa

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About

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

Security

There is no nonce reuse, no branching on secret material, and all points are validated before being operations are performed on them. Timing side challenges are mitigated via constant time arithmetic, complete point arithmetic formulas, constant time combs for base point scaling and constant time montgomery ladders for arbitrary point scaling. 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. For more details on timing analysis see the jupyter notebook at side-channel-analysis.ipynb.

The above security discussion pertains to the operations that are exposed via the fastecdsa.ecdsa and fastecdsa.eddsa modules. Timing considerations do not apply to verify as verify does not operate on secret material. Primitive operations like point arithmetic, as well as user-defined curves, have no security guarantees.

If you have any security concerns or disclosures please use the author's email found in the pyproject.toml file.

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.

Operating Systems Supported

As of v4 most flavors of Linux/MacOS/Windows are supported. v3 and below requires the GMP library which has historically made building and installing on the Windows platform difficult. Note that this package is optimized for 64bit operating systems that support 64bit arithmetic operations like multiplication in their instruction set.

Supported Primitives

ECDSA

Name Class Proposed By
P192 / secp192r1 fastecdsa.curve.P192 NIST / NSA
P224 / secp224r1 fastecdsa.curve.P224 NIST / NSA
P256 / secp256r1 fastecdsa.curve.P256 NIST / NSA
P384 / secp384r1 fastecdsa.curve.P384 NIST / NSA
P521 / secp521r1 fastecdsa.curve.P521 NIST / NSA
secp192k1 fastecdsa.curve.secp192k1 Certicom
secp224k1 fastecdsa.curve.secp224k1 Certicom
secp256k1 (bitcoin curve) fastecdsa.curve.secp256k1 Certicom
brainpoolP160r1 fastecdsa.curve.brainpoolP160r1 BSI
brainpoolP192r1 fastecdsa.curve.brainpoolP192r1 BSI
brainpoolP224r1 fastecdsa.curve.brainpoolP224r1 BSI
brainpoolP256r1 fastecdsa.curve.brainpoolP256r1 BSI
brainpoolP320r1 fastecdsa.curve.brainpoolP320r1 BSI
brainpoolP384r1 fastecdsa.curve.brainpoolP384r1 BSI
brainpoolP512r1 fastecdsa.curve.brainpoolP512r1 BSI

EdDSA

Name Module Proposed By
ed25519 fastecdsa.eddsa DJB et al.
ed448 fastecdsa.eddsa Mike Hamburg

Arbitrary Weierstrass Curves

As of version 1.5.1 construction of arbitrary curves in Weierstrass form (y^2 = x^3 + ax + b (mod p)) is supported. I advise against using custom curves for any security sensitive applications. In versions before release 4.0.0 no checks on the curve parameters are done. In 4.0.0+ some sanity checks are applied -

  • p must be a non-even (not 2) prime
  • The curve cannot be singular (discriminant equal to 0)
  • (gx, gy) must be a point on the curve
  • q must be prime

Exhaustive validation checks are not performed e.g., no check that q is actually the order of point (gx, gy) is performed.

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

Hash Functions

Any hash function in the hashlib module (md5, sha1, sha224, sha256, sha384, sha512) will work, as will any hash function that implements the same interface / core functionality as the those in hashlib. For instance, if you wish to use SHA3 as the hash function the pysha3 package will work with this library as long as it is at version >=1.0b1 (as previous versions didn't work with the hmac module which is used in nonce generation). Note that sha3_224, sha3_256, sha3_384, sha3_512 are all in hashlib as of python3.6.

Performance

You can see the times for 1,000 signature and verification operations over various curves below. These were run on a machine with a 3.6 GHz Intel Core i9-9900K.

ECDSA

Curve fastecdsa v4 ecdsa (gmpy2 backend) fastecdsa v3
P192 0.14s 0.98s 1.16s
P224 0.25s 1.21s 1.51s
P256 0.31s 1.38s 1.91s
P384 0.75s 2.25s 4.08s
P521 1.60s 3.63s 7.08s
Secp192k1 0.11s N/A 1.19s
Secp224k1 0.22s N/A 1.56s
Secp256k1 0.24s 1.34s 1.92s
Brainpoolp160r1 0.14s 0.83s 0.88s
Brainpoolp192r1 0.16s 1.01s 1.18s
Brainpoolp224r1 0.26s 1.21s 1.55s
Brainpoolp256r1 0.30s 1.35 1.88s
Brainpoolp320r1 0.52s 1.80s 2.91s
Brainpoolp384r1 0.82s 2.27s 4.08s
Brainpoolp512r1 1.60s 3.33s 7.06s

EdDSA

Curve fastecdsa v4 ecdsa (gmpy2 backend) fastecdsa v3
Edwards25519 0.20s 1.28s N/A
Edwards448 0.64s 2.53s N/A

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

$ uv run benchmark

This will use the timeit 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 to your site packages you can use the following command:

$ python -m fastecdsa.benchmark

Installing

You can use the usual tools to install / add this package to your project -

$ uv add fastecdsa
$ poetry add fastecdsa
$ pip install fastecdsa

You can also clone the repo and use $ uv run maturin develop. Note that you need to have the rust toolchain on your machine if you clone and build.

Development

This package uses uv for package management. You can install it via pip install uv. First build the rust bindings

$ uv run maturin develop -r

To run the test suite use the following command

$ uv run pytest

Install pre-commit hooks to ensure type checking and autoformatting happens before you commit your code

$ uv run pre-commit install

To build the docs use the following command, which will create a docs/_build directory with the docs built as HTML files

$ cd docs
$ uv run make html

Publishing

Note that currently only the package owner is able to publish releases to PyPI. The following steps can still be used to generate source and wheel distributions, but note that the publish command will not work.

To build a release first install all supported versions of python into the environment (double check pyproject.toml for which python versions are supported)

$ uv python install 3.11 3.12 3.13 3.14

Then build a source distribution, followed by wheels for each supported python version

$ uv run maturin sdist
$ uv run maturin build -r -i python3.x

Usage

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.

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:

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 Point 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 (+ and * respectively):

# example taken from the document below (section 4.3.2):
# https://koclab.cs.ucsb.edu/teaching/cren/docs/w02/nist-routines.pdf

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

Points on Edwards Curves

The edwards curves that underpin Ed25519 and Ed448 have their own dedicated point classes as they use different point arithmetic and representations than the Weierstrass points. See below for some example usage.

In [1]: from fastecdsa.eddsa import Ed25519Point as e25519

In [2]: G = e25519.g()

In [3]: G
Out[3]:
X: 0x216936d3cd6e53fec0a4e231fdd6dc5c692cc7609525a7b2c9562d608f25d51a
Y: 0x6666666666666666666666666666666666666666666666666666666666666658
(Affine point on curve Edwards25519)

In [4]: G * 2
Out[4]:
X: 0x36ab384c9f5a046c3d043b7d1833e7ac080d8e4515d7a45f83c5a14e2843ce0e
Y: 0x2260cdf3092329c21da25ee8c9a21f5697390f51643851560e5f46ae6af8a3c9
(Affine point on curve Edwards25519)

In [5]: G + G
Out[5]:
X: 0x36ab384c9f5a046c3d043b7d1833e7ac080d8e4515d7a45f83c5a14e2843ce0e
Y: 0x2260cdf3092329c21da25ee8c9a21f5697390f51643851560e5f46ae6af8a3c9
(Affine point on curve Edwards25519)

In [6]: e25519.scale_base(2)
Out[6]:
X: 0x5881d418cc897a3a0eb49cdb9e6859fe2c55894bcdf7f8095714be2d5cf2e79a
Y: 0x27e006a46c706615d5ef0e2ff33fb48a6fbc113c77c0bd916574e8d0e2c42106
Z: 0x1f7616052c96e0810d7ff513e392dfbad03e66e0a95c8d2796a29516b07725f4
T: 0x5dec8aec83e2c5266dd025b8da706eed6a2729713b3f035372c8624d35caa7b
(Extended projective point on curve Edwards25519)

In [7]: _.normalize()
Out[7]:
X: 0x36ab384c9f5a046c3d043b7d1833e7ac080d8e4515d7a45f83c5a14e2843ce0e
Y: 0x2260cdf3092329c21da25ee8c9a21f5697390f51643851560e5f46ae6af8a3c9
(Affine point on curve Edwards25519

Note that fastecdsa.eddsa.Ed448Point has the same interface, but it uses standard projective coordinates that omit "T".

Projective Points

As of release 4.0.0 you can also use the projective representation of points. This may be desirable if you are doing a lot of intermediate calculations with points before you need an affine representation as arithmetic on projective points is cheaper. Note that comparisons between projective points and affine points are also possible, but they are less performant than comparisons between two affine points. By default all representations and operations on points are affine. To convert a projective point back to an affine point use the normalize method.

In [1]: from fastecdsa.curve import P256

In [2]: from fastecdsa.point import Point

In [3]: x = 0xdeadc0de

In [4]: x * P256.G
Out[4]:
X: 0x32079326d26449f8b36bde4410f805eb520c0120da1585c79c369f356c8a298f
Y: 0x249daa2c57c8d6a90575630635aa5448fa56d21f5e363c155fe98c597b1c70ae
(Affine point on curve "P256")

In [5]: g_ = Point(P256.G.x, P256.G.y, P256, projective=True)

In [6]: h = x * g_

In [7]: h
Out[7]:
X: 0x15819ed127c46b11b751a1d575a6e7712fe72c03e693ea0783268e2f0d4bdfac
Y: 0x879367bd0ebcf1b97fdb0841c33333f7c321a41c2ced6ab14a1cd2cc7684f6bf
Z: 0x99682c948ddc2ddd15966aecfe83fe52b1df7d76255d460f90d7bd615bb80cfc
(Projective point on curve "P256")

In [8]: h.normalize()  # gives the same result as x * P256.G
Out[8]:
X: 0x32079326d26449f8b36bde4410f805eb520c0120da1585c79c369f356c8a298f
Y: 0x249daa2c57c8d6a90575630635aa5448fa56d21f5e363c155fe98c597b1c70ae
(Affine point on curve "P256")

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:

from fastecdsa.curve import P256
from fastecdsa.encoding.pem import PEMEncoder
from fastecdsa.keys import export_private_key, export_public_key, gen_keypair

d, Q = gen_keypair(P256)
encoder = PEMEncoder()

# save the private key to disk
export_private_key(d, curve=P256, encoder=encoder, filepath='/path/to/exported/p256.key')
# save the public key to disk (curve inferred from Q)
export_public_key(Q, encoder=encoder, filepath='/path/to/exported/p256.pub')

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:

from fastecdsa.curve import P256
from fastecdsa.encoding.pem import PEMEncoder
from fastecdsa.keys import import_public_key, import_private_key

decoder = PEMEncoder()
# returns an int
parsed_d = import_private_key('/path/to/exported/p256.key', decoder=decoder)
# returns a fastecdsa.point.Point
parsed_Q = import_public_key('/path/to/exported/p256.pub', curve=P256, decoder=decoder)

Other encoding formats can also be specified, such as SEC1 for public keys. This is done using classes found in the fastecdsa.encoding package, and passing them as keyword args to the key functions:

from fastecdsa.curve import P256
from fastecdsa.encoding.sec1 import SEC1Encoder
from fastecdsa.keys import export_public_key, gen_keypair, import_public_key

_, Q = gen_keypair(P256)
encoder = SEC1Encoder()

export_public_key(Q, encoder=encoder, filepath='/path/to/exported/p256.pub')
parsed_Q = import_public_key('/path/to/exported/p256.pub', curve=P256, decoder=encoder)

Encoding Signatures

DER encoding of ECDSA signatures as defined in RFC2459 is also supported. The fastecdsa.encoding.der provides the DEREncoder class which encodes signatures:

from fastecdsa.encoding.der import DEREncoder

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

Acknowledgements

Thanks to those below for contributing improvements:

  • boneyard93501
  • clouds56
  • m-kus
  • sirk390
  • targon
  • NotStatilko
  • bbbrumley
  • luinxz
  • JJChiDguez
  • J08nY
  • trevor-crypto
  • sylvainpelissier
  • akaIDIOT
  • Peter-Bergman
  • DimitriPapadopoulos

Much of this work is based upon the following articles, research and standards:

About

Python library for fast elliptic curve crypto

pypi.python.org/pypi/fastecdsa

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