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add description of low level elliptic curve operations
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========================= | ||
Elliptic Curve arithmetic | ||
========================= | ||
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The python-ecdsa also provides generic API for performing operations on | ||
elliptic curve points. | ||
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.. warning:: | ||
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This is documentation of a very low-level API, if you want to | ||
handle keys or signatures you should look at documentation of | ||
the :py:mod:`~ecdsa.keys` module. | ||
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Short Weierstrass curves | ||
======================== | ||
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There are two low-level implementations for | ||
:term:`short Weierstrass curves <short Weierstrass curve>`: | ||
:py:class:`~ecdsa.ellipticcurve.Point` and | ||
:py:class:`~ecdsa.ellipticcurve.PointJacobi`. | ||
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Both of them use the curves specified using the | ||
:py:class:`~ecdsa.ellipticcurve.CurveFp` object. | ||
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You can either provide your own curve parameters or use one of the predefined | ||
curves. | ||
For example, to define a curve :math:`x^2 = x^3 + x + 4 \text{ mod } 5` use | ||
code like this: | ||
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.. code:: python | ||
from ecdsa.ellipticcurve import CurveFp | ||
custom_curve = CurveFp(5, 1, 4) | ||
The predefined curves are specified in the :py:mod:`~ecdsa.ecdsa` module, | ||
but it's much easier to use the helper functions (and proper names) | ||
from the :py:mod:`~ecdsa.curves` module. | ||
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For example, to get the curve parameters for the NIST P-256 curve use this | ||
code: | ||
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.. code:: python | ||
from ecdsa.curves import NIST256p | ||
curve = NIST256p.curve | ||
.. tip:: | ||
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You can also use :py:class:`~ecdsa.curves.Curve` to get the curve | ||
parameters from a PEM or DER file. Or use the | ||
:py:func:`~ecdsa.curves.find_curve` to get a curve by specifying its | ||
ASN.1 object identifier (OID). | ||
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After taking hold of curve parameters you can create a point on the | ||
curve. The :py:class:`~ecdsa.ellipticcurve.Point` uses affine coordinates, | ||
i.e. the :math:`x` and :math:`y` from the curve equation directly. | ||
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To specify a point (1, 1) on the ``custom_curve`` you can use this code: | ||
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.. code:: python | ||
from ecdsa.ellipticcurve import Point | ||
point_a = Point(custom_curve, 1, 1) | ||
Then it's possible to either perform scalar multiplication: | ||
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.. code:: python | ||
point_b = point_a * 3 | ||
Or specify other points and perform addition: | ||
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.. code:: python | ||
point_b = Point(custom_curve, 3, 2) | ||
point_c = point_a + point_b | ||
To get the affine coordinates of the point, call the ``x()`` and ``y()`` | ||
methods of the object: | ||
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.. code:: python | ||
print("x: {0}, y: {1}".format(point_c.x(), point_c.y())) | ||
When using the Jacobi coordinates, the point is defined by 3 integers, | ||
which are related to the :math:`x` and :math:`y` in the following way: | ||
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.. math:: | ||
x = X/Z^2 \\ | ||
y = Y/Z^3 | ||
That means that if you have point in affine coordinates, it's possible | ||
to convert them to Jacobi by simply assuming :math:`Z = 1`. | ||
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So the same points can be specified as so: | ||
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.. code:: python | ||
from ecdsa.ellipticcurve import PointJacobi | ||
point_a = PointJacobi(custom_curve, 1, 1, 1) | ||
point_b = PointJacobi(custom_curve, 3, 2, 1) | ||
.. note:: | ||
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Unlike the :py:class:`~ecdsa.ellipticcurve.Point`, the | ||
:py:class:`~ecdsa.ellipticcurve.PointJacobi` does **not** check if the | ||
coordinates specify a valid point on the curve as that operation is | ||
computationally expensive for Jacobi coordinates. | ||
If you want to verify if they specify a valid | ||
point, you need to convert the point to affine coordinates and use the | ||
:py:meth:`~ecdsa.ellipticcurve.CurveFp.contains_point` method. | ||
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Then all the operations work exactly the same as with regular | ||
:py:class:`~ecdsa.ellipticcurve.Point` implementation. | ||
While it's not possible to get the internal :math:`X`, :math:`Y`, and :math:`Z` | ||
coordinates, it's possible to get the affine projection just like with | ||
the regular implementation: | ||
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.. code:: python | ||
point_c = point_a + point_b | ||
print("x: {0}, y: {1}".format(point_c.x(), point_c.y())) |
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@@ -44,6 +44,7 @@ curves over prime fields. | |
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quickstart | ||
basics | ||
ec_arithmetic | ||
glossary | ||
modules | ||
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