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RS( s9 Elliptic Curve over the field of integers modulo a prime.c C s: t | � | _ t | � | _ t | � | _ | | _ d S( su
The curve of points satisfying y^2 = x^3 + a*x + b (mod p).
h is an integer that is the cofactor of the elliptic curve domain
parameters; it is the number of points satisfying the elliptic curve
equation divided by the order of the base point. It is used for selection
of efficient algorithm for public point verification.
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The curve of points satisfying y^2 = x^3 + a*x + b (mod p).
h is an integer that is the cofactor of the elliptic curve domain
parameters; it is the number of points satisfying the elliptic curve
equation divided by the order of the base point. It is used for selection
of efficient algorithm for public point verification.
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Point on an elliptic curve. Uses Jacobi coordinates.
In Jacobian coordinates, there are three parameters, X, Y and Z.
They correspond to affine parameters 'x' and 'y' like so:
x = X / Z²
y = Y / Z³
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Initialise a point that uses Jacobi representation internally.
:param CurveFp curve: curve on which the point resides
:param int x: the X parameter of Jacobi representation (equal to x when
converting from affine coordinates
:param int y: the Y parameter of Jacobi representation (equal to y when
converting from affine coordinates
:param int z: the Z parameter of Jacobi representation (equal to 1 when
converting from affine coordinates
:param int order: the point order, must be non zero when using
generator=True
:param bool generator: the point provided is a curve generator, as
such, it will be commonly used with scalar multiplication. This will
cause to precompute multiplication table for it
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| | | | | d k S( s# Compare two points with each-other.Ni i ( R" t reader_acquiret INFINITYR$ R% R# t reader_releaseR t PointR R R R R! R, t FalseR ( R
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! c C s | j S( sE Return the order of the point.
None if it is undefined.
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Return affine x coordinate.
This method should be used only when the 'y' coordinate is not needed.
It's computationally more efficient to use `to_affine()` and then
call x() and y() on the returned instance. Or call `scale()`
and then x() and y() on the returned instance.
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Return affine y coordinate.
This method should be used only when the 'x' coordinate is not needed.
It's computationally more efficient to use `to_affine()` and then
call x() and y() on the returned instance. Or call `scale()`
and then x() and y() on the returned instance.
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Return point scaled so that z == 1.
Modifies point in place, returns self.
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:param bool generator: set to True to make the point to precalculate
multiplication table - useful for public point when verifying many
signatures (around 100 or so) or for generator points of a curve.
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d d d | j j � f \ } } } } | j } x� | j D]� \ } } | d r� | d d k r� | d d } | | | | | | d | � \ } } } q� | d d } | | | | | | d | � \ } } } q: | d } q: W| s� | r� t St | j | | | | j � S( s4 Multiply point by integer with precomputation table.i i i i ( R! R Rn R' R3 R R& ( R
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| | | | | d | � \ } } } q� q� W| ss| rwt St | j | | | | j � S( s Multiply point by an integer.i i i ( i i i ( R$ R3 R& R' Rr R+ R# R! R R RT Rn t reversedRv R ( R
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Do two multiplications at the same time, add results.
calculates self*self_mul + other*other_mul
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d � Z d � Z RS( ss A point on an elliptic curve. Altering x and y is forbidding,
but they can be read by the x() and y() methods.c C s� | | _ t rQ | o t | � | _ | o3 t | � | _ | oH t | � | _ n | | _ | | _ | | _ | j r� | j j | | � s� t � n | r� | j � d k r� | r� | | t k s� t � n d S( s@ curve, x, y, order; order (optional) is the order of this point.i N(
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j | j | j | � | } | | | j | j | } | | j | | j | } t | j | | � S( s Add one point to another point.i ( R R5 R R3 R� R( R� R� R R* R R? ( R
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! c C s* d � } | } | d k s7 | j r; | | j d k r; t S| t k rK t S| d k ra | | Sd | } t | j | j | j | j � } | | � d } | } x� | d k r%| j � } | | @d k r� | | @d k r� | | } n | | @d k r| | @d k r| | } n | d } q� W| S( s Multiply a point by an integer.c S s= | d k s t � d } x | | k r4 d | } q W| d S( Ni i i ( R( ( R Ry ( ( sH /opt/plesk/python/2.7/lib/python2.7/site-packages/ecdsa/ellipticcurve.pyt leftmost_bit� s
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c C s | | S( s Multiply a point by an integer.( ( R
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