Cryptography Reference
In-Depth Information
> solve([F, T], [x, y]);
[[x = 33/16, y = 79/64], [x = -1, y = 2], [x = -1, y = 2]]
Thus we see that the third point we were looking for is
S
=
(
33
/
16
,
79
/
64
)
.
Notice that since
P
has coordinates in
Q
, we already knew that the coordinates of
S
would also belong to
and observe also that the tangent meets the curve at
P
with
multiplicity 2 (the coordinates of
P
appear twice in the result above) so the tangent
intersects the curve at three points, two of which happen to be
P
.Nextweplotthe
process of adding
P
to itself which gives as a result 2
P
Q
=
(
33
/
16
,
−
79
/
64
)
:
> el2 := implicitplot([F, T, x-33/16], x = -2.3 .. 3.3, y = -4.5 .. 4.5,
resolution = 1000, crossingrefine = 20, gridrefine = 6, color = black);
pts2 := pointplot([[-1, 2], [33/16, 79/64], [33/16, -79/64]]);
txt2 := textplot([[-.8, 2.1, 'P'], [2.2, 1.4, 'S'], [2.3, -1.2, '2P']]);
display(el2, pts2, txt2, labels = [" ", " "], scaling = constrained);
Exercise 11.5
Prove that if an elliptic curve intersects the
x
-axis at three different
points, then the sum of any two of these points is equal to the third one.
The previous discussion on the definition of the addition operation on
E
(
K
)
may
be summarized as follows:
P
+
Q
+
R
=
O
if and only if P
,
Q
,
R are collinear.
