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Elliptic Curve Points
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Expression 12: "y" Subscript, 2 , Baseline equals StartRoot, left parenthesis, "x" cubed plus "a" "x" Subscript, 2 , Baseline plus "b" , right parenthesis , EndRoot equals 2.5 1 0 5 7 7 6 2 2 7 8
y
2
=
x
3
2
+
a
x
2
+
b
equals
=
2.5 1 0 5 7 7 6 2 2 7 8
2
.
5
1
0
5
7
7
6
2
2
7
8
12
Expression 13: "x" Subscript, 1 , Baseline equals 2.6
x
1
=
2
.
6
negative 10
−
1
0
10
1
0
13
Expression 14: "x" Subscript, 2 , Baseline equals negative 1.3
x
2
=
−
1
.
3
negative 10
−
1
0
10
1
0
14
This is the straight line connecting the two source points (step 1):
This is the straight line connecting the two source points (step 1):
15
Expression 16: "m" equals StartFraction, left parenthesis, "y" Subscript, 1 , Baseline minus "y" Subscript, 2 , Baseline , right parenthesis Over left parenthesis, "x" Subscript, 1 , Baseline minus "x" Subscript, 2 , Baseline , right parenthesis , EndFraction equals 0.0 1 3 7 9 3 2 4 2 3 5 7 1
m
=
y
1
−
y
2
x
1
−
x
2
equals
=
0.0 1 3 7 9 3 2 4 2 3 5 7 1
0
.
0
1
3
7
9
3
2
4
2
3
5
7
1
16
Expression 17: "v" equals "y" Subscript, 1 , Baseline minus "m" "x" Subscript, 1 , Baseline equals 2.5 2 8 5 0 8 8 3 7 8 4
v
=
y
1
−
m
x
1
equals
=
2.5 2 8 5 0 8 8 3 7 8 4
2
.
5
2
8
5
0
8
8
3
7
8
4
17
Expression 18: "y" equals "m" "x" plus "v" Has graph. To audio trace, press ALT+T.
y
=
m
x
+
v
18
These are the coordinates of the intersection of the line with our curve (step 2):
These are the coordinates of the intersection of the line with our curve (step 2):
19
Expression 20: "x" Subscript, 3 , Baseline equals "m" squared minus "x" Subscript, 1 , Baseline minus "x" Subscript, 2 , Baseline equals negative 1.2 9 9 8 0 9 7 4 6 4 7
x
3
=
m
2
−
x
1
−
x
2
equals
=
negative 1.2 9 9 8 0 9 7 4 6 4 7
−
1
.
2
9
9
8
0
9
7
4
6
4
7
20
Expression 21: "y" Subscript, 3 , Baseline equals "m" "x" Subscript, 3 , Baseline plus "v" equals 2.5 1 0 5 8 0 2 4 6 9 9
y
3
=
m
x
3
+
v
equals
=
2.5 1 0 5 8 0 2 4 6 9 9
2
.
5
1
0
5
8
0
2
4
6
9
9
21
Hidden Label: left parenthesis, "x" Subscript, 3 , Baseline , "y" Subscript, 3 , Baseline , right parenthesis equals left parenthesis, negative 1.2 9 9 8 0 9 7 4 6 4 7 , 2.5 1 0 5 8 0 2 4 6 9 9 , right parenthesis Has graph. To audio trace, press ALT+T.
x
3
,
y
3
Label
equals
=
left parenthesis, negative 1.2 9 9 8 0 9 7 5 , 2.5 1 0 5 8 0 2 5 , right parenthesis
−
1
.
2
9
9
8
0
9
7
5
,
2
.
5
1
0
5
8
0
2
5
22
Expression 23:
23
24
powered by
powered by
"x"
x
"y"
y
"a" squared
a
2
"a" Superscript, "b" , Baseline
a
b
7
7
8
8
9
9
over
÷
functions
(
(
)
)
less than
<
greater than
>
4
4
5
5
6
6
times
×
| "a" |
|
a
|
,
,
less than or equal to
≤
greater than or equal to
≥
1
1
2
2
3
3
negative
−
A B C
StartRoot, , EndRoot
pi
π
0
0
.
.
equals
=
positive
+
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