Cryptography Reference
In-Depth Information
isomorphic to C10 associated to the Elliptic Curve defined by
y
2
=x
3
- 58347*x + 3954150 over Rational Field
C10 denotes the cyclic group of order 10. To get a generator:
sage: E1.torsion subgroup().gen()
(3 : 1944 : 1)
The number of points on an elliptic curve mod a prime
p
has the form
p
+1
− a
p
. The value of
a
13
for
E
1
is computed as follows:
sage:
E1.ap(13)
4
Therefore, there are 13 + 1
−
4 = 10 points on
E
1
mod 13.
Let's reduce
E
1
mod 13:
sage: E3 = E2.change ring(GF(13))
sage: E3
Elliptic Curve defined by y
ˆ
2=x
ˆ
3 +10*x + 5
over Finite Field of size 13
The last command was not needed. It simply identified the nature of
E
3
.
We also could have defined a curve over
F
13
.
sage: E4 = EllipticCurve(GF(13), [10, 5])
sage: E4
Elliptic Curve defined by y
ˆ
2=x
ˆ
3 +10*x + 5
over Finite Field of size 13
sage:
E3 is E4
True
The last command asked whether
E
3
is the same as
E
4
. The answer was
yes.
We can find out how many points there are in
E
3
(
F
13
), or we can list all
the points:
sage:
E3.cardinality()
10
sage: E3.points()
[(0 : 1 : 0),
(11 : 4 : 1),
(8 : 5 : 1),
(1 : 4 : 1),
(10 : 0 : 1),
(1 : 9 : 1),
(3 : 6 : 1),
(8 : 8 : 1),
(11 : 9 : 1),
(3 : 7 : 1)]
Consider the curve
E
5
:
y
2
=
x
3
−
1over
F
229
, as in Example 4.10. We can
compute its group structure:
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