Cryptography Reference
In-Depth Information
(b) Show that the substitution
x
=4
x
1
,
y
=8
y
1
+4
x
1
changes
E
into
E
1
,givenby
y
1
+
x
1
y
1
=
x
1
+
−
β
−
α
−
1
x
1
+
αβ
16
x
1
.
4
(c) Show that the reduction mod 2 of the equation for
E
1
is
y
1
+
x
1
y
1
=
x
1
+
ex
1
∈
F
2
. This curve is singular at (0
,
0).
(d) Let
γ
be a constant and consider the line
y
1
=
γx
1
. Show that if
γ
2
+
γ
=
e
, then the line intersects the curve in part (c) to order
3, and if
γ
2
+
γ
=
e
then this line intersects the curve to order 2.
(e) Show that there are two distinct values of
γ ∈
F
2
such that
γ
2
+
γ
=
e
. This implies that there are two distinct tangent lines to the curve
E
1
mod 2 at (0,0), as in Exercise 2.20.
for some
e
We take the property of part (e) to be the definition of multiplicative
reduction in characteristic 2. Therefore, parts (a) and (e) show that
the curve
E
1
has good or multiplicative reduction at all primes. A
semistable
elliptic curve over
Q
is one that has good or multiplicative
reduction at all primes, possibly after a change of variables (over
Q
)
such as the one in part (b). Therefore,
E
is semistable. See Section 15.1
for a situation where this fact is used.
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