Cryptography Reference
In-Depth Information
For the primes where there is bad reduction, the cubic
x
3
+
Ax
+
B
has
multiple roots mod
p
. If it has a triple root, we say that
E
has
additive
reduction
mod
p
. Ifithasadoublerootmod
p
,ithas
multiplicative re-
duction
. Moreover, if the slopes of the tangent lines at the singular point (see
Theorem 2.31) are in
F
p
,wesaythat
E
has
split multiplicative reduction
mod
p
. Otherwise, it has
nonsplit multiplicative reduction
.
To treat the primes
p
=2and
p
= 3, we need to use the general Weierstrass
form for
E
. For simplicity, we have ignored these primes in the preceding
discussion. However, in the example below, we'll include them.
There are many possible equations for
E
with
A, B ∈
Z
. We assume that
A, B
are chosen so that the reduction properties of
E
are as good as possible.
In other words, we assume that
A
and
B
are chosen so that the cubic has the
largest obtainable number of distinct roots mod
p
,andthepowerof
p
in the
discriminant 4
A
3
+27
B
2
is as small as possible, for each
p
.Itcanbeshown
that there is such a choice of
A, B
. Such an equation is called a
minimal
Weierstrass equation
for
E
.
Example 14.1
Suppose we start with
E
given by the equation
y
2
=
x
3
−
270000
x
+ 128250000
.
2
8
3
12
5
12
11, so
E
has good reduction except
possibly at 2
,
3
,
5
,
11. The change of variables
The discriminant of the cubic is
−
x
=25
x
1
,
y
= 125
y
1
transforms the equation into
y
1
=
x
1
−
432
x
1
+ 8208
.
2
8
3
12
11, so
E
also has good reduction at
5. This is as far as we can go with the standard Weierstrass model. To treat
2 and 3 we need to allow generalized Weierstrass equations. The change of
variables
The discriminant of the cubic is
−
x
1
=9
x
2
−
12
,
1
=27
y
2
changes the equation to
y
2
=
x
2
−
4
x
2
+16
.
The discriminant of the cubic is
2
8
11, so
E
has good reduction at 3. Since
any change of variables can be shown to change the discriminant by a square,
this is the best we can do, except possibly at the prime 2. The change of
variables
−
x
2
=4
x
3
,
2
=8
y
3
+4
changes the equation of
E
to
y
3
+
y
3
=
x
3
− x
3
.
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