Cryptography Reference
In-Depth Information
Exercise 11.2
Show that the real curve of equation
y
2
x
3
=
−
+
3
x
2 is singular
but only one of the two points where it meets the
x
-axis is singular.
If
P
=
(
x
0
,
y
0
)
is a smooth point in the curve of equation
F
(
x
,
y
)
=
0, then the
tangent line to the curve at
P
is the line of equation:
F
x
(
x
0
,
y
0
)
·
(
x
−
x
0
)
+
F
y
(
x
0
,
y
0
)
·
(
y
−
y
0
)
=
0
.
(11.1)
In this case the coefficients of
x
and
y
are not both zero so the equation does indeed
define a line. The tangent equation comes from the Taylor expansion of
F
(
x
,
y
)
about
(
x
0
,
y
0
)
, which has the form:
F
(
x
,
y
)
=
F
(
x
0
,
y
0
)
+
F
x
(
x
0
,
y
0
)(
x
−
x
0
)
+
F
y
(
x
0
,
y
0
)(
y
−
y
0
)
+
terms of higher degree
.
The tangent line at
P
is a sort of “linear approximation” to the curve at
P
and is
obtained by taking the linear part of the Taylor expansion which, as
P
is a point of
the curve and hence
F
(
x
0
,
y
0
)
=
0, is given by Eq.
11.1
.
Exercise 11.3
Use Maple's command
plots:-implicitplot
to plot the real
curves
y
2
x
3
and
y
2
x
3
=
=
−
3
x
+
2, which allows an easy visual detection of the
singular points of both curves.
We next establish the connection between the condition 27
b
2
4
a
3
0 appearing
in the definition of elliptic curve and the smoothness of the curve. We leave the
following as an exercise:
+
=
Exercise 11.4
Consider the Weierstrass equation
y
2
x
3
=
+
ax
+
b
and let
D
=
=
i
=
1
(
=
i
<
j
(
27
b
2
4
a
3
. Show that if
x
3
2
−
(
+
)
+
ax
+
b
x
−
r
i
)
then
D
r
i
−
r
j
)
(
D
is then the
discriminant
of the polynomial
x
3
+
ax
+
b
). Deduce that
D
=
0if
and only if
x
3
+
ax
+
b
has no multiple zeros in an algebraic closure of the base field.
(Hint: Expand
i
=
1
(
x
−
r
i
)
and compare the coefficients of the resulting polynomial
with those of
x
3
+
ax
+
b
in order to express
a
and
b
in terms of two of the roots,
say
r
1
,
r
2
. Finally compute
D
and
i
<
j
(
2
to check that they are equal. This
straightforward but tedious computation may be quickly done with Maple, taking
care to use the
expand
command before doing the comparison.)
r
i
−
r
j
)
We can now show the following:
x
3
Proposition 11.1
The curve E of equation F
(
x
,
y
)
=
0
, where F
(
x
,
y
)
=
+
y
2
, is smooth if and only if
4
a
3
27
b
2
ax
+
b
−
+
=
0
(
or, equivalently, if and only
if
Δ
=
0
))
.
Δ
=
Proof
Suppose first, by contradiction, that
0 and let us show that
E
is not
smooth in this case, i.e., that
E
has a singular point. By Exercise 11.4, there exists in
