Cryptography Reference
In-Depth Information
Let
φ
be the map in Theorem 8.14. Since
x, x
−
2
,x
+2 are squares,
φ
(
x, y
)=
1. Theorem 8.14 implies that
(
x, y
)=2
P
for some point
P ∈ E
(
Q
).
Let's find
P
. We follow the procedure used to prove Theorem 8.14. In the
notation of the proof of Theorem 8.14, the polynomial
rs
T
+
r
2
t
rs
−
s
−
t
T
2
f
(
T
)=
4
rs
satisfies
f
(2) =
r
2
s
2
f
(
−
2) =
r
2
+
s
2
rs
t
rs
,
−
f
(0) =
,
.
rs
The formulas from the proof of Theorem 8.14 say that the point (
x
1
,y
1
)with
− t
)
/
4
rs
=
−
2
s
2
−s/
2
r
(
r
2
x
1
=
r
2
− t
4
rs
r
2
y
1
=
−
t
satisfies 2(
x
1
,y
1
)=(
x, y
).
The transformation
1+
2
x
3
y
2
z
=
2
x
y
,
w
=
−
maps
E
to
C
.Thepoint(
x
1
,y
1
)mapsto
s
r
w
1
=
−
1+
2
x
1
2
x
1
y
1
z
1
=
=
−
s
4
r
2
(
r
2
=
−
1
−
y
1
−
t
)
r
4
+
s
4
− r
2
t
− r
2
t
r
2
(
r
2
t
2
=
−
=
−
r
2
(
r
2
− t
)
− t
)
t
r
2
.
=
We have
t
r
2
2
=
−
s
r
4
+1
.
Therefore, the solution (
r, −s, t
) corresponds to a point
P
on
E
such that 2
P
corresponds to (
a, b, c
). Fermat's procedure, therefore, can be interpreted as
starting with a point on an elliptic curve and halving it. The height decreases
by a factor of 4. The procedure cannot continue forever, so we must conclude
that there are no nontrivial solutions to start with.
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