Cryptography Reference
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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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