Cryptography Reference
In-Depth Information
2
æ
-
÷
y
y
ç
÷
=
2
1
- -
x
ç
x
x
÷
(3.26)
ç
3
÷
1
2
ç
-
è
xx
ø
2
1
æ
-
÷
y
y
ç
÷
=
2
1
- -
y
ç
(
x
x
)
y
(3.27)
÷
ç
3
÷
1
3
1
ç
-
è
xx
ø
2
1
()
=Î
, where
P
1
¹
-
P
1
. Then
=
2. Point doubling. Let
Px
(, )
y
EK
2(
Px
,
y
),
1
1
1
1
3
3
where
2
æ
+
÷
2
3
x
a
ç
÷
=
1
-
x
ç
2
x
(3.28)
÷
ç
3
÷
1
ç
è
2
y
ø
1
and
æ
+
÷
2
3
x
a
ç
÷
=
1
- -
y
ç
(
x
x
)
y
(3.29)
÷
ç
3
÷
1
3
1
ç
è
2
y
ø
1
+¥=¥+ =
for all
Î
3. Identity.
P
P
P
PEK
( .
1
1
1
1
+-=¥
The point (
x
, -
y
) is
denoted by -
P
1
and is called the negative of
P
1
; note that -
P
1
is indeed a point
in
E
(
K
). Also, -
¥=¥
.
=Î
then
(, ) (,
4. Negatives. If
Px y
(, )
EK
( ),
xy
x
y
)
.
1
Example of addition of two points on
E
over
R
: Referring to Figure 3.4, where
P
1
=
(-2.93, -2.3) and
P
2
=
(-0.154,1.6), then
P
3
=
P
1
+
P
2
=
(5.05, -8.89).
Example of doubling of points on
E
over
R
: Referring to Figure 3.5, where
P
1
=
(-0.77, 2.88), then 2.
P
1
=
(3.6, 3.38).
Example of addition of two points on elliptic curve
y
2
=
(
x
3
+
2
x
) over
F
23
:
Referring to Figure 3.3, let
P
1
=
(4, 7) and
P
2
=
(10, 10), then
P
3
=
P
1
+
P
2
=
(15, 22).
Example of doubling of a point on elliptic curve
y
2
=
(
x
3
+
2
x
) over
F
23
: Referring
to Figure 3.3, let
P
1
=
(4, 7), then 2.
P
1
=
(1, 7).
3.5.5.2 Group Law for Nonsupersingular
EF
/
:
y
²
+=++
xy x ax b
³
²
m
2
=Î
=
Î
1. Point addition. Let
Px
(, )
y
EF
(
)
and
Px
(, )
y
EF
( ,
m
where
2
2
2
1
1
1
m
2
2
¹
Then
+=
where
PP
2
.
PP xy
(, ,
1
1
2
3
3
2
=++++
λλ
x
x
x
a
(3.30)
3
1
2
and
=+++
λ
y
(
x
x
)
x
y
(3.31)
3
1
3
3
1
Search WWH ::
Custom Search