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
()
, 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, - ¥=¥ .
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
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
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