Cryptography Reference
In-Depth Information
Set-up
: Let
G
1
be a subgroup of
E
(
F
q
), where
E
is an elliptic curve with an embed-
ding degree
k
. Let
G
T
be a subgroup of
*
k
, where
q
is a prime power, and let
F
there exist a bilinear map
e
such that
eG G
:
´
G
(4.9)
1
1
T
Î
*
Let
P
G
1
and let
αβγ
,,
Z
. Let
1
P
=
α
,
2
P
=
β
, and
3
P
=
γ
. Then
there exists a constant
v
G
1
such that
αβ
vePP ePP ePP
=
(, )
=
( , )
αβ
=
(,)
(4.10)
12
Let
H
3
be a hash function such that strings representing identities are mapped to
integers,
*
3
:{0,1}
H
(4.11)
q
and let
H
4
be another hash function such that
{0,1}
n
HG
4
:
(4.12)
T
Hence, the system parameters
PP
=
GG qeH H PPPv
,
, , ,
,
,
,
,
,
.
1
T
3
4
1
2
3
Key-Gen
: Let
id
I
, then this algorithm computes
Q
id
=
H
3
(
id
)
G
1
. Let
r
Z
p
,
then the private key is
d
=
{
Q
.
rP
+ +
αγ
P
P
,
rP
}
=
{
D
,
D
}
(4.13)
id
id
1
2
3
0
1
This private key is sent to the end user with identity
id
I
in a secure way.
Encrypt
: Let message
m
M
, then this algorithm generates the cipher text as
follows:
id
=
H
3
(
id
).
Pick a random integer
s
Z
p
and calculate the constant
k = v
s
G
T
.
Calculate
c = m
⊕
H
4
(
k
) and
C
0
=
sP.
Calculate
C
1
=
Q
id
(
sP
1
) +
sP
3
.
Hence, the cipher text
C
= (
c
,
C
0
,
C
1
).
Decrypt
: This algorithm retrieves
k
using the bilinear property as shown below:
eC D
(, )
(, )
k
=
00
(4.14)
eC D
11
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