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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