Cryptography Reference
In-Depth Information
4.3.2.1 Boneh and Franklin IBE Scheme
Boneh and Franklin explain the basic idea of the identity-based encryption scheme
through a simple scheme named
BasicIdent
(Boneh and Franklin 2001). This scheme
uses a shared secret that can be calculated by both the sender and the receiver of a mes-
sage to encrypt a plain text. Here we describe the basic scheme, which is secure against
chosen plain-text attacks and adaptive chosen-identity attacks, while the full Boneh-
Franklin IBE (Boneh and Franklin 2001) is resistant to chosen cipher-text attacks and
adaptive chosen-identity attacks.
The
BasicIdent
scheme consists of
Set-up
,
Key-Gen
,
Encrypt
,
and
Decrypt.
Set-up
: Let
k
Z
and the master secret
M
s
Z
q
. Let
G
be a bilinear Diffie-
Hellman (BDH) parameter generator. Let
G
1
,
G
2
be two groups of order
q
and
let
P
G
2
be a random generator. Let there exist a bilinear map
e
such that
e
:
G
1
×
G
1
G
2
(4.2)
Let
P
pub
=
M
s
P
.
Let the cryptographic hash functions be defined as
*
*
H
:{0,1}
G
(4.3)
1
1
n
HG
:
{0,1}
for so e
n
(4.4)
2
2
Let the message space
M
= {0,1}
n
and the cipher text
C
=
G
1
*
× {0,1}
n
.
Hence, the system parameters
=
PP
GG qenH H PP
,
, , , ,
,
,
,
.
12
1 2
pub
Key-Gen
: The algorithm computes
=Î
*
1
id
QHidG
()
(4.5)
1
where
id
I
and the private key corresponding to identity
id
I
is defined as
=
d
MQ
(4.6)
id
s
id
Encrypt
: This algorithm chooses a random integer
r
Z
q
and encrypts a message
m
M
to generate the cipher text
==Å
r
id
= Î
*
CUV rPmHg
,
,
(
)
where
g
e QP
(
,
)
G
(4.7)
2
id
id
pub
2
Decrypt
: Let
id
be the public key used to generate the cipher text
C
C
, where
the private key
Î
1
,
id
dG
then the algorithm decrypts
C
to generate
m
M
as
shown below (Figure 4.11).
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