A PAIRING-FREE IDENTITY BASED TRIPARTITE SIGNCRYPTION SCHEME

International Journal on Cryptography and Information Security (IJCIS), Vol.3, No. 4, December 2013
DOI:10.5121/ijcis.2013.3401 1
A PAIRING-FREE IDENTITY BASED TRIPARTITE
SIGNCRYPTION SCHEME
Hassan M. Elkamchouchi 1
, Eman F. Abou Elkheir2
and Yasmine Abouelseoud3
1
Elec. Eng. Dept, Fac. of Eng., Alexandria University.
2
Elec. Eng. Dept, Fac. of Eng., Kafr El-Sheikh University.
3
Math. Eng. Dept, Fac. of Eng., Alexandria University.
ABSTRACT
The certificate-based cryptosystems is traditional way in providing the system parameters. Identity-based
cryptography is more efficient than certificate-based cryptosystems. Each user in identity-based
cryptography uses any arbitrary string that uniquely identifies him as his public key. This paper proposes
a new identity-based tripartite signcryption scheme based on the elliptic curve discrete logarithm problem.
The proposed id-based tripartite signcryption scheme does not use the bilinear pairings in both the
Signcryption and unsigncryption phases. The proposed scheme used to reduce the communication over
head when three entities wants to communicate securely as in authentication protocol in GSM and in e-
commerce. The proposed scheme satisfies various desirable security properties. Also, the performance of
the proposed scheme is tested.
KEYWORDS
Tripartite, Signcryption, Without Bilinear Pairing, ECDLP, Identity Based, Security Requirements
1. INTRODUCTION
Confidentiality and authenticity are the most important security goals. Traditionally, these two
goals are separately examined, the encryption schemes supports the confidentiality and digital
signature schemes provide the authenticity [1]. The sender first digitally signs the message using
his private key then encrypts the digitally signed message using his public key, under the
consideration of using public key cryptography. The encrypted (message + signature) is then sent
together with the receiver that decrypt the message then verify the signature [2].
In 1997, Zheng [3] proposed using a single cryptographic primitive to achieve both confidentiality
and authenticity. He called this primitive 'signcryption'. A signcryption scheme typically consists
of three algorithms: Key Generation (Gen), Signcryption (SC), and Unsigncryption (USC).
Key generation phase in which each user generates his key pairs that used in the Signcryption and
unsigncryption phases, signcryption (SC) is normally a probabilistic algorithm in which the
message is signed and encrypted in a single logical step, and unsigncryption (USC) is almost
certainly to be deterministic in which the receiver authenticates the sender then decrypt the
message. Any signcryption scheme should provide the correctness, accuracy and security
requirements as it will demonstrated in the rest of the paper [4].
In 2002, Malone-Lee proposed the first ID-based signcryption scheme [5]. Figure 1 shows the
basic structure of an ID-based signcryption scheme [6]. Since then, many ID-based signcryption
schemes [7-10] have been proposed.

2
Figure 1. ID-based signcryption scheme structure
When three entities involved in an application as in e-commerce secure electronic transaction
(SET) protocol who are the merchant, the customer and the bank , the tripartite key agreement
protocols are important in providing essential security[11].
A new tripartite signcryption scheme from bilinear pairings based on elliptic curve discrete
logarithm problem developed by Y. Abouelseoud [12]. This tripartite signcryption scheme is used
to reduce the communication overhead in the secure electronic transaction protocol (SET). Also,
H. M. Elkamchouchi, E. Abou El-kheir and Y. Abouelseoud developed a tripartite signcryption
scheme without bilinear pairings [13].
In this paper, a new Id-based tripartite signcryption scheme without bilinear pairings is proposed
and its security and performance are analyzed.
The rest of the paper is organized as follows. Section 2 discusses the security requirements for
any signcryption scheme. Section 3, presents the proposed identity based tripartite scheme
without bilinear pairing is. Section 4, introduces the security analysis of the proposed scheme
followed by section 5 that discusses the proposed scheme performance. Finally, Section 6
concludes the paper.
2. SECURITY REQUIREMENTS FOR ANY SIGNCRYPTION SCHEME
Here, the security requirements for any signcryption scheme are provided [1, 13, 14 ,15]:
2.1 Confidentiality
It means that only the intended recipient of a signcrypted message should be able to read its
contents. That is, upon seeing a signcrypted message, an attacker should learn nothing about the
original message, other than perhaps its length.
2.2 Unforgeability
It refers to the inability of any entity to produce a valid message-signature pair except the
designated signer.
IDr Qr Qs IDs
m (r,c,s) m or⊥
Private Key
Generator
Signcrypt Unsigncrypt

3
2.3 Public Verifiability
It means that any third party or judge can verify that the signcrypted text is valid or not, without
any requirement for the private key of the sender or the recipient.
2.4 Non-Repudiation
The sender of a message cannot later deny having sent the message. That is, the recipient of a
message can prove to a third party that the sender indeed sent the message.
2.5 Integrity
This means that the recipient should be able to verify that the received message is the original one
that was sent by the sender and it has not been tampered with during transmission.
2.6 Authentication
The receiver needs to authenticate the sender. This identity of the sender is verified through the
key recovery process and the message integrity is checked using a suitable one-way hash
function.
2.7 Forward Secrecy
It refers to the inability of an attacker to read signcrypted messages, even with access to the
sender’s private key. That is, the confidentiality of signcrypted messages is protected, even if the
sender’s private key is compromised.
3. THE PROPOSED PAIRING FREE ID-BASED TRIPARTITE SCHEME
3.1 Setup
Given security parameter l (usually 160), the PKG chooses q a large prime number with k
2
q > ,
(a, b) is a pair of integers which are smaller than q and satisfy 0
q
mod
)
b
27
a
4
( 2
3
≠
+ . E is the
selected elliptic curve over the finite field q
mod
)
b
ax
x
(
y
:
F 3
2
q +
+
= . P is the base point or
generator of a group of points on E, denoted as G . Also, O is the point at infinity and n is the
order of the point P, with n being a prime number, O
P
.
n = and k
2
n > . The PKG selects a
cryptographic one way hash function q
*
Z
}
1
,
0
{
:
H → . The PKG selects a random number
PKG
mk as the master key and computes the master public key P
.
mk
R PKG
= . The PKG keeps
PKG
mk secret and publishes the system parameters params: }
H
,
R
,
G
,
P
,
E
,
k
{
3.2 Key generation
The PKG generates the secret and public key pairs for the three communicating parties. The PKG
sends the secret keys via a secrete channel and publishes the public keys with user the identities.
The PKG calculates the secret keys of the three communicating parties as follows:
q
mod
)
mk
).
ID
(
H
(
d PKG
a
a = , q
mod
)
mk
).
ID
(
H
(
d PKG
b
b =

4
and q
mod
)
mk
).
ID
(
H
(
d PKG
c
c = .
The PKG calculates the public keys for entities A, B and C as follows: R
.
d
Q a
a = , R
.
d
Q b
b =
and R
.
d
Q c
c = respectively. Figure 2 shows the id-based tripartite signcryption and
unsigncryption phases.
3.3 Signcryption Phase
A wants to send a message 1
m to B and a message 2
m to C. A signcrypts the messages as
follows:
The sender A generates a random number ]
1
q
,
1
[
w −
∈ and computes:
R
.
w
.
ID
k a
1 = , b
b
2 Q
.
w
.
ID
k = , and c
c
3 Q
.
w
.
ID
k = , the key used is the x-coordinate
value of the points 3
2
1 k
,
k
,
k
)
( 1
1 2
m
E
c k
= , and )
( 2
2 3
m
E
c k
=
)
||
(
),
||
||
||
||
( 2
1
1 c
c
c
k
c
ID
ID
ID
Hash
r c
b
a =
=
q
mod
)
d
.
r
w
(
s a
−
=
A sends )
s
,
c
,
r
( to both A and B
3.4 Unsigncryption
Receiver B proceeds as follows:
The receiver B uses his/her secret key b
d to recover the encryption key
b
b
a
b
b
2
2 Q
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
.
d
k
;
k =
+
= .
B recovers 1
k where R
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
k a
a
a
1 =
+
= . Thus any third party can
authenticate the sender.
B computes )
||
||
||
||
( 1
k
c
ID
ID
ID
Hash
r c
b
a
=
−
, if the equation r
r =
−
holds B accepts the
signcrypted-text.
B computes )
( 1
2
1 c
D
m k
=
The receiver C does the same steps as B:
The receiver C uses his/her secret key to recover the encryption
key c
c
a
c
c
3
3 Q
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
.
d
k
;
k =
+
= .
C recovers 1
k where R
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
k a
a
a
1 =
+
= .
Then C computes )
k
||
c
||
ID
||
ID
||
ID
(
Hash
r 1
c
b
a
=
−
, if the equation r
r =
−
holds B
accepts the signcrypted-text.
Finally, C computes )
( 2
2 3
c
D
m k
=

5
3.5 Signature Verification by Any Third Party
Any third party can recover 1
k where R
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
k a
a
a
1 =
+
= without using any short
or long term secret keys. Then, the third party computes )
k
||
c
||
ID
||
ID
||
ID
(
Hash
r 1
c
b
a
=
−
, if
r
r =
−
accepts the signcrypted-text.
Figure 2 The proposed Id-based tripartite signcryption scheme.
4. SECURITY ANALYSIS OF THE PROPOSED SCHEME
4.1 Correctness
• The correctness of the signature verification:
R
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
k a
a
a
1 =
+
=
)
Q
.
r
R
).
d
.
r
w
.((
ID
k a
a
a
1 +
−
=
R
.
w
.
ID
)
Q
.
r
R
.
d
.
r
R
.
w
.(
ID
k a
a
a
a
1 =
+
−
=
• For the receiver B:

6
)
Q
.
r
R
.
s
.(
ID
.
d
k a
b
b
2 +
=
)
Q
.
d
.
r
R
.
d
).
d
.
r
w
.((
ID
k a
b
b
a
b
2 +
−
=
b
b
a
b
a
b
b
b
2 Q
.
w
.
ID
)
Q
.
d
.
r
Q
.
d
.
r
Q
.
w
.(
ID
k =
+
−
=
• For the receiver C:
)
Q
.
r
R
.
s
.(
d
.
ID
k a
c
c
3 +
=
)
Q
.
d
.
r
R
.
d
).
d
.
r
w
.((
ID
k a
c
c
a
c
3 +
−
=
c
c
a
c
a
c
c
c
3 Q
.
w
.
ID
)
Q
.
d
.
r
Q
.
d
.
r
Q
.
w
.(
ID
k =
+
−
=
4.2 Security Properties
This section demonstrates that the proposed id-based tripartite signcryption scheme provides the
seven security functions that defined in section 2. The security of the proposed scheme relies on
the elliptic curve discrete logarithm problem (ECDLP) that considered hard computational
problem till now [16].
Definition 1: The Elliptic Curve Discrete Logarithm Problem (ECDLP) is defined as follows. Let
G and Q be two points on an elliptic curve and G is of order n and n is a prime. The point
G
.
k
Q = , where n
k . Given these two points G and Q , find the discrete logarithm of Q to
the base G; that is, k [16].
4.2.1 Confidentiality
If an adversary wants to derive the original message, he must be able to recover 3
2 k
,
k to encrypt
the messages or the key 1
k . However, to obtain the secret keys 3
2
1 k
,
k
,
k is equivalent to solving
the ECDLP. Assume that an adversary tries to compute any of the keys
R
.
w
.
ID
)
Q
.
r
R
.
s
.(
ID
k a
a
a
1 =
+
= , )
Q
.
r
R
.
s
.(
ID
.
d
k a
b
b
2 +
= , )
Q
.
r
R
.
s
.(
d
.
ID
k a
c
c
3 +
= , he
must be able to derive the random number w to get the correct 1
k , the receiver's secret key b
d ,
where R
.
d
Q b
b = , and the receiver's secret key c
d , where R
.
d
Q c
c = . Therefore to derive
c
b d
,
d
,
w one needs to solve the ECDLP. Without knowing the secret key of the receiver, no one
can recover the message encryption key. It is only the valid receiver with valid identity with
secret key b
d , c
d who can recover the key and unsigncrypt the message.
4.2.2 Unforgeability
The signcrypted text is generated using the sender's secret key a
d . Also, the sender's secret key is
computed as R
.
d
Q a
a = , but computing a
d is another elliptic curve discrete logarithm problem
under Definition 1.So, no one can forge signcrypted text without knowing the sender's secret key
a
d .
If an adversary wants to forge a signcrypted text he proceeds as follow:
• Generate random number w'
• R
'.
w
.
ID
'
k a
1 = , b
b
2 Q
'.
w
.
ID
'
k = , and c
c
3 Q
'.
w
.
ID
'
k =
• )
'
(
' 1
'
1 2
m
E
c k
= , and )
'
(
' 2
'
2 3
m
E
c k
=

7
• )
'
c
,
'
c
(
'
c
),
'
k
||
'
c
||
.
ID
||
.
ID
||
.
ID
(|
Hash
'
r c
b
1
c
b
a =
=
• q
mod
)
d
'.
r
'
w
(
'
s adv
−
= , adv
d is the attacker secrete key
• The attacker sends )
'
s
,
'
c
,
'
r
( to both A and B
The receiver B and C unsigncrypts the message by recovering the key 3
2 '
k
,
'
k respectively as
follows:
For the receiver B: b
b
a
b
b
2 Q
.
w
.
ID
)
Q
'.
r
R
'.
s
.(
d
.
ID
'
k =
+
=
)
Q
'.
r
.
d
Q
).
d
'.
r
'
w
.((
ID
)
Q
'.
r
R
'.
s
.(
d
.
ID
'
k a
b
b
adv
b
a
b
b
2 +
−
=
+
=
)
Q
.
w
.
ID
k
(
)
Q
.
d
'.
r
Q
.
d
'.
r
Q
'.
w
.(
ID
'
k b
b
2
a
b
b
a
b
b
2 =
≠
+
−
=
Then B computes )
( 1
'
1 2
c
D
m k
≠ . Also , the same steps are carried out by receiver C. Without
knowing the sender's secret key, no one can generate a valid signcrypted text. Therefore, the
proposed scheme achieves unforgeability.
4.2.3 Authentication
The receiver needs to authenticate the sender. The receiver authenticates the sender through the
key recovery process and the message integrity is checked using a suitable one-way hash
function.
4.2.4 Public Verifiability
Any third party can recover 1
k without using any secret keys as demonstrated in Section 3.5.
4.2.5 Non-Repudiation
The sender cannot deny sending the signcrypted text because any third party can make sure that
the original sender is the one who signcrypted the message. So, the public verifiability property
solves the problem of non-repudiation.
4.2.6 Integrity
The alteration or modification in the ciphertext by any third party can be easily detected because
of the signature part that will need to be changed accordingly.
4.2.7 Forward Secrecy
If the attacker tries to derive the plaintext m, he has to decrypt the associated ciphertext c using
the corresponding secret key. This secret session key involves a random number w that appears in
the computation of the three keys: R
.
w
.
ID
k a
1 = , b
b
2 Q
.
w
.
ID
k = , and c
c
3 Q
.
w
.
ID
k = . Without
knowing the random number w, even if the long term key of the sender is known, the encryption
key cannot be recovered. In other words, he cannot decrypt the signcrypted text to get a previous
message m. Even if the sender's private key is compromised, the proposed scheme supports the
forward secrecy of message confidentiality.

8
5. PERFORMANCE OF THE PROPOSED SCHEME
First, in Table 1, the time abbreviations are listed as will be used in the performance evaluation
table that follows. The performance of the proposed scheme is examined in Table 2, it shows the
number of the computationally operations involved in the proposed identity based tripartite
scheme. In the proposed scheme, the public key generator in set up phase performs 4 scalar point
multiplications over an elliptic curve, 3 hash operations and 3 multiplications over a finite field.
For the sender (A) , it performs 3 scalar point multiplications over an elliptic curve, 2 encryptions,
one hash operation and 4 multiplications over a finite field in the signcryption phase. The
receivers (B , C) , each receiver in unsigncryption phase performs 3 scalar point multiplications
over an elliptic curve, one decryption operation, one hash operation , one addition operation over
an elliptic curve and 2 multiplications over a finite field.
6. CONCLUSION
This paper introduces a new identity based tripartite signcryption scheme without bilinear
pairings. The proposed scheme is efficient in case of sending two different messages. The security
analysis have been proved. Also, the performance of the proposed scheme have been discussed.
The proposed scheme may be used in various applications such as mobile communication , secure
electronic transaction (SET) protocols and e-cash protocol to reduce computations and timing
cost . The proposed Id-based tripartite signcryption scheme can be used between the mobile
communication entities which will reduce the signaling overhead and the computations required.
Table.1Time Abbreviations
Symbol Operation
TEC-mult time required for executing the point multiplication operation on elliptic curve E
TEC-add time required for executing the point addition operation on elliptic curve E
Tmult time required for executing modulus multiplication in a finite field
Th time required for executing one way dispersed row function operation
Tencr time required by the system for executing encryption operation
Tdecr time required by the system for executing decryption operation
TABLE.2 The performance of the proposed id-based tripartite Signcryption scheme.
Phase The proposed ID-based tripartite scheme
Set Up 4TEC-mult+3Th+ 3Tmult
Signcryption 3TEC-mult+1Th+ 4Tmult+ 2 Tencr
Unsigncryption
(for each receiver)
3TEC-mult+1TEC-add+2Tmult+ 1Th+ 1 Tdecr
Total 10TEC-mult+1TEC-add+5Th+9Tmult+2 Tencr + 1 Tdecr

9
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Technology, Volume 2, Issue4, April-2013 ISSN 2278-7763
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(Signature) + Cost (Encryption), Advances in Cryptology, LNCS, Vol. 1294. Springer-Verlag,
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[14] http://en.wikipedia.org/wiki/Authentication
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