A New Pairing Free ID Based Certificate Less Digital Signature (CL-DS) Scheme Using Elliptic Curve Cryptography

A New Pairing Free ID Based Certificate Less Digital Signature (CL-DS)
Scheme Using Elliptic Curve Cryptography
Sarvesh Tanwar
Dept. of Computer Science & Engineering
Mody University of Science & Technology, MUST
Laxmangarh, India
s.tanwar1521@gmail.com
Anil Kumar ((Senior IEEE Member)
Dept. of Computer Science & Engineering
Mody University of Science & Technology, MUST
Laxmangarh, India
Abstract— Certificate less public key cryptography is a design
that is secure against key escrow issue and remove loopholes of
ID based cryptography. Lots of work has been done on CL –DS
yet they depend on bilinear pairing that required more time to
perform pairing operations. Bilinear pairing is executed with
super-singular EC group which is tedious. In this paper we
propose a pairing free ID based Certificate Less Digital Signature
(CL-DS) scheme utilizing elliptic curve cryptography, which
maintain a strategic distance from tedious operations required in
bilinear matching. We improve the security of the previously
proposed scheme with less computation time with time stamp.
Keywords- Certificate Less, Elliptic curve, Bilinear pairing,
Digital Signature, Security
I. INTRODUCTION
IBE evacuates the requirement for a PKI, by supplanting it
with PKG that process user's private key. IBE is more
efficient, yet it experiences key escrow issue. Key escrow
means that PKG who is trusted third party register the private
keys for the users it implies PKG can read the messages of
users in the framework. In IBE, there is a significant practical
problems associated with it is to handle key revocation of
users. Al-Riyami and Paterson [25] in 2003, proposed new
scheme for public key encryption that removes the
disadvantages of both public key encryption and IBE keeping
in mind the end goal to determine the key escrow issue. The
new scheme is known as Certificate less –Public Key
encryption (CL-PKE). CL-PKE is best among the two. The
fundamental reason to utilize CL-PKE is to determine key
escrow issue acquired from IBE without the use of certificates
as in PKI. Today gadgets having constrained computational
resources and communication bandwidth discover CL public
key cryptography extremely appealing and imperative to
reduce stack on the system. To wipe out the need of certificate
Shamir proposed ID based cryptosystem in which PKG is
responsible to take client's identity as input and process
relating private key. However this method has two
disadvantages – 1) If PKG is compromised, all messages and
public private key pair protected with him also be
compromised and 2) As PKG knows the private keys of users,
he can use user’s private key to decrypt messages on behalf of
them. Thus it suffers major drawback of key escrow problem.
Subsequently to prevent from key escrow issue CL - DS came
into light. In CL cryptography, PKG computes a partial private
key for each user using his master private key . The
user combines this partial key with some user selected secret
information to generate a complete private key. Thus PKG
has not knowledge of user’s private key. The private/ public
key of user is computed from user’s private key and PKG’s
public parameters . The public key is made available
by transmitted it with the message. Thus public key does not
required any authentication mechanism. So, it is better than
certificate based signature because public key can’t be
computed from user’s identity alone.
Signer’s public key is attached as part of digital signature in
CL digital signature. It provides some benefits over ID based
signature. There are many proposal on CL digital signature
[5][14][16][17][18][19][20][21].
The rest of the paper is presented in the following sections:
Motivation, Related work and security notations are explained
in section 2. Section 3 explained the proposed approach
followed by security analysis in section 4. Section 5 explains
the conclusion of the paper.
II. MOTIVATION AND RELATED WORK
A. Motivation
Number of researchers working on CL digital signature. But
they proposed schemes which are based on bilinear pairing.
There are disadvantage of bilinear pairing as far as calculation
overhead. Although the scheme proposed by Chen, Yu-Chi, et
al [33] is secure but security level of this scheme can be
increased and decrease the computation cost.
The motivation of this paper is to propose Certificate Less
Digital Signature (CL-DS) scheme and prove its security.
B. Related Work
In PKI certificates are issued by Certificate Authority (CA). It
depends on CA. A user can compute his private and public
key, then submit certificate request to CA to issue a certificate.
A user name and his public key are bind on a certificate signed
by a CA. But there are two problems in PKE-(1) Certificate
revocation is crucial and (2) Verification of certificate is costly
to check whether certificate is valid or expired.
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To avoid these problems, in 1984 Shamir proposed the idea of
identity based cryptosystem based on inter factorization
problem to simplify key management procedures of PKI [14].
ID based cryptosystem is advantageous over traditional PKI. It
is a public key system where public key can be represented by
an arbitrary string such as name, contact number, email
address, network or IP address. Rather than creating arbitrary
private/public key, Shamir proposed mathematically
producing beneficiary public key from receiver's identity
information. PKG has maser public and private keys. Any user
can generate a public key with reference to the identity by
combing the identity attribute with master public key. He
utilizes his master private key to produce the private key for
the user [1][6].
The main motivation to propose this approach is to eliminate
the need of the certificate and issues related to them such as
whether certificate is valid or not, key/certificate revocation,
trust on CA and maintenance of public key directory. As the
public key is derived from the publically available information
then there is no need for public key directory and certificate
management.
Ai-Riyani and Paterson [25] in 2003 proposed the concept of
certificate less public key cryptosystem. Lots of work has been
done on CL public key cryptosystems by
[5][2][7][14][16][17][18][19][20][21].
C. Security Notions
• Setup ( ): PKGrandomly arbitrarily picks a secret
value and generates master secret key , master
public key and generates public parameters
of system required by signature schemes.
• Set Secret Key: User executes this algorithm on
( , , ) to generate partial private key
.
• Partial Private Key Extract: PKG runs this
algorithm only once.The input of this algorithm is
, , , ). It returns .
• Set Private Key: for
given ( , , , , ) return
complete private key as .
• Set Public Key: given ( , , , )
return complete public key as .
• Certificate less Digital Signature: for
given( , , , ) outputs signature on
message . It is run by signer or user.
• Certificate less Digital Signature Verification: for
given ( , , , , ) returns if
the signature is valid, otherwise returns .
D. Drawback of previous work
The schemes [2][5][6][14][33] are based on bilinear pairing
and does not consider key escrow issue in ID based
cryptography. We propose a new scheme that takes less time
as compared to [2][6]. The security of our scheme is based on
ECDLP assumptions. It could be efficient in terms of
computation, communication cost and security [1].
III. PROPOSED CL DS SCHEME
In this section, we propose an efficient pairing free Certificate
Less Digital Signature scheme based on ECC.
The followings steps are required for CL-DS:-
Setup: For given security parameter , PKG generates
system‘s parameter in the following way:
• To determine the tuple { , / , , } , PKG
selects a prime number .
• Selects a secure random number ∈ ∗
as the
master private key and computes master public
key = .
• Selects one way hash functions , , , ∶
{0,1}∗
→ {0,1} .
• Publish system parameters = { , / , , , ,
, , , } .
Set Secret Key: Let Alice is the sender and Bob is receiver.
PKG computes private and public key pairs of the sender and
the receiver with identities of , ∈ {0,1}∗
.
The sender Alice with select a secure random number
∈ ∗
as their secret key and computes the scalar
multiplication = for mathematical related public key
Partial Private Key Extract: After computing , Alice
send ( , ) ) to the PKG. PKG chooses a random number
∈ ∗
and computes = P. PKG also computes secret
key = + where
= ( ∥ ∥ )
= ( ∥ ∥ )
The private key and are send to the sender Alice with
by PKG via secure channel. The corresponding ID based
public key of Alice is computed = + . Now
the private/public key ( , ) pair can be verified by
checking the equation = + = .
Set Private Key: The private key is set = ( , ) for
Alice with . Similarly Bob’s private key with is
set = ( , ).
Set Public Key: The public key is set = ( , )for
Alice with and for Bob with public key will be
= ( , ) .
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A. CL-DS-Sign
In order to generate the signature on the message Alice
performs the following steps:-
( , , , , , ) : The algorithm works as
follows:-

• Alice sends = ( ) to Bob.
• Alice selects a random number ∈ ∗
and
computes = .
• Alice Computes
= ( ) ,
= ( , , , , , )
= ( , , , , , , , ) where is
time when message signed.
• Computes = ( + + )
• Alice Computes
= ( ∥ )⨁
=
And returns signature = ( , , ) on message .
B. CL-DMS Verification
To verify signature is ( , , ) on message , the verifier
Bob performs the following steps:
• Bob decrypts = ( ) to extract message
signature time.
• Computes = ( )
• ⨁ = ( ∥ )
• Before verification Bob first of all computes
, ,
= ( , , , , , )
= ( , , , , , , )
= +
where = ( ∥ ∥ )
And then verifies the following equation
= + +
Checks whether the equation = . If it verifies the
verifier accepts the signature = ( , , ) otherwise rejects it.
IV SECURITY AND PERFORMANCE ANALYSIS
A. Security Analysis
Here we shall prove that the proposed scheme is secure against
foregeability, key replacement and key escrow attack which
shows how it full fill the security requirements confidentiality,
unforegeability. Verifier can verify = and check the
validity of the signature.
a. Correct and consistent
The proposed scheme is correct and consistent

= ( + + )
= + +
= + +
Hence the theorem is proved.
b. Origin Authentication
The proposed scheme consists of chain of digests in which ID
of user is hashed e.g.
= ( , , , , , , , ) . After receiving
message receiver will compute ′ , ′ ′ . If ′ = , ′ =
′ = , then only be signature verified, which
provides authentication for both parties.
c. Time Stamp
First of all receiver check when the message is signed by
computing = ( ) . Then computes ′ which contains
the time when message is signed, =
( , , , , , , , ) . If extracted t is same as
calculated in , that will ensure message is new and is only
sent by the sender.
d. Confidentiality
Although Adversary knows σ, Y, Z still he is not be able to
get partial key information of the message m because
adversary would have to know the receiver’s private key as
well as two prime number and its factor known only to
PKG, signer and receiver. It is not feasible to derive a factorial
from a large prime number [1].
A CL DS scheme is secure against attacks. Master secret key
and complete private (Secret) key = ( , ) will
not be accessed by Adversary .Adversary is not allowed to
extract secret key and public key = ( , ).
e. Unforgeability
An adversary can be able to forge any DS only if he is in
possession of sender’s key = ( , ) which is required
to sign the message. He can only obtain the message by
decrypting it using corporate public key ( , ).
f. Key Replacement Attack
Private/public key pair can be replaced by adversary. But in
this scheme he does know the user’s partial private key issued
by PKG. An adversary who knows only the public key
but not the corresponding secret key will not be able to
forge a signature and message . The identity based public
key of the user is ( , ).
= +
= +
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= +
= , = , = +
=( + )
=
Hence demonstrated that public key is derived from secret of
the user e.g = . Thus this scheme secure against key
replacement attack.
g. Key Escrow attack
PKG only knows the partial key but not the additional secret
key generated by user. He is not able to do any cryptographic
operations as the user [13].
h. Secure against Type-I attack
In Type –I attack the adversary knows the public keys of the
PKG. To know the master secret key , adversary has
to solve the Discrete Logarithm Problem (DLP) which is
impossible as well as infeasible from DLA. The proposed
scheme is secure against Type -1 attack.
i. Secure against Type-II attack
In Type –II attack adversary is dishonest PK. G, who
knows the partial private and public key and can attempt to
find user’s private key. In proposed scheme user’s private key
, is generated from ∗
and can be known only if DLP
can be solved. Thus proposed scheme is secure against Type-
II attack.
B. Performance Analysis
In this section we compare the number of operation required in
proposed scheme and existing schemes [2][6]. Table 2 shows
how proposed scheme is more efficient in term of operations.
It does not require bilinear pairing thus it reduce the
operational cost required to perform exponential operations in
bilinear pairing.
Notation Description
Time required for executing modular
exponentiation 240
Time required for executing an elliptic
curve scalar point multiplication,
29
Time required for executing bilinear
pairing, 87
Time required for addition of two elliptic
curve points, 0.12
Time required for executing one way hash
function
Time required for executing modular
multiplication
Table 1: Definition and conversion of various operation units
Operation Proposed
Scheme
Chen, Yu-
Chi, et al
[33]
He,
Debiaoet
al.[ [2]
Harn’
s
Sche
me [6]
Exponential - 3 3 7
Hash 4 4 4 3
Multiplicati
on
6 4 4 3
Addition 6 6 3 -
Total time 174.72 836.72 836.36 1767
Table 2 : Comparison with existing CL-DS
V. CONCLUSION
In this paper, we have proposed secure and efficient CL-DS
scheme which is secure against Type-I and Type –II attacks as
well as other attacks [2][33]. Its computational cost and
communication overhead is also less and can be used for
limited communication and computational devices. Hence
proved that proposed scheme is taking less time as compared
to [2][6]. The proposed scheme uses for the validity of the
message which prevents the scheme from the replay attack.
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