ID-Based Directed Multi-Proxy Signature Scheme from Bilinear Pairings

B.Umaprasada Rao & P.Vasudeva Reddy
International Journal of Computer Science and Security (IJCSS), Volume (5) : Issue (1) : 2011 107
ID-Based Directed Multi-Proxy Signature Scheme
from Bilinear Pairings
B.Umaprasada Rao buprasad@yahoo.co.in
Research scholar
Dept. of Engineering Mathematics
A.U. College of Engineering
Andhra University
Visakhapatnam. A.P, INDIA.
Dr.P.Vasudeva Reddy vasucrypto@yahoo.com
Associate Professor
Dept. of Engineering Mathematics
A.U. College of Engineering
Andhra University
Visakhapatnam, A.P, INDIA.
Abstract
In a multi-proxy signature scheme, an original signer delegates his signing power to a group of
proxy signers. Then the group of proxy signers cooperatively generates a multi-proxy signature
on behalf of the original signer; and any one can verify the validity of the multi-proxy signature.
But, when the signed message is sensitive to the signature receiver, it is necessary to combine
the concepts of multi-proxy signatures with directed signatures. In this paper, we propose an
identity based directed multi-proxy signature scheme using bilinear pairings. This scheme allows
a group of proxy signers to generate a valid multi-proxy signature to a designated verifier. The
designated verifier can only directly verify the multi-proxy signature generated by a group of proxy
signers issued to him on behalf of the original signer and, in case of trouble or if necessary, he
can convince any third party about the validity of the signatures. Finally, we discuss the
correctness and security analysis of the proposed scheme.
Keywords: Public Key Cryptography, Proxy Signature Scheme, Multisignature Scheme, Proxy
Signature Scheme, Bilinear Pairing, CDH Problem.
1. INTRODUCTION
Proxy signature, as an important cryptographic primitive, was firstly introduced by Mambo,
Usuda, and Okamoto in 1996 [1]. In a proxy signature scheme, an original signer is allowed to
delegate his signing power to a designated person called the proxy signer and the proxy signer is
able to sign the message on behalf of the original signer. There are three types of delegation: full
delegation; partial delegation and delegation by warrant. In full delegation, the original signer
gives its private key to the proxy signer. In partial delegation, the original signer produces a proxy
signature key from its private key and gives it to the proxy signer. The proxy signer uses the
proxy signature key to sign. As far as delegation by warrant is concerned, warrant is a certificate
composed of a message part and a public signature key. The proxy signer gets the warrant from
the original signer and uses the corresponding private key to sign.
Since the proxy signature concept was proposed, various kinds of proxy signature schemes have
been proposed such as threshold proxy signatures [2, 3, 4, 5, 6], multi proxy signatures [7, 8, 9],
proxy multisignatures [10, 9, 8], proxy blind signatures [11, 12 ], multi proxy multi signatures [13,
14], ordered multi proxy [15], multi proxy multi signcryption [16,17] etc. In threshold proxy
signature schemes, a group of n proxy signers share the secret proxy signature key. To produce
a valid proxy signature on the message m, individual proxy signers produce their partial

signatures on that message, and combine them into a full proxy signature on m. In a (t, n)
threshold proxy signature scheme, the original signer authorizes a proxy group with n proxy
members. Only the cooperation of t or more proxy members is allowed to generate the proxy
signature. Threshold signatures are motivated both by the demand which arises in some
organizations to have a group of employees agree on a given message or document before
signing, and by the need to protect signature keys from attacks of internal and external
adversaries.
In 1999, Sun proposed a threshold proxy signature scheme with known signers [4]. Then Hwang
et al. [3] pointed out that Sun's scheme was insecure against collusion attack. By the collusion,
any t - 1 proxy signers among t proxy signers can cooperatively obtain the secret key of the
remainder one. They also proposed an improved scheme which can guard against the collusion
attack. After that, [2] showed that Sun's scheme was also insecure against the conspiracy attack.
In the conspiracy attack, t malicious proxy signers can impersonate some other proxy signers to
generate valid proxy signatures. To resist the attack, they also proposed a scheme. Hwang et al
pointed out [18] that the scheme in [3] was also insecure against the attack by the cooperation of
one malicious proxy signer and the original signer.
As a special case of the threshold proxy signature, the multi-proxy signature scheme was first
introduced by Hwang and Shi [7]. In a multi-proxy signature scheme, an original signer could
authorize a group of proxy members and only the cooperation of all the signers in the proxy group
can generate the proxy signatures on behalf of the original signer. Multi proxy signature scheme
can be regarded as a special case of the ( , )t n threshold proxy signature scheme [5] fort n= . It
plays an important role in the following scenario: Suppose a president of a company needs to go
on a business trip, during the trip he will receive many important documents must be signed by
him. Some may need to be responded to quickly. To solve this problem, before going on a trip,
the president can delegate his signing power to every department manager of the company. Then
the document must be signed jointly by these department managers authorized by the president
of the company. One solution to the case of this problem is to use a multi-proxy signature
scheme.
A contrary concept, called proxy-multisignature is introduced by Yi et al. in 2000 [10], where a
designated proxy signer can generate the signature on behalf of a group of original signers.
Hwang and Chen [13] introduced the multi-proxy multi-signature scheme. Only the cooperation of
all members in the original group can authorize a proxy group; only the cooperation of all
members in the proxy group can sign messages on behalf of the original group.
Some designated verifier multi proxy signatures are also proposed in the literature [19]. In these
schemes, an original signer could authorize a group of proxy members and only the cooperation
of all the signers in the proxy group can generate the proxy signatures to a designated verifier on
behalf of the original signer. The designated verifier only can directly verify the multi-proxy
signature issued to him. In these schemes, the designated verifier cannot convince any third party
about the validity of the multi-proxy signatures. To solve this problem, it necessary to combine the
concepts of multi-proxy signatures with the directed signatures [20, 21, 22, 23].
Plenty of multi-proxy signature schemes have been proposed under the CA-based public key
systems. The concept of ID-based public key system, proposed by Shamir in 1984 [24], allows a
user to use his identity as the public key. It can simplify key management procedure compared to
CA-based system, so it can be an alternative for CA-based public key system in some occasions,
especially when efficient key management and moderate security are required. Many ID-based
schemes have been proposed after the initial work of Shamir, but most of them are impractical for
low efficiency. Recently, the bilinear pairings have been found various applications in
cryptography, more precisely; they can be used to construct ID-based cryptographic schemes
[25, 26, 27, 28, 29].

Motivated by the mentioned above, in this paper, based on Hess ID-based signature scheme
[28], a directed multi-proxy signature scheme is proposed. In the proposed scheme, the
designated verifier can only directly verify the multi-proxy signature generated by a group of proxy
signers issued to him, on behalf of the original signer, and he can convince any third party about
the validity of the signatures. To the best of our knowledge there is no existing scheme on this
concept. The proposed scheme can provide the security properties of proxy protection,
verifiability, strong identifiability, strong unforgeability, strong nonrepudiability, distinguishability,
and prevention of misuse of proxy signing power.
The rest of the paper is organized as follows. Section 2 briefly explains the bilinear pairings and
some computational problems on which of our scheme is based. The syntax and security model
of ID-based Directed Multi Proxy Signature Scheme is given in Section 3. We then present our
ID-based Directed Multi Proxy Signature (ID-DMPS) Scheme in Section 4. The correctness and
security analysis of the proposed scheme is given in Section 5. Section 6 concludes this paper.
2. PRELIMINARIES
In this section, we will briefly review the basic concepts on bilinear pairings and some related
mathematical problems.
2.1 Bilinear Pairings
Bilinear pairing is an important cryptographic primitive and has been widely adopted in many
positive applications in cryptography.
Let 1G be a additive cyclic group generated by P, whose order is a prime ,q and 2G be a
multiplicative cyclic group of the same order .q A bilinear pairing is a map 1 1 2:e G G G× → with
the following properties:
1. Bilinear: ( ) ( ), , ,
ab
e aP bQ e P Q= for all
*
1, and all , qP Q G a b Z∈ ∈ .
2. Non –degenerate: There exists 1,P Q G∈ such that ( ), 1e P Q ≠ .
3. Computable: There is an efficient algorithm to compute ( ) 1, for all , .e P Q P Q G∈
Such a pairing may be obtained by suitable modification in the Weil-pairing or the Tate-pairing on
an elliptic curve defined over a finite field [25].
2.2 Computational Problems
Now, we give some computational problems, which will form the basis of security for our scheme.
Decisional Diffie-Hellman Problem (DDHP): For
*
, , ,R qa b c Z∈ given ,P ,aP ,bP cP in 1G ,
decide whether mod .c ab q≡
Computational Diffie-Hellman Problem (CDHP): For
*
, , ,R qa b c Z∈ given ,P ,aP bP in 1G
Compute abP .
Bilinear Diffie-Hellman Problem (BDHP): For
*
, , ,R qa b c Z∈ given ,P ,aP ,bP cP in 1G ,
compute ( , )abc
e P P in 2G .
Gap Diffie-Hellman Problem: A class of problems, where DDHP can be solved in polynomial
time but no probabilistic algorithm exists that can solve CDHP in polynomial time.
Such groups can be found in supersingular elliptic curve or hyperelliptic curve over finite fields,
and the bilinear pairings can be derived from the Weil or Tate pairings. For more details, see [25].

3. SYNTAX AND SECURITY REQUIREMENTS FOR ID-DMPS SCHEME
In this section, we give formal model and some security requirements for our ID-based directed
multi-proxy signature scheme (ID-DMPS).
3.1 Syntax of ID-Based Directed Multi-Proxy Signature Scheme
Our scheme has five phases described as follows:
In our identity-based multi-proxy signature scheme, there is an original signer and a group of
proxy signers. Let O be the original signer and { }1 2, ,....., nL PS PS PS= be the group of proxy
signers designated by O. Sometimes there may be a clerk or a chairman of the group.
For {1,2,.., }i n∈ , iPS has an identity iIDs , O has an identity oID .
Our ID-DMPS scheme consists of the following five algorithms.
• Setup: This algorithm is run by the PKG on input a security parameterl N∈ , and
generates the public parameters of the scheme and a master secret .s< > The PKG
publishes system parameters as params and keeps the s< > as secret.
• Extract: Given an identity ID, params, this algorithm generates the private key IDd
of ID . The PKG will use this algorithm to generate private keys for all participants in the
scheme and distribute the private keys to their respective owners through a secure
channel.
• Generation of the Proxy Key: This is a protocol between the original signer and all
proxy signers. All participants input their identities ,0 ,isID i n≤ ≤ the proxy signers also
take as input their private keys ,1 ,si
IDd i n≤ ≤ and the delegation warrantω which
includes the type of the information delegated, the period of delegation etc. The original
signer also inputs his secret key IDod . As a result of the interaction, every proxy signer
outputs a partial proxy signing key (1 ).iSKP i n≤ ≤
• Multi-proxy Signature Generation: This is a randomized algorithm. Every iPS takes
input his partial signing key (1 ),iSKP i n≤ ≤ the warrant mω ,the designated verifier’s
identity VID , and the message
*
{0,1} .M ∈ In the end, outputs a directed multi-proxy
signatureσ on the message M on behalf of the original signer.
• Multi-proxy Direct Verification: It is a deterministic algorithm. It takes input the
identities ,0 ,isID i n≤ ≤ the warrant ω , the message M and a directed multi-proxy
signature σ for M, the algorithm outputs 1 if σ is a valid multi-proxy signature for M by
the proxy signers on behalf of the original signer, and outputs 0 otherwise.
• Multi-proxy Public Verification: It is a deterministic algorithm. It takes identity of the
original signer oID , identities of the proxy signers iIDs , identity of the designated
verifier VID , message M, warrantω , Aid provide by VID or Clark and multi-proxy
signature σ as input, outputs 1ifσ is valid or 0 otherwise.
3.2 Security Requirements of ID-Based Directed Multi Proxy Signature

The following are general security requirements of the proposed scheme.
• Verifiability: From the proxy signature, the verifier can be convinced of the original
signer’s agreement on the signed message.
• Strong Identifiably: Anyone can determine the identity of the corresponding proxy
signer from the proxy signature.
• Strong Undeniability: Once a proxy signer creates a valid proxy signature of an original
signer, he cannot repudiate the signature creation.
• Distinguishability: Proxy signatures are distinguishable from normal signatures by
everyone.
• Prevention of Misuse: The proxy signer cannot use the proxy key for other purposes
than generating a valid proxy signature. That is, he cannot sign, with the proxy key,
messages that have not been authorized by the original signer.
• Strong Unforgeability: A designated proxy signer can create a valid proxy signature for
the original signer. But the original signer and other third parties who are not designated
as a proxy signer cannot create a valid proxy signature.
• Strong Designated Verifiability: The designated verifier uses his/her secret key to
verify the proxy signature generated by a proxy signer on behalf of the original signer to
designated verifier. So, only the designated verifier can verify the proxy signature issued
to him.
4. PROPOSED SCHEME ID-BASED DIRECTED MULTI-PROXY SIGNATURE
SCHEME FROM BILINEAR PAIRINGS
The proposed scheme involves four roles: the private key generator (PKG), the original signer, a
set of proxy signers { }1 2, ,....., nL PS PS PS= and the verifier. It consists of the following Six
algorithms.
Setup: Given security parameterl , the PKG chooses groups 1G and 2G be additive and
multiplicative groups of prime order 2l
q > with a bilinear pairing 1 1 2:e G G G× → and a
generator P of 1G . PKG then selects
*
qs Z∈ randomly and compute the public key pubP sP= ,
also picks cryptographic hash functions { }
* *
1 2 1, : 0,1H H G→ and { }
* *
2: 0,1 qh G Z× → . The
private key generator PKG now publishes system parameters as
params 1 2 1 2, , , , , , , ,pubG G q e P P H H h=< > , and keeps s< > secret as the master secret key.
Private key Extraction: Let the original signer identity oID and his private key
( )1o oID ID od sQ sH ID= = , and { }iPS be the proxy signers with identity{ }iIDs and their
corresponding private key ( )1 ,i iIDs IDs id sQ sH IDs= = for 1 i n≤ ≤ .
Generation of the Proxy Key: To delegate the signing power to proxy signers, the original
signer uses Hess’s ID-based signature scheme [28] to generate the signed warrant wm and each
proxy signer iPS computes his proxy key iSKP .

• The original signer computes ( ), oK
oU e P P= , where
*
R qk Z∈ ,
( )2 , ,o o w oH H ID m U= , ( ),o o oV h U H= and then computes oo o ID oW V d k P= + .
• The signature on wm is the warrant , ,w o om W V then he sends , ,w o om W V to each
proxy signer in the proxy group L.
• Each iPS L∈ verifies the validity of the signature on wm by computing
( ) ( ), ,
O
o
V
o o ID pubU e W P e Q P
−
= and ( )2 , ,o o w oH H ID m U= . Accepts the signature if
and only if ( ),o o oV h U H= .
If the signature valid, each iPS computes the proxy key iSKP as ii o IDs oSKP V d W= + .
Multi-Proxy Signature Generation: Suppose the proxy group L want to sign a delegated
message m, on behalf of the original signer, to the designated verifier V. Each proxy signer
iPS generates the partial signature and an appointed clerk C, who is one of the proxy signers,
combines the partial proxy signatures to generate the final multi-proxy signature.
• Each iPS randomly selects two integers
*
,i i R qk r Z∈ , computes ( ), ,i
i
k
PU e P P=
,i iP i IDsR rQ= ( ),i iP IDs i IDvL e d rQ= and broadcast ,i iP PU L to the remaining (n-1)
signers.
• Each iPS computes
1
i
n
P P
i
U U
=
= ∏ ,
1
i
n
P P
i
L L
=
= ∏ ,
1
i
n
P P
i
R R
=
= ∑ , ( ),P P PV h U H=
and broadcast to the clerk .
• Each proxy signer also computes ( ),i iP P PV h U H= and ,iP P i iW V SKP k P= +
where ( )2 ,P PH H M L= .
Finally the individual proxy signature of message m is , ,i i iP P PV W R .
• All the proxy signers send their partial signatures to the clerk C. The clerk verifies
each individual signature by checking the equality
( ) ( ) ( )( )2 , , , ,
P
o
i i o i
VV
P P P ID IDs pub oV h H M L e W P e Q Q P U
−
 
= + 
 
.
Once all individual proxy signatures are correct, the clerk C computes
1
i
n
P P
i
W W
=
= ∑ .
The valid directed multi-proxy signature is the tupleσ = , , , , ,w P P p om m V W R U .
Direct Verification: The designated verifier VID first evaluate
( ) ( )1
, , .
P
O
o i
VVn
n
P P ID IDs pub o
i
U e W P e Q Q P U
−
=
  
= +     
∑ and ( )( )2 , ,VP ID PH H m e d R= . He then
accepts the signature if and only if ( ),P P PV h H U= .
Public Verification: In case of trouble or if necessary, any third party T can verify the validity of
multi-proxy signature with the help of the ( ),VID P PAid e d R L= = provided by either the clerk C

or the designated verifier VID . Now with this Aid, T computes
( ) ( )1
, , .
P
O
o i
VVn
n
P P ID IDs pub o
i
U e W P e Q Q P U
−
=
  
= +     
∑ and 2 ( , )PH H m Aid= . T accepts the
signature if and only if ( ),P P PV h H U= .
5. ANALYSIS OF THE PROPOSED SCHEME
In this section first we discuss proof of correctness and then security analysis of the ID-DMPS
scheme.
5.1 Proof of Correctness
The following equations give the proof of correctness for individual proxy signer’s signature.
( ) ( )( ), ,
P
o
i O i
VV
P ID IDs pub oe W P e Q Q P U
−
+
( ) ( )( ), ,
P
o
i o i
VV
P ID IDs pub oe W P e Q Q P U
−
= +
( ) ( )( ), ,
P
o
o i
VV
P i i ID IDs oe V SKP k P P e d d P U
−
= + +
( ) ( ) ( )( ), , ,
PV
P i i i o o o oe V SKP P e k P P e SKP W W k P P U
−
= − + −
( ) ( ) ( ) ( ), , , ,
VPP i P PV k V V
i i o oe SKP P e P P e SKP P e k P P U
−− −
= −
( ), i
i
k
Pe P P U= = .
The following equations give the proof of correctness for multi-proxy signature.
( ) ( )1
, , .
P
o
o i
VVn
n
P ID IDs pub o
i
e W P e Q Q P U
−
=
  
+     
∑
( )1 1
, ,
P
o
i o i
VVn n
n
P ID IDs pub o
i i
e W P e Q Q P U
−
= =
    
= +         
∑ ∑
( ) ( )1 1
, ,
P
o
o i
VVn n
n
P i i ID IDs o
i i
e V SKP k P P e d d P U
−
= =
    
= + +         
∑ ∑
( ) ( )
1 1
, ,
PV
n n
n
P i i i o o o o
i i
V SKP k P P e SKP W W k P P U
−
= =
    
= + − + −    
    
∑ ∑
( ) ( )
1 1 11
, , , ,
P P P
Pi
V V Vnn n n Vk n
i i o o
i i ii
e SKP P e P P e SKP P e k P P U
− −
−
= = ==
     
= −     
     
∑ ∑ ∑∏
( )
1
, i
n
k
P
i
e P P U
=
= =∏
5.2 Security Analysis
Our ID-DMPS scheme satisfies the following security requirements which are stated in section
3.2.

Strong Identifiability: Because identity pubic key iIDsQ of all proxy signers are involved in the
verification of the proxy signature, anyone can identify all the proxy signers.
Strong Undeniability: The clerk verifies the individual proxy signature of each proxy signer, so
no one can be deniable of his signature.
Distinguishability: This is obvious, because there is a warrant wm in a valid multi-proxy
signature, at the same time, this warrant wm and the public keys of the original signer and the
proxy signers must occur in the verification process.
Prevention of Misuse: Due to using the warrant mω , the proxy signers can only sign messages
that have been authorized by the original signer.
Strong Unforgeability: In general, there are mainly three kinds of attacks: outsiders, who are not
participating in the issue of the proxy signature; some signers who play an active in the signing
protocol and the user (signature owner). Furthermore, some of these attackers might collude. The
outsider-attack consists of the original signer attack and any third adversary attack. We assume
that the third adversary can get the original signer’s signature on warrant mω (So, our scheme
needs not the secure channel for the delivery of the signed warrant). Even this, he forges the
multi-proxy signature of the message
'
m for the proxy group L and the original signer, this is
equivalent to forge a Hess’s ID-based signature with some public key. On the other hand, the
original signer cannot create a valid multi-proxy signature since each proxy key includes the
private key iIDsd of each proxy signer.
In our scheme, the clerk is one of the proxy signers, but he has more power than other
proxy signers. Assume that the clerk wants the proxy group to sign the false message
'
m . He can
change his iPU , therefore PU can be changed, but from the security of the basic ID-based
signature scheme and public one-way hash function 2H , it is impossible for the clerk to get
' '
andP PV W such that , , , , ,w P P p om m V W R U is a valid multi-proxy signature. Also, the attack
of some signers collude can be prevented for the identity of each proxy signer is involved in the
verification of the signature.
Finally, the user can not forge the multi-proxy signature because he can not obtain more
information than the Clerk.
Designated Verifiability: The designated verifier VID has to use his secret key VIDd at the time
of verification of the multi-proxy signature. So, only the designated verifier can directly verify the
validity of the proxy signature. No one can verify the validity of the multi-proxy signature without
the help of either the designated verifier VID or the designated Clark.
5.3 Performance Analysis
Performance of signature scheme protocols can be approximated in terms of computation and
communication overheads. In this section, we mainly discuss the performance of pro posed ID-
DMPS scheme.
For convenience, the following notations are used to analyze the computation and communication
complexity. smulT represents the time for one scale multiplication in 1G , pairT denotes the total

one pairing computation; mhashT define the time for one Map-to-Point hash function; tN denotes
the total number of transmissions and bN denotes the total number of broadcasts. Note that the
times for other computations or operations are ignored, since they are much smaller than smulT ,
pairT and mhashT .
We summarize the computation and communication overheads of our proposed ID-DMPS
scheme in Table1. As shown in Table1, The computation complexity for Setup, Extract,
Generation of proxy key, Multi signature generation, Direct verification, Public verification
algorithms are 1 ,smulT ( ) ( )1 1 ,smul mhashn T n T+ + + ( )2 1 pairn T+ ( )2 2 mhashn T+ + +
( )2 smuln T+ , ( )4 3 1 4pair mhash smulnT n T nT+ + + , 3 2pair mhashT T+ and 2 2pair mhashT T+
respectively. Also the total communication overheads for generation of Proxy Key and Multi-Proxy
signature generation algorithms are tnN , 2b tnN nN+ respectively in our ID-DMPS scheme.
Computation overheads
Communication
overheads
System Setup 1 smulT --
Key Extract ( ) ( )1 1smul mhashn T n T+ + + --
Generation of proxy key ( ) ( ) ( )2 1 2 2 2pair mhash smuln T n T n T+ + + + + tnN
Multi-Proxy Signature
Generation
( )4 3 1 4pair mhash smulnT n T nT+ + + 2b tnN nN+
Multi-Proxy Direct
Verification
3 2pair mhashT T+ --
Multi-Proxy Public
Verification
2 2pair mhashT T+ --
TABLE 1: Computation and Communication Overheads
for ID-DMPS Scheme
6. CONCLUSION
Proxy signature is an indispensable mechanism in the modern e-business and e-government
infrastructures. Many variants of proxy signatures have been proposed in the literature. In this
paper, we propose an ID-based directed multi-proxy signature scheme using bilinear pairings.
This scheme allows only a designated verifier to directly verify the multi-proxy signature,
generated by a group of proxy signers on behalf of the original signer, issued to him. In case of
trouble or if necessary the designated verifier can prove the validity of the multi-proxy signature to
any third party. Our scheme satisfies the security requirements such as strong identifiability,
strong undeniability, distinguish ability, prevention of misuse of proxy signing power, strong
unforgeability and designated verifiability. The proposed scheme is suitable for some applications
where the signed message is personally or commercially sensitive to the signature receiver.
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