Certificateless One-Time Proxy Signature
2010-11-22HONGDongzhao
HONG Dong-zhao
(College of Science, Hangzhou Normal University, Hangzhou 310036, China)
1 Introduction
In 1996, Mambo et al[1-2]proposed the concept of proxy signature. After that, several new proxy signature[3-8]schemes have been studied. To limit the power of the proxy signer, the notionof one-time proxy signature was introduced[9]. In this paradigm, the proxy signer uses the same proxy key to sign message more than once, then his private key will be leaked out. And then many one-time proxy signature schemes have been proposed[10-15]. Almost all one-time proxy signature schemes available in the literature are based on the traditional public key cryptography or the identity-based cryptography.
In 1984, Shamir[16]introduced the concept of Identity-based signature which avoids the necessity of certificates. However, an inherent problem of ID-based cryptosystems is the key escrow. To overcome the key escrow problem, Al-Riyami and Paterson[17]introduced the concept of certificateless signature (CLS) in 2003. Huang et al[18]formalized the security definition of certificateless signature. The security model of CLS schemes was further developed in[19-21].
In this paper, a certificateless one-time proxy signature scheme is proposed. The security model of certificateless one-time proxy signature scheme is formalized. The security model captures the notion of existential unforgeability of certificateless one-time proxy signature against Strong Type I and Super Type II adversaries[21]. The proposed scheme is provably secure in this model.
2 Definition and Security Model for CLOTPS
2.1 Definition
Definition1(Certificatelessone-timeproxysignature). Given an integerk, a certificateless one-time proxy signature scheme CLOTPS=(Setup, Partial-Private-Key-Extract, UserKeyGen, PSign, PVerify) with security parameterkis defined by the following:
1) Setup,KGCtakes as input a security parameterkand outputs public system parametersparamsand the master secret keymsk.
2) Partial-Private-Key-Extract,KGCtakes as inputparams,msk, and a user’s identityID, returns the partial private keysDID.
3) UserKeyGen, the user takes as inputparamsandDID, outputs the user’s public keyPKIDand corresponding secret keySKID.
4) PSign, the user takes as inputparams,msk, two identitiesIDo,IDp∈{0,1}*of the original signerUoand the proxy signerUp, a messagem, and a pair of public keys {PKIDo,PKIDp}, outputs a proxy signing keyoskforUp, at last, outputs a certificateless one-time proxy signatureσ.
5) Verify, the verifier takes as input a proxy signatureσ, returns 1 (accept) or 0 (reject).
For certificateless one-time proxy signature has also the following properties[22]: Correction, One timeness, Unforgeability, Identifiability, Undeniability, Prevention of misuse.
2.2 Security Model
Similar to the adversaries against certificateless signature scheme defined in [21], for the security of certificateless one-time proxy signature schemes, two types of adversaries are introduced, namely Strong Type I Adversary, Super Type II Adversary[23].
Strong Type I Adversary: A Type I AdversaryAIdoes not have access to the master-key, butAIis allowed to replace the public key of any user with his choice.
Super Type II Adversary: A Type II AdversaryAIIhas access to the master-key, butAIIisn’t allowed to replace the target user’s public key.
Two games are defined, one forAIand the other forAII.
GameI:The first game is performed between a challengerCand a Type I adversaryAIfor a certificateless one-time proxy signature scheme as follows.
Setup:Cruns the Setup algorithm of the CLOTPS scheme, takes as input a security parameterkto obtain amskandparams.Cthen sendsparamsto the adversaryAI.
Attack:The adversaryAIcan query the following oracles which are controlled byC.
Create-Useroracle: On input an identityID. IfIDhas not been created it runs the algorithms Partial-Private-Key-Extract, UserKeyGen to obtain the partial private keyDID, private keySKIDand public keyPKID. Then it adds (ID,*,DID,SKID,PKID) to the listL. In this case,IDis said to be created. In both cases,PKIDis returned.
Partial-Private-Keyoracle: On input an identityID, which has been created, returns the partial private keyDID; otherwise, returns ⊥.
Public-Key-Replacementoracle: On input an identityIDand a new public keyPKID′ and private keySKID′ if userIDhas been created,the oracle updates the corresponding information in the listL; otherwise, dose nothing.
Hashoracle:AIcan request the hash values for any input.
Public-Keyoracle: On input an identityID, which has been created, the oracle returns the public keyPKID.
Private-Keyoracle: On input an identityIDexcept the challenged identityID, which has been created, the oracle returnsSKID.
Proxy-Signoracle: On input a messagem, a warrantmw, an original signer’s identityIDo, a proxy signer’s identityIDp, the oracle outputs a proxy signatureσas answer.
Game2: The first game is performed between a challengerCand a Type II adversaryAIIfor a certificateless one-time proxy signature scheme as follows.
Setup:Cruns the Setup algorithm of the CLOTPS scheme, takes as input a security parameterkto obtainmskandparams.Cthen sendsparamsandmskto the adversaryAII.
Attack: The adversaryAIIcan query Create-User oracle, Hash oracle , Public-Key oracle, Public-Key-Replacement oracle, Private-Key oracle, Proxy-Sign oracle which are the same as that in Game I.
Due to the proxy warrantmwbinded in the signature, if only the unforgeability satisfies, other security requirements, such as identifiability, undeniability and prevention of misuse, obviously follow [14].
3 The New CLOTPS
3.1 Description of the New of CLOTPS
The new scheme CLOTPS which modifying the key generation algorithms of Lu et al.’s ID-based one-time proxy signature scheme is proposed as follows.
2)Partial-Private-Key-Extarct: GivenUo’s identityIDoandUp’sIDp∈{0,1}*.KGCsetsQIDo=H(IDo),DIDo=sH(IDo),QIDp=H(IDp) andDIDp=sH(IDp) and sends the partial private keyDIDotoUoandDIDptoUpvia some secure channel.
4)PSign: (identical with Lu et al.’s scheme)
a)Uomakes a warrantmwand sends it toUp.
d) To sign a messagem,Upcomputesσp=r2+r1·H2(R1,R2,m‖mw) modq. Finally,Upoutputs the proxy signatureσ=(σo,opk,mw,σp,m).
5)PVerify: (similar to Lu et al.’s scheme) Upon receiving the proxy signatureσ=(σo,opk,mw,σp,m), verifier checks that the following equations
Hold. If not, rejected. Otherwise, accepted.
The correctness of the scheme can be verified directly.
e(Yo,H(IDo))·e(Ro,H1(Ro,opk‖mw))=e(P,sxoH(IDo))·e(P,roH1(Ro,opk‖mw))=
e(P,sxoH(IDo)+roH1(Ro,opk‖mw))=e(P,SIDo+roH1(Ro,opk‖mw))=e(P,So)
3.2 Security Proof
Theorem1(One-timeness). The new CLOTPS scheme has the property of one-timeness.
This proof is similar to the proof of Theorem 1 in [14].
Theorem2Our CLOTPS scheme is secure against existential forgery under adaptively chosen message and chosen ID attacks in the random oracle model with the assumptions that CDH problem inGis intractable.
This theorem follows from the following Lemmas 1 and 2.
Lemma1Our CLOTPS scheme is unforgeable against a Strong Type I adversary in the random oracle model assuming the CDH problem is intractable.
Lemma2Our CLOTPS scheme is unforgeable against a Super Type II adversary in the random oracle model assuming the CDH problem is intractable.
Due to page limitation, the proof will be presented in the full version of this paper.
4 Conclusion
In this paper, an appropriate security model and a concrete construction of certificateless one-time proxy signature scheme are proposed. The security model takes into account the strong adversaries in certificateless public key settings. The security of the proposed scheme is provable secure in the random model with the intractability of the Computational Diffie-Hellman problem. The proposed scheme is as efficient as Lu et al.’s scheme.Due to its efficiency and certificateless, it can be widely used in areas such as E-cheque, etc.
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