Recently, Yoon et al. and Wu proposed two improved remote mutual authentication and key agreement schemes for mobile devices on elliptic curve cryptosystem. In this paper, we show that Yoon et al.’s protocol fails to provide explicit key perfect forward secrecy and fails to achieve explicit key confirmation. We also point out Wu’s scheme decreases efficiency by using the double secret keys and private/public pair, and is vulnerable to the password guessing attack and the forgery attack

Chen Jianhua

He  Debiao

Hu Jin

Cryptology ePrint Archive

 1Weakness of two ID-based remote mutual authentication with key agreement protocols for mobile devices  He Debiao*, Chen Jianhua, Hu Jin Mathematics & Statistics School of Wuhan University, Wuhan, Hubei province, 430072, China  Abstract: Recently, Yoon et al. and Wu proposed two improved remote mutual authentication and key agreement schemes for mobile devices on elliptic curve cryptosystem. In this paper, we show that Yoon et al.’s protocol fails to provide explicit key perfect forward secrecy and fails to achieve explicit key confirmation. We also point out Wu’s scheme decreases efficiency by using the double secret keys and private/public pair, and is vulnerable to the password guessing attack and the forgery attack. Key words: ID-based; Mutual authentication; Key agreement; Elliptic curve Cryptosystem; Perfect forward secrecy; Modular multiplication 1. Introduction With the rapid development of electronic technology, various mobile devices (e.g., cell phone, PDA, and notebook PC) are produced and people’s life is made more convenient. More and more electronic transactions for mobile devices are implemented on Internet or wireless networks. In electronic transactions, remote user authentication in insecure channel is an important issue. For example, when one user wants to login a remote server and access its services, such as on-line shopping and pay-TV, both the user and the server must authenticate the identity with each other for the fair transaction.  Generally, the remote user authentication can be implemented by the traditional public-key cryptography (Rivest et al., 1978; ElGama, l985). The computation ability and battery capacity of mobile devices are limited, so traditional public-key cryptograph, in which the computation of modular exponentiation is needed, can’t be used in mobile devices. Fortunately, Elliptic curve cryptosystem (ECC) (Miller, 1986; Koblitz, 1987) has significant advantages like smaller key sizes, faster computations compared with other public-key cryptography. Thus, ECC-based authentication protocols are more suitable for mobile devices than other cryptosystem. However, like other public-key cryptography, ECC also needs a key authentication center (KAC) to maintain the certificates for users’ public keys. When the number of users is increased, KAC needs a large storage space to store users’ public keys and certificates. In addition, users need additional computations to verify the other’s certificate in these protocols To solve the above problems, several ID-based authentication protocols on ECC are proposed (Abichar et al., 2007; Choie et al., 2005; Cao et al., 2008; Chen and Song, 2007; Jiang C et al., 2007; Jia Z. et al., 2006; Tian et al., 2005; Wu et al.,2005). But there are some disadvantages in the previous user authentication protocols on ECC (Yang et al. 2009). That is, some of these *Corresponding author E-mail: hedebiao@163.com, Tel:+0086015307184927, Fax: +008602787817667  2protocols do not provide the mutual authentication (Chen and Song, 2007; Jiang et al., 2007; Jia et al., 2006; Wuet al., 2005) or the session key agreement (Cao et al., 2008; Chen and Song, 2007; Jia et al., 2006;Wuet al., 2005) between the user and the server. For some applications, the user and the server need a session key to encrypt the secret information for the subsequent communications after they authenticate with each other. In 2009, Yang et al. propose the first ID-based remote mutual authentication with key agreement protocol on ECC (Yang et al., 2009). Based upon the ID-based concept, the protocol does not require public keys for users so that the additional computations for certificates can be reduced. Moreover, the protocol not only provides mutual authentication but also supports a session key agreement between the user and the server. Recently, Yoon et al. (Yoon et al., 2009) found Yang et al.’s protocol is vulnerable to an impersonation attack and does not provide perfect forward secrecy. At the same time, Wu (Wu, 2009) pointed out Yang et al.’s protocol depends solely on a long-term private key stored in the mobile device, does not provide perfect forward secrecy and does not consider personal privacy problem. Nevertheless, we find Yang et al.’s protocol does not provide perfect forward secrecy and fails to achieve forward secrecy. We also find Wu’s protocol is vulnerable to the password guessing attack and the forgery attack. In addition, Wu’s protocol decreases efficiency by using the double secret keys. In this paper, we propose an efficient ID-based remote mutual authentication with key agreement protocol for mobile devices on elliptic curve cryptosystem. Compared with that of Yang et al., Yoon et al. and Wu, the proposed protocol is more secure, efficient, and more suitable for mobile devices. The rest of our paper is organized as follows. Section 2 gives the some basic concept. Section 3 reviews the protocols of Yoon et al. and Wu. Section 4 analyzes the security of the protocols of Yoon et al. and Wu. Section 5 and Section 6 propose our protocol and the security of the proposed protocol. Finally, Section 7 concludes the paper. 2. Preliminaries 2.1 Notations We first introduce common notations used in this paper as follows.  pF : a finite field;  E : an elliptic curve defined on finite field pF  with large order;  G : the group of elliptic curve points on E ;  P : a point on elliptic curve E  with order n , where n  is a large prime number;  1( )H ⋅ : a secure one-way hash function, where *1 :{0,1}H G→ ;   2 ( )H ⋅ : a secure one-way hash function, where * *2 :{0,1} pH Z→ ;   3( )H ⋅ : a secure one-way hash function, where * *3 :{0,1} pH Z→ ;   3 4 ( )H ⋅ : a secure one-way hash function, where * *4 :{0,1} pH Z→ ;   U : the user;  S : the server;  UID : the identity of the user U ;  SID : the identity of the server S ;  ( ,S Sq Q ): the server S ’s private/public key pair, where S SQ q P= ⋅ . 2.2 Background of elliptic curve cryptograph We will just give a simple introduction of elliptic curve defined on prime field pF . The knowledge of elliptic curve defined on binary field can be found in (Miller, 1986; Koblitz, 1987). Let the symbol / pE F  denote an elliptic curve E  over a prime finite field pF , defined by an equation  baxxy ++= 32 ， pFba ∈,              (1) and with the discriminant  3 24 27 0a bΔ = + ≠ .                      (2) The points on / pE F  together with an extra point O  called the point at infinity form a group  {( , ) : , , ( , ) 0} { }pG x y x y F E x y O= ∈ = ∪ .  (3) Let the order of G  is ,n  G is a cyclic additive group under the point addition “+” defined as follows: Let ,P Q G∈ , l  be the line containing P  and Q  (tangent line to / pE F  if P  = Q ), and R , the third point of intersection of l  with / pE F . Let l′  be the line connecting R  and O . Then P  “+” Q  is the point such that l′  intersects / pE F  at R  and O  and P “+” Q. Scalar multiplication over / pE F  can be computed as follows:  (  )tP P P P t times= + + +…              (4).  43. Review of Two Protocols 3.1 Yoon et al.’s Protocol Yoon et al.’s protocol consists of three phases: system initialization phase, user registration phase, and mutual authentication with key agreement phase.  System initializing phase In this phase, S generates parameter of the system. 1). S chooses an elliptic curve E over a finite field pF . Let ( )pE F denote the set of all the point on E . 2). S chooses a point ( )pP E F∈ , such that the subgroup generated by P  has a large order n . 3). S chooses three hash functions 1 2 3( ), ( ), ( )H H H⋅ ⋅ ⋅  described in section 2.1. 4). S publishes the parameter 1 2 3( , , , , ( ), ( ), ( ))p E G n H H H⋅ ⋅ ⋅ .  User registration phase In this phase, everyone who wants to register at the server should obtain a smart card. The user U  begins his registration at the server S  as shown in Fig 1.  Fig. 1. User registration phase of Yoon et al.’s protocol  Mutual authentication with key agreement phase In this phase, the user U  sends a login request message to the server S  whenever U  wants to access some resources upon S . Then the server S  verifies the authenticity of the login message requested by the user U . At the same time, a session generation between U  and S  is generated. The detailed of the phase is illustrated in Fig 2.  5 Fig. 2. Mutual authentication with key agreement phase of Yoon et al.’s protocol 3.2 Wu’s Protocol Wu’s protocol also consists of three phases: system initialization phase, user registration phase, and mutual authentication with key agreement phase.  System initializing phase In this phase, S generates parameter of the system. 1). S chooses an elliptic curve E over a finite field pF . Let ( )pE F denote the set of all the point on E . 2). S chooses a point ( )pP E F∈ , such that the subgroup generated by P  has a large order n . 3). S chooses three hash functions 2 3 4( ), ( ), ( )H H H⋅ ⋅ ⋅  described in section 2.1. 4). S  computes private/public key pair ( ,S Sq Q ) and selects a private key Sd . 5). S publishes the parameter 2 3 4( , , , , ( ), ( ), ( ))p E G n H H H⋅ ⋅ ⋅ .  User registration phase In this phase, everyone who wants to register at the server should obtain a smart card. The user U  begins his registration at the server S  as shown in Fig 3.  Fig. 3. User registration phase of Wu’s protocol  6 Mutual authentication with key agreement phase In this phase, the user U  sends a login request message to the server S  whenever U  wants to access some resources upon S . Then the server S  verifies the authenticity of the login message requested by the user U . At the same time, a session generation between U  and S  is generated. The detailed of the phase is illustrated in Fig 4.  Fig. 4. Mutual authentication with key agreement phase of Wu’s protocol 4. Analysis of Two Protocols 4.1 Analysis of Yoon et al.’s Protocol This section shows that Yoon et al.’s protocol does not provide perfect forward secrecy and does not achieve explicit key confirmation.  Failure to provide explicit key perfect forward secrecy Perfect forward secrecy is one of desirable attributes of key agreement protocols, it means that if the long-term private keys of one or more entities are compromised, the secrecy of previous session keys, which was established by honest entities, is not affected (Blake-Wilson S. et al., 1997). We find that Yoon et al.’s protocol could not provide perfect forward secrecy. The following is our reasons. In Yoon et al.’s protocol, since all transcripts are transmitted over an open network, a benign (passive) adversary can easily obtain a valid information pair 1{ , , , }UU UID M R T  and  72{ , , }S kM M T .  If the long-term private key UIDA of the user is compromised, and the key is derived by the attacker A , then A  can compute all the session key generated between the user and the server as follow. 1) A  computes 1 2 1( )t H T= , and 1( , ) UU U U U IDR x y M t A= = − . 2) A  computes 2 2 2( )t H T= , and 2( , ) US S S S IDR x y M t A= = − . 3) A  computes U SDH x x P= ⋅ ⋅ . 4) The attacker A  get the session key 2 ( , );Uk H ID DH= . If the long-term private key sq of the server S  is compromised, the attacker A  can compute the session key at the same way, because the attacker A  can get the long-term private key UIDA of any user U  by computing 1( )UID s UA q H ID= ⋅ . Since the attacker can get the session key through the method described above, then we can conclude that Yoon et al.’s protocol does not provide perfect forward.  Failure to achieve explicit key confirmation A key agreement scheme is said to provide the explicit key confirmation if one entity is assured that the second entity has actually computed the session key (Blake-Wilson S. et al., 1997). In many applications, it is highly desirable for a key agreement scheme to provide the explicit key confirmation. We can see that the scheme of Yoon et al. merely provides the implicit key confirmation, because S  cannot confirm U  has correctly computed the session key after the Mutual authentication with key agreement phase. However, in general, key agreement scheme can provide the explicit key confirmation. Hence, the scheme of Yoon et al. is not practical for application. 4.2 Analysis of Wu’s Protocol This section shows that Wu’s protocol decrease the efficiency by using double secret keys and is vulnerable to the password guessing attack and the forgery attack.  Inefficiency of double secret keys We can see that the scheme of Wu requires S  to keep two keys secret, i.e., the secret key Sd  and the private key Sq  for the elliptic curve algorithm. In common sense, it is possible to only use one secret key for achieving the user authentication and key agreement service. Therefore, two secret keys mean more overheads without the security enhancement for the whole authentication system. Furthermore, we need to point out the drawback of using the elliptic-curve algorithm in the scheme of Wu. Since S  uses the private/public key pair ( ,S Sq Q ), this elliptic-curve algorithm is a public key algorithm, which may involve the certificate mechanism,  8e.g., X.509 (ITU-T, 2005). To maintain the certificate framework, the public key infrastructure incurs a nontrivial level of system complexity and implementation costs.  Vulnerable to password guessing attack and forgery attack We assume that an attacker A  has total control over the communication channel between the user U  and the remote server S , which means that he can insert, delete, or alter any messages in the channel. According to the researches in (Kocher et al.’s, 1999; Messerges et al.’s 2002), all existing smart cards are vulnerable since the secret values stored in a smart card could be extracted by monitoring its power consumption. Therefore, we further assume that the attacker A  can steal the user’s smart card and extract the values stored in the smart card. Under these two assumptions, we will examine some security flaws of Wu’s remote user authentication method. The server S  stores UB into the smart card of the user U  in the registration phase. If the attacker A steals the smart card and extracts the secret values from the smart card as in (Kocher et al.’s, 1999; Messerges et al.’s 2002 ), he can then easily figure out U ’s password as follow. 1) A  get a message 1 { , , }M X SID α=  transmitted between U  and S . 2) A selects a password PW ′ s  from a uniformly distributed dictionary D . 3) A computes 3( )UID UA B H PW′ ′= ⊕ . 4) A computes 4 ("1", , , )UIDH X SID Aα′ ′=  5) A then verify the correctness of PW ′ s  by checking that α  is equal to α′ . 6) A repeats steps 1, 2, and 3 of this phase until the correct password if found. After the adversary has obtained the password UPW  (using the above method), since she has also UB , A  can compute 3( )UID U UA B H PW= ⊕ . In this way he can impersonate U  by forging her login message { , , }X SID α  and { }γ . Therefore, Wu’s scheme is vulnerable to forgery attacks. 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Weakness of two ID-based remote mutual authentication with key agreement protocols for mobile devices

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