How to Authenticate a Device? Formal Authentication Models for M2M Communications Defending against Ghost Compromising Attack

In Machine-to-Machine (M2M) communications, authentication of a device is of upmost importance for applications of Internet of Things. As traditional authentication schemes always assume the presence of a person, most authentication technologies cannot be applied in machine-centric M2M context. In this paper, we make the first attempt to formally model the authentication in M2M. We first model four attacking adversaries that can formulate all possible attacks in M2M, which are channel eavesdropping attack, credential compromise attack, function compromise attack, and ghost compromise attack. Next, we propose four models to tackle those corresponding adversaries, namely, credential-based model, machine-metrics-based model, reference-based model, and witness-based model. We also illustrate several concrete attacking methods and authentication approaches. We proof the authentication security for all proposed models and compare them for clarity. Our models present soundness and completeness in terms of authentication security, which can guide the design and analysis of concrete authentication protocols. Particularly, we construct a uniform authentication framework for M2M context and point out all possible authentication mechanisms in M2M.


Introduction
Machine-to-Machine (M2M) communication is a typical communication fashion in the Internet of Things (IoT).It has been envisioned as one of the most promising Internetaccessing approaches in the IoT for long-distance remote devices.M2M networking interface enables wide area communications for mobile or static devices, so that it is quit convenient and easy to deploy remote devices rapidly.M2M thus becomes a typical communication method for most IoT applications such as remote environmental sensing, longdistance controlling, moving object locating, and tracing.Besides, M2M communication further incorporates various novel applications such as smart grid [1], tele-medicine [2], and smart vehicles [3].
As remote devices are always located faraway in unattended or malicious environments, it is very likely that the devices may be hacked by attackers.For example, the software system in devices may be injected or infected by certain malicious codes, which may modify or fabricate forthcoming outgoing data.Moreover, the devices may perform arbitrary (Byzantine) misbehavior after being compromised.Thus, the data sent by remote sensing devices must be authenticated.Otherwise, remote receivers in the other end will receive a lot of garbage information and consequently respond falsely.
We note that the authentication in M2M context is quite different from traditional authentication.Roughly speaking, traditional authentication usually assume the entity being authenticated is a person, or human is involved in authentication procedures.Simply speaking, the authentication methods usually rely on three aspects: "what you know, what you have, and who you are." As the entity authentication in M2M is not a person, the traditional methods such as "what you know and who you are" may not be applied.
For example, password-based authentication (by manually inputting password) cannot be applied in M2M context.The biometric-based authentication such as fingerprint recognition cannot be applied in M2M context neither.
Furthermore, traditional authentication methods for wireless sensor networks (WSNs) may not be able to smoothly migrate to M2M situations.That is, the authentication methods in WSNs usually rely on certain secrete credentials in cryptographic building blocks, for example, secrete keys for symmetric key encryptions, or private keys for digital signatures.Those secrete credentials are stored in devices or chips attached to devices (e.g., smart cards).However, such an authentication method still cannot defend against all aforementioned attacks.For example, when devices are compromised, those credentials will be revealed by attackers, by hacking the devices or injecting malicious codes.It again shows the distinction between M2M authentication and person-based authentication where entities (i.e., persons) usually are assumed uncompromising.
Therefore, the entity authentication in M2M context must be reconsidered and reinspected.We also need to explore the tailored authentication methods in M2M context, to guarantee a stronger authentication (that will be formally defined later), in case the devices are compromised and credentials are exposed.Besides, the formal models for authentication are appealing, as the formal models can guarantee the security strength and clarify the core part out from various application details.Unfortunately, such problem has not been explored thoroughly until now, to the best of our knowledge.
Currently, although several related works start to concentrate on M2M security problems [4][5][6][7][8], the strong authentication methods in M2M have not been thoroughly explored yet [9][10][11][12][13].In this paper, we make the first attempt to figure out the abstract models for M2M context.We adapt a formal and rigorous method used in modern or theoretic cryptography, to strictly state, present, and analyze the security of authentication.More specifically, we firstly formulate attack models regarding to device compromising in M2M context by using interactive Turing machine.We next categorize the classifications in M2M authentication and their security specification.Finally, we propose several abstract authentication models to address different attacking patterns and proof their security.All our presentations strictly follow the formal expressions for better clarity and rigorous generality.
The contributions of the paper are listed as follows: (1) we strictly formulate the possible attacks and adversaries in M2M, which facilities to clearly locate the security fragile point.For example, we point out credential compromising attack, function compromising attack, and ghost compromising attack; (2) we formulate the general and abstract authentication models with provable security in M2M context, which figures out the fundamental characteristics of all possible authentication methods in M2M to guide the further design and security analysis in practices; (3) we point out several concrete attacking methods and propose corresponding authentication approaches to illustrate our models.
The rest of the paper is organized as follows.In Section 2, we discuss the basic assumption and models used throughout the paper.Section 3 provides the detailed description of our proposed models and analysis.Section 4 gives an overview on relevant prior work.Finally, Section 5 concludes the paper.

Problem Formulation
2.1.Network Model.In most M2M communications, there exists two major entities: devices, denoted as Dev, and a central server.
Dev is defined to equip following components.
( In this paper, we focus on above typical fashion in M2M communications-from devices to central servers.It is without loss of generality, because communications between devices and devices far away in M2M are usually relayed by a central server.As the devices are equipped with wide area communication capabilities, multihop M2M relay within remote devices rarely happen.The relay is usually unnecessary, as the devices can upload reports directly to central servers. Even though multihop relay between devices happens, it is usually local area wireless communication such as wireless personal area network (WPAN), for example, Zigbee, or wireless local area network.That is, the reasonable architecture has two tiers: the communication between devices and gateways is WPAN, but the communication between gateways and a central server is M2M.In this case, the authentication of devices in the former tier has been explored in WSN communities and previous solutions can be migrated; the authentication of gateways in the latter tier is our focus.That is, in this scenario, we look on gateways as devices in our discussion.
In addition, no matter in which kind of authentication scenarios, we always assume the peer who authenticates (i.e., verifier) is trusted or secure.It is a baseline for the further meaningful discussion.We thus focus on the peer being authenticated (i.e., prover), which is a device in M2M scenarios.That is, how to authenticate a device in M2M.

Attack Model.
It is required to consider the situation that devices may be compromised, as the devices may be always located in unattended environments.From the viewpoint of security strength, such assumption for the existence of stronger adversary will result in stronger security guarantee, which is mandatory for certain critical applications, such as gas emission monitoring and back-bone smart grid.According to the modeling of networks and devices, we classify the attack models into four folders as follows.
It is a straightforward attack that adversaries sniff communication channels and try to subvert authentication methods, for example, revealing credentials.
(2) Credential Compromising Attack, denoted as Attack cc .In this attack, the credential module is hacked, exposed, modified, cloned, or replaced by adversaries.
In this attack, the functional module is hacked, exposed, modified, cloned, or replaced by adversaries.
(4) Ghost compromising attack, denoted as Attack gc .In this attack, the whole device can be hacked, exposed, modified, cloned, or replaced by adversaries.It can be looked as the combination of Attack cc and Attack fc .
Note that, the Attack cc and Attack fc may not be concurrent.For example, when tamper-proof hardware is applied for credential module, adversary's malicious code can only compromise function module, not credential module.

Security Definition.
Roughly speaking, the secure authentication in this paper is defined as the interactive proof between two probabilistic polynomial interactive Turing machine (ITM) [14].
The ITM being authenticated is called Prover (denoted as P); the ITM authenticating Prover is called Verifier (denoted as V).The P and V both have one outgoing communication tape, one incoming communication tape, one input computing tape, one output computing tape, and one inner working tape.They also have other auxiliary tapes for interaction and security: one identity tape, one security parameter tape, one random tape, and one-bit activation tape.In this paper, the outgoing communication tape of P is the same with the incoming communication tape of V.For simplicity, this tape is called interaction tape, denoted as T  .
The adversary (denoted as A) is also modeled as an ITM.In different attack models, adversaries have different corresponding capabilities.We list them from weaker one to stronger one incrementally.The stronger one inherently has the capability of the weaker one.
(1) In Attack ce , the adversary is denoted as A ce , which can read and write the interaction tape.
(2) In Attack cc and not in Attack fc , the adversary is denoted as A cc , which can read and write the work tape.We assume it inherently has the capability of A ce .
(3) In Attack fc and not in Attack cc , the adversary is denoted as A fc , which can also read and write input tape and output tape.We assume it inherently has the capability of A ce .
(4) In Attack gc , the adversary is denoted as A gc , which can also read and write all tapes.
As we have already stated, V needs to be secure (or honest) in authentication semantics.
Environmental ITM (denoted as Z) exists.Z can read and write input tapes of P and V.The protocol result is the output of Z.
The one interaction between P and V can be modeled as follows.
The sequence can be interchangeable.The times of interaction may be more than once.
Next, we state the definition of authentication and its security as follows: Definition 1 (Authentication).From the transcripts on tape T  by V, V can believe the data is indeed from P (and data is not modified).More specifically, from tag in the transcripts {id, data, tag} on tape T  , V can believe data is indeed from the device P with the id without any change.
Definition 2. Security of Authentication of Protocol Π in the presence of adversary A. From tag  in {id, data  , tag  } in the protocol Π's transcripts that data are tampered to data  by adversary A, the probability that the adversary can fool V to believe data  is from the device P with id without any change is negligible.
Define attack experiment AuthFool A,Π () as follows: (1) run protocol Π in the presence of adversary A, where P and V are both ITM with security parameter ; (2) V witnesses A tampered transcripts on tape T  − {id, data  , tag  }, V output 1.That is, V believes that data  come from the device P with id without any change.Otherwise, output 0; (3) if and only if V output 1, the experiment output 1.
Definition 3. Authentication protocol Π is secure, if for any ITM adversary A, it exists a negligible function negl satisfying Indeed, authentication can be further divided into two types: entity authentication and message authentication.Entity authentication is the concentration in this paper.Message authentication is however usually required also in communication context and can be attained with entity authentication together.We thus later do not explicitly split those two.Definition 4 (Entity Authentication (Message Source Authentication)).From tag in the transcripts {id, data, tag} on tape T  , V can believe data is from the device P with id.
Definition 5 (Message Authentication (Message Integrity Authentication)).From tag in the transcripts {id, data, tag} on tape T  , V can believe data is transferred without any change.We can similarly define the attack experiment and corresponding security for entity authentication and message authentication.

Proposed Authentication Models
In this section, we propose a family of models to solve the authentication problem in M2M context.
We list all major notations used in the remainder of the paper in Table 1.

Defending Channel Eavesdropping Attack: Credential-Based Model (CBM).
To defend Attack ce , the simplest authentication model is credential-based model, where the authentication relies on the knowledge or possession of certain credentials, for example, secrete keys.The credentialbased model is described as follows: MSG cbm : P → V : {id, data, tag =  (cred ‖id‖ data)} , V : Verify (id, data, tag) where the Verify() is a tag verification function Verify(tag); MSG cbm is a message from P to V; cred is a credential such as a secret key; () is a function with following properties.
(1) One-wayness.From (cred‖id‖data), it is computationally infeasible to compute cred, even if id, data and () are public.By using a formal notation, that is where negl() is a negligible function with parameter .Pr{ | } means given  the probability of event  (i.e., guessing right ).That is, any polynomial probabilistic Turing machine (PPTM) with parameter  cannot guess cred with noneligible probability, no matter whether (cred‖id‖data), id, data, () are given or not.
(2) Second-Preimage Resistance.Without cred, it is computationally infeasible to find another id  (id  ̸ = id) such that (cred‖id  ‖data) = (cred‖id‖data), even if data, id and () are public.Similarly, without cred, it is computationally infeasible to find another data Proof.It can be proofed by similar methods in Universally Composable security [14]; hereby only sketch is given.The security of the proposed model is guaranteed by the security of ().The rationale is from the Turing Test.There exist a reality model and an ideal model.From the viewpoint of environmental machine, if the reality model and ideal model are indistinguishable, the reality model will reach the security of the ideal model.In reality model, there exist adversary, prover, and verifier running protocol; In ideal model, there exist ideal function that emulates the protocol, dummy prover and dummy verifier, and simulated adversary that emulates all possible interactions and attacks.Ideal function is always secure (it is regarded as an imaged trusted third party).If the ideal model and reality model are indistinguishable from the viewpoint of environmental machine, the security of protocol will be guaranteed.
In other words, if environmental machine can distinguish the reality model and the ideal model, a new adversary can be created to subvert a certain security assumption by revoking the environmental machine as a subfunction.In the ideal function, () is perfectly secure.In the reality model, () is assumed to satisfy certain security requirements (e.g., onewayness and preimage resistance).If the ideal model is indistinguishable with reality model by environmental machine, the adversary will subvert the assumption by consulting the environmental machine.Finally, the proposition proofs the soundness of credential-based model.

Proposition 9. Credential-based model has completeness for defending Channel Eavesdropping Attack.
Proof.The proof has two folders: the credential is required; the credential is presented properly.On the one hand, as devices need to distinguish themselves with others, they need to show their secretly possessed knowledge-credential-to proof their identity.Thus, there must exist a credential in the tuples in MSG cbm on the transcripts   .On the other hand, the credential must be secretly and properly presented, as the interaction tape   can be tampered by eavesdropping adversary A ce .Thus, the presence of credential relies on () that has one-wayness and second-preimage resistance.
Besides, multicredential-based model is securer than one credential-based model.That is, the exposure of one credential by eavesdropping adversary A ce will only result in subverting the authentication security guaranteed by that credential.If P has multiple credentials, the exposure of credentials will become more difficult.It is natural to extend the security discussion to following statement.

Definition 10 (Forward Authentication Security). That is, if current credential is exposed by eavesdropping adversary
A ce , the authentication security before exposure is still guaranteed.

Proposition 11. In Attack 𝑐𝑒 , multiple credentials that derive from one-way function can guarantee forward authentication security.
Proof.By using one-way function , cred  = (cred −1 ), (1 ≤  ≤ ), where cred 0 = cred.The sequence of using cred is from cred  to cred 0 .Due to the one-wayness of (), it can guarantee the confidentiality of previous credentials, even if current used credential is revealed by A ce .

Proposition 12. Credential-based model
Proof (Straightforward).The authentication security of credential-based model relies on the secrecy of credentials.The presence of A cc will reveal all credentials.Thus, the authentication security is broken.

Defending Credential Compromising Attack: Machine-Metrics Based Model (MBM).
If the attack model is strengthened to Attack cc , the credential-based model will not be able to guarantee the security of authentication, as it is proofed in previous proposition.
For simple illustration of Attack cc , we first point out an attack method called relocation attack.
Definition 13 (Credential Relocation Attack).It is an attack that the credentials are relocated to anther device.Although the received {id‖data‖tag} is verified valid at V, the data do not come from the device with claimed id; indeed, the data come from another device with different id.
To tackle the Attack cc , we propose a machine-metric based model (MBM).In this MBM model, the physical characteristics of devices are collected as the identification of devices.Suppose the physical characteristic space is , which consists of  characteristics.That is,  = {char 1 , char 2 , . . ., char  }.Software program prgm  () returns char  ∈ , ( = 1, . . ., ).The machine-metrics based model is described as follows: MSG mbm : P → V :{id, data, , tag = (prgm  () ‖id‖ data)} , V : Verify (id, data, tag) where Verify() is a tag verification function (we assume V securely possesses a table Tbl = ⟨, char  ⟩, ( = 1, . . ., ), and fetches char  according to index  to verify received tag); MSG mbm is a message from device P to V; prgm  () is securely computed at functional module in P and it returns a characteristic char  as a credential of P; () is a function with one-wayness and second-preimage resistance; note that () can be securely computed by functional module.
If we look char  as a credential in CBM model, we will have similar discussion as follows.prgm  () and (prgm  ()‖id‖data), ( = 1, . . ., ) can be easily computed, especially for resource constraint devices in M2M context.() can be instantiated with cryptographic primitives such as one-way function with second-preimage resistance, pseudorandom function, and trapdoor permutation.
Machine-metrics based model can be further divided into two folders: one Machine-Metrics based model and multiple Machine-Metrics based model.As it is named, multiple Machine-Metrics based model use multiple prgm  () in each MSG mbm message or distinct prgm  () in multiple MSG mbm messages.

Analysis Proposition 14. Machine-Metrics based model
Proof.As the function module is secure, adversary cannot compute prgm  (), ( = 1, . . ., ).It can be looked as a computed credential to guarantee the authentication security.

International Journal of Distributed Sensor Networks
The proof thus is reduced to the proof of the Proposition 8.It proofs the soundness of Machine-Metrics based model.

Proposition 15. Machine-Metrics based Model has completeness for defending Credential Compromising Attack.
Proof.The proof can be reduced to the proof of Proposition 9.
Concretely, the proof has two folders: a new credential is required; the new credential is presented properly.On the one hand, as devices need to distinguish themselves with others, they need to show their secretly possessed knowledgecredential-to proof their identity.As the credential module can be compromised, the credential must come from the functional module.Thus, there must exist a credential in the tuples in MSG mbm on the transcripts   .On the other hand, the credential must be secretly and properly presented, as the interaction tape   can be tampered by eavesdropping adversary A ce .Thus, the presentation of credential relies on  that has one-wayness and second-preimage resistance.Besides, if the credential is online computed (not stored) by functional module, the security will be enhanced, which is similar to the case of multiple credentials.
Similarly, multiple Machine-Metrics based model is securer than one Machine-Metrics based model.The discussion is similar to the one in the previous section.In summary, Machine-Metrics based model can be looked as an analog of biometric-based authentication for human.
We give three special illustrations on Machine-Metrics based model for defending Attack cc (e.g., Credential Relocation Attack).
Definition 17 (Computation-Based Authentication).Each time V sends a one-time random computational puzzle, P counts the computation duration time in microsecond or Central Process Unit (CPU) cycles as the result of prgm  ().
Example 18. V sends a computational puzzle   mod , where  is a prime number; 2 <  <  is an integer;  <  is a positive integer.P counts the computation duration time in microsecond as the computation characteristic, in other words, the returning result of prgm  ().
Definition 19 (Location-Based Authentication).V sends an one-time random location puzzle.P returns computed location values.V verifies the location characteristic of P. That is, prgm  () returns location values of P.
Example 20.V sends an one-time random location puzzle: the distance from P to a randomly chosen point.P computes the distance value according to its location values such as global positions and latitudes, as the location characteristic, in other words, the returning result of prgm  ().
Similarly, we can further define authentication methods by requesting other physical characteristics such as memory size, hardware fingerprints.Note that such kind of requesting must be fresh and generated at real time.

Defending Function Compromising Attack: Reference-Based Model (RBM). If the attack model is strengthened to
Attack fc , the Machine-Metrics based model will not be able to guarantee the security of authentication, which is proofed in previous proposition.
To illustrate Attack fc , we first point out two concrete attack methods called Characteristic Replication Attack and Data Pollution Attack.
Definition 21 (Characteristic Replication Attack).The set  = {char  , . . ., char  } is replicated to another device by attackers, and they will choose corresponding char  by elements upon revoking of prgm  ().
Definition 22 (Data Pollution Attack).Data is polluted by attackers, although the credential is valid to make the data being authenticated.That is, P → V : {id, data  , tag = (cred‖id, ‖data  )}, where data is changed to data  by adversary Attack fc after functional module is compromised.
To tackle the Attack fc , we propose a reference-based model (RBM).In RBM model, the reference behavior of P (e.g., from previous data or other devices) will be checked for the trustworthiness of data.
The reference-based model is described as follows: MSG rbm : P → V : {id, data, tag =  (cred ‖id‖ data)} , V : Verify () where the Verify() includes a tag verification function Verify(tag) and especially, an extra verification function for data, called Verify(data).
We give two special illustration on reference-based model in the following.
Definition 23 (History-Based Authentication).The authenticator V verifies the history behaviors (e.g., data) to authenticate the trustworthiness of current behavior of P, for example, the trustworthiness of data.Concretely, the data filter method can be applied at V, where data come from the history.
Example 24.Suppose the history set is , which consists of  values.That is,  = { 1 ,  2 , . . .,   }.Suppose the history set is trustworthy, the detection of abnormal data can be done by the following verification function: where th is an alert threshold.
Definition 25 (Neighbor-Based Authentication).The authenticator V verifies the other P's behaviors (e.g., neighbors' reporting data) to authenticate the trustworthiness of current behavior of P, for example, the trustworthiness of data.
Concretely, the data filter method can be applied at V, where the data come from neighbors.
Example 26.Similar to above example, suppose the neighbor set is , which consists of  values.That is,  = { 1 ,  2 , . . .,   }.Suppose the neighbor set is trustworthy, the detection of abnormal data can be done by the following verification function: where th is an alert threshold.
In the neighbor-based authentication, the reference set  may not be always trustworthy.That is, the values in  may come from other Ps that are untrustworthy.To deal with this issue, we propose extra two methods in the following.
Definition 27 (Trustworthy Stunt Authentication).The  is from other trustworthy Ps that are predeployed.That is, the authenticator V verifies the trusted stunt's report (e.g., data) to authenticate the trustworthiness of current behavior of P, for example, the trustworthiness of data.The verification function in this situation is similar to historybased Authentication.
Definition 28 (Threshold Stunt Authentication).Suppose there exists at least  nodes in  that are trustworthy (i.e., not compromised) in neighbor-based authentication.Simply speaking, the verification function can be designed as follows: select the median  −  values in  and then use verification function similarly.

Analysis
Proof.Proof.As functional module can be compromised by A gc , computational credential (namely, prgm  () or char  ) may be revealed, which is so-called characteristic replication attack.Besides, even though credentials in credential module can grantee that data cannot be modified by adversaries in the channel, the data can still be modified at the devices after the functional module is compromised, which is so-called data pollution attack.Thus, credentials in credential module defend against the former one; history-based authentication and neighbor-based authentication defend against the latter one.
Proposition 31.Reference-based model Π  is not secure in Attack  .That is, Proof (Straightforward).As the credential module can be compromised in A gc , the authentication security is broken.

Defending Ghost Compromising Attack: Witness-Based Model (WBM).
If the adversary has the most powerful strength, namely, the Attack gc .We point out it cannot be defended against by any authentication enhancement method only from P.
Proposition 32.Attack  cannot be defended only by P itself.
Proof (Straightforward).As the credential module and functional module can both be compromised, any security enhancement will be also compromised.Thus, Attack gc cannot be defended only by P itself merely.
To tackle the Attack gc , we propose a witness-based model (WBM).In WBM model, environmental characteristics and device characteristics are both collected for the final authentication decision.
system.Bartoli et al. [16] studied secure aggregation in smart grid M2M networks.They included the security design in the physical layer and MAC layer.Saied et al. [8] proposed a key establishment solution for heterogeneous M2M communications.Other works explored the security in a broader domain-IoT, which is relevant to M2M.For example, Alam et al. [17] studied the interoperability in security attributes between different administrative domains in IoT.They proposed a layered architecture of IoT framework.As the M2M is still undergoing development, several works studied M2M standards [6,[18][19][20].For example, Bartoli et al. [6] reviewed the current undergoing standards for M2M communications.

Conclusions
In this paper, we made the first attempt to propose a family of formal models to authenticate devices in M2M context.We first modeled four attacking adversaries that can include all possible attacks in M2M.Next, we proposed four models to tackle corresponding adversaries.We also illustrated several concrete attacking methods and authentication approaches.We proofed the authentication security for all proposed models and compared them for clarity.Our models presented soundness and completeness with respect to necessary and sufficient conditions for authentication security, which can guide the design and analysis of concrete authentication protocols.Especially, we constructed a uniform authentication framework for possible various authentication approaches.Our model also pointed out all possible authentication mechanisms or sufficient solutions for authentications in M2M.
(data  ̸ = data satisfying (cred‖id‖data) = (cred‖id‖data  ), even if data, id and () are public.That is, Pr {id  ̸ = id,  (cred      id       data) =  (cred ‖id‖ data) | data, id,  () } ≤ negl () , Pr {data  ̸ = data,  (red ‖id‖ data  ) =  (cred ‖id‖ data) | data, id,  () } ≤ negl () .(cred‖id‖data) can be easily computed if cred, id, and data are available, especially for resource constraint devices in M2M context.() can be instantiated with cryptographic primitives such as one-way function with second-preimage resistance, pseudorandom function, and trapdoor permutation.Credential-based model can be further divided into two folders: one credential-based model and multicredentialbased model.As it is named, multicredential-based model use multiple credentials in each MSG cbm message or distinct credential in multiple MSG cbm messages.
cbm : P → V : {id, data, tag = (cred‖id)}; cbm : P → V : {id, data, tag = (cred‖data)}.AnalysisDefinition 6 (Soundness).The designed protocol can guarantee the required security.That is, the designed protocol is the sufficient condition of the required security.Definition 7 (Completeness).The required security needs the designed protocol to guarantee.That is, the designed protocol is the necessary condition of the required security.Proposition 8. Credential-based model Π  is secure in Attack  .That is,