Evaluation of effective hyperelastic material coefficients for multi-defected solids under large deformation

1.   Introduction

With the rapid growing of wireless networks, there is a corresponding increase in demand for multimedia-supported services, such as internet protocol television (IPTV) [1,2] video conference, and 3D holographic displays. Such services can be offered via voice over IP (VoIP) [2,3] services using session initial protocol (SIP) [3-7], a text-based signalling protocol. SIP has also been deployed for IP multimedia implementations [8,9], smart home and network management [10,11] and mobility management [12-14].

There are, however, open security and privacy challenges when transmitting voice packets over an open network. For example, designing a provably secure and efficient authentication protocol for SIP remains a challenging task. The original authentication mechanism in SIP was based on hypertext transport protocol (HTTP) [15] digested authentication, designed to provide only data authentication. Yang, Wang and Liu [16] introduced the first Diffie-Hellman authenticate key agreement protocol (AKAP) for SIP using the client-server model. However, it was later found to be vulnerable to server spoofing and off-line password guessing attacks [16]. Several elliptic curve based SIP AKAP has also been presented in the literature [17-24], due to the advantages elliptic curve based protocols offered over Diffie-Hellman based protocols (e.g., reduced computational and storage costs). For example, the key length for elliptic curve cryptography (ECC) is much smaller than RSA and ElGamal cryptosystem at the same level of security. Similar to the troubled history of Diffie-Hellman-based protocols, several published ECC-based protocols were found to be insecure after their publication. For example, the protocol of Wu, Zhang and Wang [25] was found to be vulnerable to a range of attacks in [26]. An improved protocol was then presented.

To minimize the impact of a security breach at the server, one of the many desired security features in SIP authentication is to ensure that users' passwords (which may be stored as plaintext) are securely stored. To withstand stolen verifiers attack at the SIP server, a number of smart card-based AKAPs were proposed in the literature, and some of them were subsequently found to be flawed [24,27-31].

More recently in 2016, Zhang, Tang and Zhu [32] introduced an energy-efficient AKAP for SIP. In this paper, we demonstrate that the proposed protocol is vulnerable to key-compromise impersonation attacks, in violation of their security claims. We then propose an ECC-based AKAP, and demonstrate its correctness and security respectively using BAN logic and the Automated Validation of Internet Security Protocols and Applications (AVISPA) simulation tool [33,34]. We also evaluate the performance of the proposed protocol.

2.   Preliminaries

The basic knowledge on ECC and some notations are introduced as follows [35].

$\mathbf{\textbf{Definition 1.}}$ $\mathbf{E}_{p}(a, b): y^{2} = (x^{3}+ax+b)mod p$ be the form of an elliptic curve $\mathbf{E}_{p}(a, b)$ over $\mathbf{F}_{p}$ with $a, b\in \mathbf{F}_{p}$ satisfying $(4a^{3}+27b)mod p\neq0$.

$\mathbf{\textbf{Assumption 1.}}$ Elliptic curve discrete logarithm problem (ECDLP): Known two points $aP$ and $P$ over $\mathbf{E}_{p}(a, b)$, to determine the random number $a \in \mathbf{Z}_q^{*}$. It is a hard problem in polynomial time such that the security is achieved.

$\mathbf{\textbf{Assumption 2.}}$ Elliptic curve Diffie-Hellman problem (ECDHP): Known two points $aP$ and $bP$ over $\mathbf{E}_{p}(a, b)$, to calculate the point $abP$, where the random numbers $a, b\in \mathbf{Z}_q^{*}$. $abP$ can not be solved with non-negligible probability in polynomial time.

3.   Review of Zhang, Tang and Zhu's protocol

Zhang, Tang and Zhu's protocol consists of four phases: initiation, registration, authentication, and password change.

3.1.   Initiation

$S$ selects its private key as $s$, gets its public key as $sP$. Also, $S$ releases the $\{P_{pub}, P, h(), E(\mathbf{F}_{p})\}$, where $h()$ be the hash function.

3.2.   Registration

Step 1. $U_i$ computes $C_1 = h(PW_i\oplus r)$ and sends $\{ID_i, C_1\}$ to $S$ via a secure channel. In the formula, $r$ is a random number.

Step 2. $S$ calculates $C_3 = h(ID_i\oplus s)\oplus C_1$ and sends back a smart card including $\{C_3\}$ to $U_i$.

Step 3. $U_i$ writes $r$ into the smart card, which has stored the values $\{C_3, r, h()\}$.

3.3.   Authentication

Step 1. The smart card asks $U_i$ to enter: identity $ID_i$ and password $PW_i$. Based on the two values, the smart card derives $h(ID_i\oplus s)$ by computing $C_3\oplus h(ID_i\oplus PW_i)$. Subsequently, the card picks two random numbers $r_1$, $r_2$, and computes $C_4 = r_1P$, $C_5 = r_1C_2P_{pub}$, and $C_6 = h(C_5)\oplus (h(ID_i\oplus s)\oplus r_2, (C_5)_{x}, (C_5)_{y})$. In the formula, $(C_5)_{x}$ and $(C_5)_{y}$ are x- and y-coordinate values of point $C_5$. Then, $U_i$ delivers a request message REQUEST $\{ID_i, C_4, C_6\}$ to $S$.

Step 2. $S$ calculates $C_2 = h(ID_i\oplus s)$ and retrieves $(h(ID_i\oplus s)\oplus r_2, (C_5)_{x}, (C_5)_{y})$ by computing $h(sC_2C_4)\oplus C_6$. Then $S$ checks whether $(C_5)_{x}, (C_6)_{y}\stackrel{?} = (sC_2C_4)_{x}, (sC_2C_4)_{x}$. If this holds, $S$ derives $r_2$ by means of computing $C_2\oplus h(ID_i\oplus s)\oplus r_2$. Then, $S$ picks two random numbers $r_3$, $r_4$ and computes $C_7 = r_3P$, $SK = h(C_4, r_3C_4, C_7)$, and $Auth_s = h(h(ID_i\oplus s), r_2, (SK)_{x}, (C_5)_{x}, (SK)_{y}, (C_5)_{y})$. Next, $S$ replays a challenge message CHALLENGE $\{realm, C_7, Auth_s, r_4\}$.

Step 3. $U$ computes $SK = h(C_4, r_1C_7, C_7)$ and verifies whether $h(C_2, r_2, (SK)_{x}, (C_5)_{x}, (SK)_{y}, (C_5)_{y})\stackrel{?} = Auth_s$. If this holds, $U$ computes $Auth_u = h((SK)_{x}, (r_4+1), (SK)_{y})$ and transmits a response message REPONSE $\{realm, Auth_u\}$ to $S$.

Step 4. Once $S$ verifies whether $h((SK)_{x}, (r_4+1), (SK)_{y})\stackrel{?} = Auth_u$. If the equation is correct, $S$ and $U$ successfully shares a common session key $SK = r_1r_3P$.

3.4.   Password change

When $U_i$ plans to change a password into a new one, he needs a helping of $S$. The process is as follows.

Step 1. $U_i$ keys his old password $PW_i$ and retrieves $Z = h(ID_i\oplus s)$ by computing $h(PW_i\oplus r)\oplus C_3$ and $V = Enc_{(SK)_{x}}(h(PW_i^{*}\oplus r^{*}), ID_i, R, Z)$. In the formula, $PW_i^{*}$ and $r^{*}$ are the new password and the random number. Next, $U_i$ submits $\{V\}$ to $S$.

Step 2. $S$ computes $h(ID_i\oplus s)$, and and examines whether it is equivalent to the decrypted value $Z$ from $V$, using x-coordinate of point $SK$. Subsequently, $S$ computes $C_3^{*} = h(PW_i^{*}\oplus r^{*})\oplus h(ID_i\oplus s)$ and $W = Enc_{(SK)_{x}}(C_3^{*}, h(C_3^{*}, (R+1)))$. $S$ then returns $\{W\}$ to $U_i$.

Step 3. $U_i$ checks whether $h(C_3^{*}, R+1)$ is equivalent to the derived one which comes from received message $W$. If it is true, $U_i$ uses the values $\{C_3^{*}, r^{*}\}$ instead of the old one.

4.   Cryptanalysis of Zhang, Tang and Zhu's protocol

Before performing security analysis on the protocol of Zhang, Tang and Zhu, we assume that $\mathbb{A}$ could intercept any packets which are delivered in the authentication phase. Also, assume $\mathbb{A}$ can perform the corresponding computation.

Resilience to key compromise impersonation (KCI) attacked is considered as an important, and a desired security attributed to a key exchange protocol. It means an adversary $\mathbb{A}$ once has obtained one party's long-term private key. The saboteur may disguise another entity to the corrupted one. However, the three-round protocol of Zhang, Tang, and Zhu does not provide KCI-resilient. We show more detail on how $\mathbb{A}$ launches such an attack and achieves its goal. It breaks the confidentiality of the session key established.

Step 1. $\mathbb{A}$ compromises the perpetual secret key $s$ of the server $S$. And $\mathbb{A}$ obtains the user's identity $ID_i$ utilizing intercepting the communication messages;

Step 2. Based on the two parameters, $\mathbb{A}$ calculates $C_2 = h(ID_i\oplus s)$ and then generates two random numbers $r_1^{'}$, $r_2^{'}$;

Step 3. $\mathbb{A}$ computes $C_4 = r_1P$, $C_5 = r_1^{'}C_2P_{pub}$ and $C_6 = h(C_5)\oplus (h(ID_i\oplus s)\oplus r_2^{'}, (C_5)_{x}, (C_5)_{y})$ and transmits the REQUEST message $\{ID_i, C_4, C_6\}$ to $S$;

Step 4. $S$ calculates $C_2$ and verifies $(C_s5)_{x}, (C_5)_{y}\stackrel{?} = (sC_2C_4)_{x}, (sC_2C_4)_{y}$. If this holds, $S$ computes $C_7 = r_3P$, $SK = h(C_4, r_3C_4, C_7)$, and $Auth_s = h(h(ID_i\oplus s), r_2^{'}, (SK)_{x}, \\(C_5)_{x}, (SK)_y, (C_5)_y)$. In the formula, $r_3$ and $r_4$ are the random numbers. Next, $S$ transmits the CHALLENGE message $(realm, C_7, Auths, r_4)$ to $U_i$;

Step 5. $\mathbb{A}$ calculates $SK = h(C_4, r_1C_7, C_7)$ and verifies whether $h(C_2, r_2, (SK)_{x}, \\(C_5)_{x}, (SK)_{y}, (C_5)_y)\stackrel{?} = Auth_s$. Also, $\mathbb{A}$ computes $Auth_u = h((SK)_x, (r_4+1), (SK)_y)$ and sends the RESPONSE message $(realm, Auth_u)$ to $S$.

Step 6. $S$ needs to check whether $h((SK)_x, (r_4+1), (SK)_y)$ is equivalent to the received $Auth_u$. If it is false, $S$ discards the packets; Otherwise, $S$ accepts $\mathbb{A}$, and agrees on the session key $SK = r_1^{'}r_3P$ as their subsequent shared key.

More seriously, $\mathbb{A}$ could even guess the correct password $PW_i$ once the secret key of $S$ is compromised. $\mathbb{A}$ is by checking whether $h(PW_i^{*}\oplus r)\oplus h(ID_i\oplus s)$ until the equation holds. In this formula, $PW_i^{*}$ is an arbitrary element that is selected from the password candidate set.

5.   The proposed protocol

To solve the security problems found in Zhang, Tang and Zhu's protocol, we develop a secure SIP authentication protocol with a key agreement facility. Unlike Zhang, Tang and Zhu's design, our proposal comprises of five phases: initialization, registration (Table 1), login (Table 2), key agreement (Table 3) and password update, where the password update process is concise and convenient, that lying in it does not require interaction with the server. The proposed protocol is explained below along with Figure 1.

5.1.   Initialization

$S$ selects an additive group $G$, with a generator $P$ of large prime order $q$, including points of an elliptic curve $E$ over a finite field $\mathbf{F}_{p}$. $S$ publishes the public parameters $\{h(), sP, P, E, q \}$, where $s\in \mathbf{F}_{p}, P\in \mathbf{E}_{p}(a, b)$.

5.2.   Registration

Once a new user $U_i$ attempts to access services, he first selects his identity $ID_i$ and password $PW_i$ of his/her choice.

Step 1. The user $U_i$ picks out a random number $r$, and calculates $h(PW_i, r)$. $U_i$ then transmits the registration request message $\{ID_i\}$ to the proxy server, through a private channel.

Step 2. The proxy sever $S$ calculates $X_1 = h(ID_i, s)P$ and $X_2 = h(ID_i)nsP$. In the formula, $n$ is a random number. Next, $S$ personalizes $U_i$'s smart card which remains the values $\{X_1, X_2\}$ to $U_i$ via a secure channel.

Step 3. $U_i$ computes $X_3 = h(PW_i, r)h(ID_i, s)P$ and $X_4 = h(h(ID_i)\oplus h(PW_i, r))$. $U_i$ finally stores $r$ into his smart card. Note that the smart card includes the information $\{r, X_2, h(), X_3, X_4\}$.

5.3.   Login

$U_i$ enters his identity $ID_i$ and password $PW_i$ after putting the smart card into the card slot.

Step 1. The smart card verifies whether the condition $h(h(ID_i)\oplus h(PW_i, r))\stackrel{?} = X_4$ holds.

Step 2. If the equality holds, the smart card then gets $h(ID_i, s)P$ by calculating $h(PW_i, r)^{-1}X_3$ and $nsP = h(ID_i)^{-1}X_2$. Also, a random number $r_1$ is generated, the value $X_5 = Enc_{nsP}(ID_i, r_1P, sP)$ to verify.

Step 3. $U_i$ delivers the login request message REQUEST $\{X_5\}$ to $S$ through a public path.

5.4.   Key agreement

Step 1. $S$ uses its private key $s$ and secrets parameter $n$ to retrieve $(ID_i, r_1P, sP)$.

Step 2. $S$ proceeds to generate two random numbers $r_2$, $r_3$, and calculates the temporary key $SK = h(r_2r_1P, ID_i)$ to be shared with the user $U_i$. Also, the smart card calculates $X_6 = Enc_{h(ID_i, s)P}(r_2P, r_1P)$ and $Auth_s = h(SK, r_1P, r_3)$. And then the smart card responds with the challenge message CHALLENGE $\{realm, X_6, Auth_s, r_3\}$ to $U_i$.

Step 3. The smart card gets the values $(r_2P, r_1P)$ by decrypting $X_6$. And then, the temporary key $SK = h(r_2r_1P, ID_i)$ is gotten by $U_i$ and to be shared with the server $S$. Verifying whether $h(SK, r_1P, r_3)\stackrel{?} = Auth_s$. If the condition holds, $U_i$ deems $S$ as the legitimate server.

Step 4. The smart card computes $Auth_u = h(SK, r_2P, r_3+1)$, and transmits a response message RESPONSE $\{realm, Auth_u\}$ to $S$ through a public channel.

Step 5. $S$ examines the verification condition $h(SK, r_2P, r_3+1)\stackrel{?} = Auth_u$. If this equation holds, $S$ ensures $U_i$ as authentic and agrees on the session key $SK$ as valid key.

5.5.   Password updating

The following mechanism achieves altering the password of a legal user $U_i$ without interacting $S$.

Step 1. $U_i$ puts his smart card into the card slot and waits for commands of the terminal to provide the identity $ID_i$ and password $PW_i$. The smart card verifies that if the condition $h(h(ID_i)\oplus h(PW_i, r))\stackrel{?} = X_4$ holds. If the validation does not validate, the session is quitted promptly. Otherwise, the smart card derives $h(ID_i, s)P$ by computing $X_3h(PW_i, r)^{-1}$, and requests a new password.

Step 2. $U_i$ picks his new password $PW_i^{*}$ and the random number $r^{*}$. The smart card calculates $X_3^{*} = h(PW_i^{*}, r^{*})h(ID_i, s)P$ and $X_4^{*} = h(h(ID_i)\oplus h(PW_i^{*}, r^{*}))$.

Step 3. The smart card discards $X_3$ and $X_4$ but keeps $X_3^{*}$ and $X_4^{*}$ in its memory for renewal.

6.   Security analysis of the proposed protocol

We confirm our proposal could achieve a mutual handshake using well-popular BAN logic [36]. Also, the robustness of our proposal is validated via the universally applicable simulation tool-AVISPA [33,34]. In addition, we provide informal cryptanalysis so as to demonstrate our proposal is well protecting against relevant security attacks.

6.1.   Verification of the proposal under BAN Logic

BAN logic is a well-known formal method used to strictly prove the authentication protocols' security, or find security vulnerabilities. Subsequently, we will introduce more details about BAN logic. BAN logic includes basic logical notations and some logic postulates. According to these preliminaries, we show the desired goals, idealized form, assumptions for our protocol. And we finally demonstrate its correctness.

$\mathbf{\textbf{Notations}}$

$\cdot$ $P|\equiv X$: $P$ deems $X$ is true;

$\cdot$ $P \triangleleft X$: $P$ observes $X$;

$\cdot$ $P |\sim X$: $P$ ever have sent $X$;

$\cdot$ $P\Rightarrow X$: $P$ judges $X$;

$\cdot$ $\# X$: $X$ is fresh;

$\cdot$ $P \stackrel{K}\leftrightarrow Q$: share a key $K$ between $P$ and $Q$;

$(X, Y)$: $X$ or $Y$ is one portion of the formula $(X, Y)$;

$\cdot$ $< X, Y > _K$: $K$ is the key to encrypt $X$ and $Y$.

$\mathbf{\textbf{ Postulates }}$

$\cdot$ Message-meaning rule: $\frac{A|\equiv A\stackrel{K}\leftrightarrow B, \ A\triangleleft < X > _K}{A|\equiv |B\sim X}$;

$\cdot$ Fresh conjuncatenation rule: $\frac{A|\equiv \#(X)}{A|\equiv\#(X, \ Y)}$;

$\cdot$ Belief rule: $\frac{A|\equiv X, \ A|\equiv Y}{A|\equiv (X, \ Y)}$;

$\cdot$ Nonce-verification rule: $\frac{A|\equiv \#(X), \ A |\equiv B|\sim X}{A|\equiv B|\equiv X}$;

$\cdot$ Jurisdiction rule: $\frac{A|\equiv B\Rightarrow X, \ A |\equiv B|\equiv X}{A|\equiv X}$.

$\mathbf{\textbf{Goals}}$

$\cdot$ $Goal_1$: $S|\equiv ID_i$

$\cdot$ $Goal_2$: $S|\equiv U_i\stackrel{SK}\longleftrightarrow S$

$\cdot$ $Goal_3$: $S|\equiv U_i |\equiv U_i \stackrel{SK}\longleftrightarrow S$

$\cdot$ $Goal_4$: $U_i|\equiv U_i\stackrel{SK}\longleftrightarrow S$

$\cdot$ $Goal_5$: $U_i|\equiv S |\equiv U_i\stackrel{SK}\longleftrightarrow S$

$\mathbf{\textbf{Idealized form}}$

$\cdot$ $U_i\rightarrow S$: $\{X_5, Auth_u \}$

$X_5$: $< ID_i, r_1P, sP > _{U_i\stackrel{nsP}\longleftrightarrow S}$,

$Auth_u$: $< r_2P, r_3+1, SK > _{U_i\stackrel{SK}\longleftrightarrow S}$;

$\cdot$ $S\rightarrow U_i$: $\{X_6, Auth_s \}$

$X_6$: $< r_2P, r_1P > _{U_i\stackrel{h(ID_i, s)}\longleftrightarrow S}$,

$Auth_s$: $< r_1P, r_3, SK > _{U_i\stackrel{SK}\longleftrightarrow S}$

$\mathbf{\textbf{Assumptions}}$

$\cdot$ $A_1$: $U_i|\equiv ID_i$;

$\cdot$ $A_2$: $U_i|\equiv \# r_1$;

$\cdot$ $A_3$: $S|\equiv \# n$;

$\cdot$ $A_4$: $S|\equiv \# s$;

$\cdot$ $A_5$: $U_i |\equiv (U_i\stackrel{nsP} \leftrightarrow S)$;

$\cdot$ $A_6$: $S |\equiv (U_i\stackrel{nsP} \leftrightarrow S)$;

$\cdot$ $A_7$: $U_i |\equiv (U_i\stackrel{h(ID_i, s)} \leftrightarrow S)$;

$\cdot$ $A_8$: $S|\equiv U_i\Rightarrow ID_i$

$\cdot$ $A_9$: $S|\equiv U_i\Rightarrow r_1$

$\cdot$ $A_{10}$: $S|\equiv \# r_2$

$\cdot$ $A_{11}$: $U_i|\equiv S\Rightarrow r_2$

$\cdot$ $A_{12}$: $S|\equiv \# r_3$

$\cdot$ $A_{13}$: $U|\equiv \# r_3$

$\mathbf{\textbf{Proofs}}$

$\cdot$ According to message $X_5$, we have

$P_1$. $S\triangleleft < ID_i, r_1P, sP > _{U_i\stackrel{nsP}\longleftrightarrow S}$

$\cdot$ By $P_1$, $A_6$ and message-meaning rule, we obtain

$P_2$. $S|\equiv U_i|\sim(ID_i, r_1P, sP)$

$\cdot$ From $P_2$, $A_4$ and fresh conjuncatenation rule, we derive

$P_3$. $S|\equiv \#(ID_i, r_1P, sP)$

$\cdot$ Since $P_2$, $A_3$ and nonce-verification rule, we get

$P_4$. $S|\equiv U_i|\equiv(ID_i, r_1P, sP)$

$\cdot$ According to $P_4$ and belief rule, we get

$P_5$. $S|\equiv U_i|\equiv ID_i$, $S|\equiv U_i|\equiv r_1P$

$\cdot$ From $A_8$, $A_9$ and jurisdiction rule, we derive

$P_6$. $S|\equiv r_1P$, $Goal_1$. $S|\equiv ID_i$

$\cdot$ Since $SK = h(r_1r_2P, ID_i)$, $Goal_1$, and $P_6$, we have

$Goal_2$. $S|\equiv U_i\stackrel{SK}\longleftrightarrow S$

$\cdot$ According to message $Auth_u$, we have

$P_7$. $S\triangleleft < r_2P, r_3+1, SK > _{U_i\stackrel{SK}\longleftrightarrow S}$

$\cdot$ By $P_7$, $Goal_2$ and message-meaning rule, we obtain

$P_8$. $S|\equiv U_i|\sim (r_2P, r_3+1, U_i\stackrel{SK}\longleftrightarrow S)$

$\cdot$ Since $P_8$, $A_{10}$, $A_{11}$ and nonce-verification rule, we have

$Goal_3$. $S|\equiv U_i |\equiv U_i \stackrel{SK}\longleftrightarrow S$

$\cdot$ From message $X_6$, we attain

$P_9$. $U_i\triangleleft < r_2P, r_1P > _{U_i\stackrel{h(ID_i, s)}\longleftrightarrow S}$

$\cdot$ According to $A_7$, $P_9$ and message-meaning rule, we get

$P_{10}$. $U_i|\equiv S |\sim (r_2P, r_1P)$

$\cdot$ By $A_2$ and fresh conjuncatenation rule, we have

$P_{11}$. $U_i|\equiv \#(r_2P, r_1P)$

$\cdot$ According to $P_{10}$, $P_{11}$ and nonce-verification rule, we attain

$P_{12}$. $U_i|\equiv S|\equiv r_2P$

$\cdot$ Since $P_{12}$, $A_{11}$ and jurisdiction rule, we derive

$P_{13}$. $U_i|\equiv r_2P$

$\cdot$ From $SK = h(r_1r_2P, ID_i)$, $A_1$, $A_2$, $P_{13}$, we have

$Goal_4$. $U_i|\equiv U_i\stackrel{SK}\longleftrightarrow S$

$\cdot$ By message $Auth_s$, we attain

$P_{14}$. $U_i\triangleleft < r_1P, r_3, SK > _{U_i\stackrel{SK}\longleftrightarrow S}$

$\cdot$ Since $P_{14}$, $Goal_4$ and message-meaning rule, we have

$P_{15}$. $U_i|\equiv S|\sim (r_1P, r_3, U_i\stackrel{SK}\longleftrightarrow S)$

$\cdot$ According to $P_{13}$, $P_{15}$, $Goal_4$ and nonce-verification rule, we attain

$Goal_5$. $U_i|\equiv S |\equiv U_i\stackrel{SK}\longleftrightarrow S$

6.2.   Formal security analysis

Theorem 1. The probability that an attacker $\mathbb{A}$ breaks the AKE security of our AKAP is

$Adv_{P}^{Ake}(\mathbb{A})\leq \frac{q_h^{2}}{2^{l-1}}+\frac{2q_{send}}{|D|}$

where $q_{send}$, $q_h$ and $D$ denote the number of Send queries, Hash queries, and a uniformly distributed dictionary, respectively.

Proof: Game $G_i(i = 0, 1, 2)$ defines three games. Game $G_0$ is the factual attack, and game $G_3$ concludes a breach of the AKE security of our AKAP is asymptotically optima:

Game $G_0$: This game corresponds to the actual attack.

$Adv_{P}^{Ake}(\mathbb{A}) = |2Pr[Succ_0] -1|$.

Game $G_1$: This game simulates the eavesdropping attack by querying $Execute(U^{i}, S^{j})$ oracle, and then by querying $Test(P^{i})$ oracle. It decides whether the result of Test is the real session key $SK$ or a random value. We know that $r_1P$ is derived by the server's secret key $s$, and secrets parameter $n$. That is, $\mathbb{A}$ has no way to compute $r_1P$ through eavesdrop on the communication channel unless $S$ is compromised. Also, $r_2P$ is not impossible to obtain, unless it possesses both the smart card and password. Hence, intercepting is not probable for helping $\mathbb{A}$ to win in this game. Thus,

$Pr[Succ1] = Pr[Succ0]$.

Game $G_2$: This game models $Send(M, P^{i})$ query, in which $\mathbb{A}$ can eavesdrop or alter the information from the transcripts. Then, games $G_2$ and $G_1$ are undistinguishable unless the collision occurring in $G_2$. Thus,

$|Pr[Succ2] - Pr[Succ1]| \leq q_{h}^{2}/2^{l}$.

Game $G_3$: This game models $Corrupt(SC)$ query, in which $\mathbb{A}$ has obtained the smart card to simulate the smart card breach attack. Since the password $PW_i$ is protected by a cryptographic one-way function, where $X_3 = h(PW_i, r)h(ID_i, s)P$ and $X_4 = h(h(ID_i)\oplus h(PW_i, r))$. This implies that $\mathbb{A}$ has no way to check the password excepts possession of user's identity, or corrupts the server to get $s$. Hence, $|Pr[Succ3] - Pr[Succ1]| \leq q_{send}/|D|$.

6.3.   Verifying protocol using AVISPA tool

AVISPA is a simulation engine for the automated validation of Internet security protocols and applications. Upon Dolev and Yao model, four model back-ends, called OFMC (On-the-fly Model-Checker), CL-AtSe (Constraint-Logic-based Attack Searcher), SATMC (SAT-based Model-Checker), and TA4SP (Tree Automata-based Protocol Analyzer) (Figure 1) are utilized for the validation using HLPSL (High-Level Protocol Specification Language). The HLPSL presentation of the protocol is compiled to IF by the translator-HLPSL2IF. IF is an entrance of the four different back-ends. The output OF is exported by using one of the four back-ends, which shows the conclusion if the AKAP is secure or insecure.

During the protocol execution, each entity act a role, which is a feature of AVISPA. We show the role specifications in HLPSL of Appendix the initiator, responder, session, and environment and goal in Appendix Figures 1–4. In our implementation, we assume $S$'s private key is a public parameter. The privacy of three parameters and client-server authentication is verified:

$\cdot$ The secrecy_of subs1: the user confidential parameters $ID_i$ is gotten only $U_i$ and $S$.

$\cdot$ The secrecy_of subs2: the user confidential parameters $PW_i$ is gotten only $U_i$.

$\cdot$ The secrecy_of subs3: the session key is gotten only $U_i$ and $S$.

$\cdot$ Authenticaion_si_ui_auths: $U_i$ validates $S$ by receiving $r_2$ securely, $r_2$ is a ephemeral number of S.

$\cdot$ Authenticaion_si_ui_authu: $U_i$ validates $S$ by receiving $r_1$ securely, $r_1$ is a ephemeral number of $S$.

After running the program under two back-ends CL-AtSe and OFMC, Figures 2 and 3 show that our proposal realizes the session security without imperfection.

6.4.   Informal security analysis

We demonstrate that our proposal holds many security attributes, such as mutual authentication, anonymity, privileged insider attack, perfect forward secrecy, KCI-resistance, etc., under a condition. The condition is that $\mathbb{A}$ extracts all the data stored inside a user's smart card, or/and eavesdrops on all the messages involved in an authentication-key agreement session [37,38].

6.4.1.   Mutual authentication

Note that $r_1P$ can be only decrypted by the legal server. Thus, after $U_i$ receives CHALLENGE message depending upon the result of decryption test, $U_i$ verifies the legitimacy of $S$ by checking the equivalence $h(SK, r_1P, r_3)\stackrel{?} = Auth_s$. Simultaneously, only the legal $U_i$ can derive $r_2P$ from CHALLENGE message. And hence, $S$ assures that he is communicating with the legitimate $U_i$, employing checking the equivalence $h(SK, r_2P, r_3+1)\stackrel{?} = Auth_u$ after receiving RESPONSE message. Therefore, $\mathbb{A}$ cannot cheat any of the communicating entity. And thus the proposed protocol achieves proper mutual authentication.

6.4.2.   Internal-privileged attack

$U_i$'s password $PW_i$ is hashed by the random number $r$ during registration phase. And it is not delivered to $S$. Therefore, an honest but curious insider has no ability, to know the real password $PW_i$ of $U_i$. In other words, the proposed protocol indicates good capability of defeating insider attack.

6.4.3.   User anonymity

The identity $ID_i$ of $U_i$ is disguised for dual protection, involved in all the transmitted messages. For one thing, the random number $n$ is picked excepts the private key $s$ of $S$, based on the ECDLP assumption. For another, the plain-text $ID_i$ is hashed by double times along with the secret parameters $r_1P$, and $r_2P$. And the two parameters are encrypted by $nsP$. That is, to identify two parameters are equate as determine ECDLP problem. The problem is one of the well-known difficult problems within polynomial time. In a word, the proposed protocol supports high user anonymity.

6.4.4.   Perfect forward and backward secrecy

Suppose that perpetual privacy information $s$ of $S$ is compromised by $\mathbb{A}$, he is incapable of computing the current as well as the future session keys. It is noteworthy that the session key is related with three important parameters, i.e, $U_i$'s identity, two random numbers $r_1$ and $r_2$ generated by $U_i$ and $S$, which are present in the form of $r_1P$ and $r_2P$, respectively. With the purpose of acquiring $ID_i$ and $r_1P$ from intercepted REQUEST message $\{X_5 = Enc_{nsP}(ID_i, r_1P, sP)\}$, another random number $n$ is also needed. Unfortunately, no one but $S$ knows what is the real random number $n$. More seriously, $r_2P$ can not be derived through $X_6 = Enc_{h(ID_i, s)P}(r_1P, r_2P)$ without knowing $ID_i$ of $U_i$. Next, assume that the current session key is corrupted by $\mathbb{A}$. He plans to deduce the next negotiatory key. The cause of failure of extraction the secret $SK = h(r_1P, r_2P, ID_i)$ lies in the irreversible property of hash function, he total has no way to compute the previous and future session keys thereby. In a conclusion, our protocol preserves perfect forward secrecy property.

6.4.5.   Resist modification attacks

We assume $\mathbb{A}$ intercepts the REQUEST message $\{X_5\}$ submitted to $S$, he attempts to modify the parameter $r_1P$ and transmit the forged message $\{X_5\}$ to $S$. However, he has no way to attain $ID_i$ and $n$ to compute the symmetric key $nsP$, and thus encrypt $ID_i$. Even with the smart card security breached, he is not strong enough to gain the exact values of $ID_i$, and $s$ without the knowledge of $PW_i$. On the other hand, without knowing $ID_i$, $n$ and $s$, it is impossible for $\mathbb{A}$ to learn $r_1P$ picked by $U_i$ intercepting CHALLENGE message. That is, $\mathbb{A}$ could neither impersonate as an authorised user nor masquerade as a legitimate sever, through eavesdropping the raw messages to tamper with them. In a word, our proposal is is resistance to modification attacks.

6.4.6.   Resist off-line password attack

We suppose $\mathbb{A}$ gets $\{r, X_2, X_3, X_4, h()\}$ and all the public messages $\{ X_5, X_6, Auth_s, r_3, Auth_u\}$ from the communication channel. First and foremost, the password $PW_i$ of $U_i$ do not involve in transcription between $U_i$ and $S$. Hence, $\mathbb{A}$ has no way to ensure whether the guesses password is true or false. Secondly, to derive the correct password $PW_i$ from $X_3$ is a computationally infeasible task for $\mathbb{A}$, in condition that he is unaware of the identity $ID_i$ of $U_i$, and the private key $s$ of $S$. Eventually, $\mathbb{A}$ is not entirely sure what are the real values of $ID_i$, and $PW_i$ by $X_4$, because the two personal information are hidden in two-layer hash function. In conclusion, the proposed protocol could thwart off-line password attack successfully.

6.4.7.   Resist key-compromise impersonation attack

The whole design presumes that the privacy information $s$ is considered as an open parameter. In this case, $\mathbb{A}$ has no way to impersonate as an authorised user thereby accessing service resources. The resources are provided by the secure server. Let's analyze the cases. Aiming at playing a valid user, the identity of the real user $U_i$ is urgent required, since the server will detect the attack while checking $Auth_u$. To get the real value of $ID_i$, another random number $n$, which only the legal server knows it, is also needed. Thus, it is not possible for $\mathbb{A}$ to try to impersonate as a legitimate user.

6.4.8.   Session key agreement

Two parties independently negotiate an ephemeral session key $SK = h(r_1r_2P, ID_i)$. They keep communicate securely on the strength of it for a subsequent communication. Upon this ability, each entities can encrypt the following packets, to preserve the security of the handshake. Moreover, the negotiatory key is fresh for each session. The reason is that random numbers are different based on the property of hash function. As a consequence, deriving the session key through the eavesdropped information is a challenging task for $\mathbb{A}$.

6.4.9.   Stolen-verifier attacks

The service provider does not create the password verification table of the service requester. Even though the service provider's database is available by $\mathbb{A}$, he still cannot steal and modify user passwords, and thus attain the authentication information of users. Thus it can be said that our proposal can withstand stolen-verifier attacks.

7.   Performance considerations and functionality comparison

This part evaluates and examines the performance of our proposed scheme, and compare it with five related protocols relies basically on the ECC, one-way hash functions. In order to evaluate the entire computing cost for each protocol more accurately, the arithmetic mean for each cryptographic computation timings after running 1000 times are shown in Table 4. The processor is Intelr Pentiumr CPU G3250, 3.20GHz with 4.0GB of RAM running Windows 10. We use the jPBC library primitives timings. The processor is Intelr Pentiumr CPU G3250, 3.20GHz with 4.0GB of RAM running Windows 10. We use the jPBC library(2.0.0) [41], a Java port of the PBC library written in C [39,40], the Java Development Kit used is the Oracle jdk 1.8.0 65. We used the Type $A$ curves with the prime order $q$ defined as $E(Fq): y^2 = x^3 + x$, hash function as SHA-3 [19,42], and symmetric encryption algorithm as AES [43]. The calculation expense and execution time of the registration and key agreement phases with the revelent protocols [24,27-30,32] are listed in Table 5. Also, we compare the security attributes with the related protocols [24,27-30,32] (Table 6). The entire running time of our AKAP is lower than Tu et al. [24]. And the time is lower than Yeh, Chen and Shih's protocol[28] too. Table 5 has shown the results. However, one thing is ignored by these protocols [24,27-30,32] from Table 6. It is to analyze whether the design has ability to conquer key compromise impersonation attack. Additionally, the related protocols seem not to be considered as a SIP authenticated key agreement, which can be perfect or ideal. The reason is that the protocol lacks some essential security properties. In general, our proposal takes a better tradeoff between computational cost and security attributes, while comparing with the protocols [24,27-30,32].

8.   Conclusions

Despite the role that AKAP plays in ensuring the security of communication in an open network, designing secure and efficient protocols, including in a VoIP environment, remains challenging. For example, in this paper, we revisited and revealed vulnerabilities in the design of Zhang, Tang and Zhu's protocol [32]. We also presented an improved protocol and demonstrated its correctness and security, as well as demonstrating its utility in terms of performance efficiency. Future research will include implementing a prototype of the protocol and evaluating it in a real-world deployment.

Acknowledgments

The authors would like to thank all the editors and anonymous reviewers for their helpful advice. This paper is supported by the National Natural Science Foundation of China (No. 61802276), and the Fundamental Research Funds for the Central Universities of China (No.3122021027).

Conflict of interest

The authors declare that they have no known competing financial interest or personal relationship that could have appeared to influence the work reported in this paper.

Appendix