A Key Caching Mechanism for Reducing
WiMAX Authentication Cost in Handoff
Shih-Feng Hsu and Yi-Bing Lin, Fellow, IEEE
Abstract—The IEEE 802.1X is utilized in mobile Worldwide Interoperability for Microwave Access (WiMAX) authentication. This procedure incurs a long delay in WiMAX handoff. To re-solve this issue, this paper proposes a key caching mechanism to eliminate the nonnecessary IEEE 802.1X authentication cost in WiMAX handoff. This mechanism is investigated through analytic and simulation modeling. Our study indicates that the key caching scheme can effectively speed up the handoff process.
Index Terms—Authentication, authorization, and account-ing (AAA), handoff, mobile Worldwide Interoperability for Microwave Access (WiMAX).
I. INTRODUCTION
T
HE IEEE 802.16e mobile Worldwide Interoperability for Microwave Access (WiMAX) provides broadband wire-less services with wide service coverage, high data throughput, and high mobility. To support security network access, the au-thentication, authorization, and accounting (AAA) mechanism is exercised in WiMAX [3]. Fig. 1 shows the AAA architecture and protocol stack for WiMAX. In this architecture, the access service network [see ASN; Fig. 1 (2)] consists of base stations [see BSs; Fig. 1 (4)] and ASN gateways [see ASN-GWs; Fig. 1 (5)]. An ASN-GW controls several BSs. A BS provides WiMAX radio access for mobile stations [see MSs; Fig. 1 (1)] after the MSs are authenticated by the AAA server [see Fig. 1 (6)] in the connectivity service network [see CSN; Fig. 1 (3)]. In the WiMAX AAA architecture, the ASN-GW serves as the authenticator for the MS. The authenticator is responsible for forwarding authentication messages between the MS and the AAA server and for maintaining the MS-related information (e.g., encryption keys) after authentication. We assume that the subscriber identity module (SIM)-based extensible authen-tication protocol (EAP) is utilized for AAA [1]. Note that this approach reuses the authentication mechanism in mobile telecommunications [6]. In the authentication procedure, an EAP-SIM message [see Fig. 1(a)] is encapsulated in an EAP message [see Fig. 1(b)]. The MS then encapsulates the EAP Manuscript received September 27, 2008; revised January 23, 2009 and March 14, 2009. First published May 2, 2009; current version published October 2, 2009. The work of Y.-B Lin was supported in part by the Na-tional Science Council, Taiwan, under Grant NSC 97-2221-E-009-143-MY3 and Grant NSC 97-2219-E-009-016, by Intel, by Chunghwa Telecom, by the Industrial Technology Research Institute/National Chiao Tung University Joint Research Center, and by the Ministry of Education under the Aim for Top University project. The review of this paper was coordinated by Prof. J. Li.S.-F. Hsu is with the Department of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected]).
Y.-B. Lin is with the Department of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan, and also with the Institute of Information Science, Academia Sinica, Taipei 115, Taiwan (e-mail: [email protected]).
Digital Object Identifier 10.1109/TVT.2009.2022079
message in privacy key management protocol version 2 [see PKMv2; Fig. 1(c)] before it is transmitted to the BS. The BS exercises the authentication relay protocol [see AuthRelay; Fig. 1(d)] to forward the received EAP message to the authen-ticator (i.e., the ASN-GW). Upon receipt of an EAP message, the authenticator translates it into a remote authentication dial-in user service [see RADIUS; Fig. 1(e)] message. Then, the RADIUS message is sent to the AAA server. Upon receipt of the RADIUS message, the AAA server utilizes the mobile application part [see MAP; Fig. 1(f)] of the signaling sys-tem number 7 [see SS7; Fig. 1(g)] protocol to communicate with the home location register (HLR)/authentication center (AuC) [see Fig. 1 (7)]. The HLR is the mobility database of the GSM/UMTS mobile telecommunication networks [2], [6]. The AuC maintains the secret keys of the MSs and provides the authentication information to the AAA server.
II. WIMAX INITIALNETWORKENTRYPROCESS By using the protocols described in Fig. 1, the WiMAX authentication works as follows. Supposing that an MS first connects to the WiMAX network, the following steps are executed for the initial network entry process (see Fig. 2).
Step 1) The MS, the BS, and the ASN-GW (authenticator) negotiate the security policy (i.e., to select the encryp-tion and decrypencryp-tion algorithms) and the authorizaencryp-tion policy, specifically to select the message-authentication-code (MAC) type.
Step 2) The authenticator sends an EAP request message to the MS. This message initiates the IEEE 802.1X authentication procedure by requesting the user identity. Steps 3) and 4) The MS replies with an EAP response message with the user identity to the authenticator. The user identity consists of two elements: the AAA server address AAA-addr and the user account User-acct. In the SIM-based EAP authentication, the user account is set to the international mobile subscriber identity (IMSI) of the MS [2]. According to AAA-addr, the authenticator forwards the EAP response message to the AAA server. Step 5) Upon receipt of the user identity, the AAA server performs the SIM-based EAP authentication with the MS as follows.
Step 5.1) The AAA server issues an EAP request message with type “Start” to the MS.
Step 5.2) The MS replies with the EAP response message containing a random number, i.e., MS-RAND. This ran-dom number is used to derive the encryption keys in Steps 5.3 and 5.5 below.
Fig. 1. WiMAX AAA architecture and protocol stack.
Fig. 2. WiMAX initial network entry process.
Step 5.3) Based on the IMSI received in Step 4), the AAA server communicates with the HLR/AuC to obtain the authentication information, including a random number RAND, a signed result SRES, and a cipher key Kc. Both the MS and the HLR/AuC utilize the RAND and the secret key Ki (stored in the SIM card and the HLR/AuC) to execute the A3 and A8 algorithms to derive the signed result SRES and the cipher key Kc [9]. Then, the AAA server utilizes Kc and MS-RAND (received in Step 5.2) to derive the master session key (MSK) and the EAP integrity key KEAP.
Step 5.4) The AAA server sends a challenge EAP request message with the RAND and the MAC. This MAC is derived from KEAPand is used to ensure the integrity of
this message.
Step 5.5) Upon receipt of the EAP request message, the MS utilizes RAND, MS-RAND (generated in Step 5.2), and Ki (stored in the SIM card) to generate SRES∗, Kc, MSK, and KEAP. With KEAP and the received
RAND, the MS verifies the received MAC. If the MAC
Fig. 3. WiMAX key derivation tree.
is correct, the AAA server is successfully authenticated by the MS. Then, the MS replies with a challenge EAP response message with a code MAC∗derived from KEAPand SRES∗.
Step 6) The AAA server verifies MAC∗ by using KEAP
(generated in Step 5.3) and SRES (received in Step 5.3). If MAC∗ is correct, the MS is successfully authenticated by the AAA server. The AAA server sends the EAP success message to the authenticator containing MSK (generated in Step 5.3), the MSK lifetime, and the MS authorization profile (e.g., service restrictions and supplementary services). The MSK lifetime is the period that the MS is authorized to access the ASN-GW. When the MSK lifetime has expired, the MS should execute the IEEE 802.1X authentication with the AAA server again.
Step 7) The ASN-GW stores MSK, the MSK lifetime, and the authorization profile. Then, it derives the authentication key (AK) by using the MSK and the BS address. This AK is shared between the MS and the BS.
Step 8) The ASN-GW forwards the EAP success message to the BS with AK. The BS passes the EAP success message to inform the MS that the authentication is suc-cessful. Upon receipt of this message, the MS generates its version of AK.
Step 9) The BS generates the final encryption key, i.e., the traffic encryption key (TEK). This encryption key is used to provide data integrity and confidentiality for a communication session between the MS and the BS. The BS passes the generated TEK (encrypted by AK) to the MS.
The relationship of WiMAX encryption keys and the loca-tions maintaining these keys are shown in Fig. 3.
Fig. 4. Relationship of the MSK lifetime and the MS movement.
If the MS moves from the old BS to the new BS connecting to a different authenticator (ASN-GW), a new MSK must be generated in this inter-ASN-GW handoff process, which is the same as the initial network entry process described in Fig. 2. In this case, the authenticator (ASN-GW) of the old BS will remove the MS key record (i.e., MSK, the MSK lifetime, and the MS authorization profile). When the MS moves back to the old ASN-GW again, another inter-ASN-GW handoff process should be performed, which may incur a long delay.
III. KEYCACHINGMECHANISM
To speed up the inter-ASN-GW handoff process, we propose a key caching mechanism. The idea is simple: When the MS moves from the old ASN-GW to the new ASN-GW, the old ASN-GW still keeps the MS key record. If the MS returns to the old ASN-GW before the MSK lifetime expires, it can reuse the MSK without executing the IEEE 802.1X authentication. That is, only Steps 1) and 9) in Fig. 2 are executed to speed up the inter-ASN-GW handoff process. In Fig. 2, Step 1) contains two message exchanges, and Step 9) contains five message exchanges [3]. Therefore, the caching mechanism speeds up the process by saving 50% (= 7/14) of the message exchanges between the MS and the BS.
Although the key caching mechanism may effectively avoid the execution of IEEE 802.1X authentication, it consumes extra storage to keep the MS key records at the old ASN-GW, where a stored key record includes 512 or 1024 bits for the MSK, 32 bits for the MSK lifetime, and 512 or 1024 bits for the MS authorization profiles. Therefore, it is desirable to select an appropriate MSK lifetime to eliminate the IEEE 802.1X authentication without consuming too much extra storage in the ASN-GW. We investigate the effect of the MSK lifetime on the caching performance by an analytic model described below.
Fig. 4 illustrates the relationship between the movement of an MS and its MSK lifetime. In this figure, the IEEE 802.1X authentication is executed at time τ0 [see Fig. 4 (1)], and the
MSK lifetime expires at time τ3 [see Fig. 4 (4)]. At time τ1
[see Fig. 4 (2)], the MS moves from the old ASN-GW to the new ASN-GW. The residual MSK lifetime is tK = τ3− τ1. If
the MS does not return to the old ASN-GW before the MSK lifetime expires, we call this tK period the unused key period.
At time τ2 [see Fig. 4 (3)], the MS returns to the old
ASN-GW. Let tM = τ2− τ1 be the period between when the MS
leaves the old ASN-GW and when it returns. If the MS returns
before the MSK lifetime expires, the MS can reuse the MSK for period t∗K= tK− tM without executing the IEEE 802.1X
authentication. Period t∗Kis referred to as the reused key period. We make the following assumptions.
1) We consider two distributions for the MSK lifetime T . That is, T is either an exponential period with rate μ or a fixed period.
2) The MS residence time tM in new ASN-GWs has the
density function f (tM) with mean 1/λ and variance VM.
Three output measures are evaluated in our study:
1) α: the probability that the MS returns to the old ASN-GW before the MSK lifetime expires;
2) E[tK|tM ≥ tK]: the expected unused key period under
the condition that the MS does not return to the old ASN-GW before the MSK lifetime expires (therefore, the cached MSK will not be reused);
3) E[t∗K|tM ≤ tK]: the expected reused key period under
the condition that the MS returns to the old ASN-GW before the MSK lifetime expires (the cached MSK is reused).
We derive the above output measures for exponentially dis-tributed tM with fixed T and then generalize the derivation for
generally distributed tM with exponentially distributed T .
A. Derivation for Exponentially DistributedtM and FixedT
Suppose that the departure of the MS from the old ASN-GW is a random observer to the MSK lifetime. For the fixed MSK lifetime T , from the residual life theorem [4], tKhas a uniform
distribution over 0≤ tK ≤ T . Then, α is derived as
α = P r[tM ≤ tK] = T tK=0 1 T × ⎛ ⎝ tK tM=0 λe−λtMdt M ⎞ ⎠ dtK =e −λT + λT − 1 λT . (1) E[tK|tM ≥ tK] is expressed as E[tK|tM ≥ tK] = E[tKand tM ≥ tK] P r[tM ≥ tK] (2) where E[tKand tM ≥ tK] = T tM=0 λe−λtM× ⎡ ⎣ ⎛ ⎝ tM tK=0 tK× 1 T dtK ⎞ ⎠ ⎤ ⎦ dtM + ∞ tM=T λe−λtM× ⎡ ⎣ ⎛ ⎝ T tK=0 tK× 1 T dtK ⎞ ⎠ ⎤ ⎦ dtM = 1− e −λT λ2T − e−λT λ . (3)
From (1)–(3), we have E[tK|tM ≥ tK] = E[tKand tM ≥ tK] P r[tM ≥ tK] = 1− e−λT λ2T − e−λT λ × 1 1− α =1 λ− T e−λT 1− e−λT. (4)
Similarly, E[t∗K|tM ≤ tK] is expressed as
E [t∗K|tM ≤ tK] = E [t∗K and tM ≤ tK] P r[tM ≤ tK] (5) where E [t∗K and tM ≤ tK] = T tK=0 1 T × ⎡ ⎣ tK tM=0 (tk− tM)λe−λtMdtM ⎤ ⎦ dtK =T 2 − 1 λ+ 1− e−λT λ2T . (6)
From (1), (5), and (6), we have E [t∗k|tM ≤ tK] = T 2 − 1 λ+ 1− eλT λ2T × 1 α = λT 2 2(λT + e−λT − 1) − 1 λ. (7)
B. Derivation for Generally DistributedtM and
ExponentialT
Since the departure of the MS from the old ASN-GW is a random observer to the MSK lifetime, from the residual life theorem, tK is exponentially distributed with mean E[T ] =
1/μ. Let tM have an arbitrary distribution with density function
f (tM) and Laplace transform f∗(s). Then, α is derived as
α = ∞ tK=0 μe−μtK× ⎡ ⎣ tK tM=0 f (tM)dtM ⎤ ⎦ dtK = f∗(μ). (8) E[tKand tM ≥ tK] is expressed as
E[tKand tM ≥ tK] = ∞ tM=0 f (tM)× ⎛ ⎝ tM tK=0 tKμe−μtKdtK ⎞ ⎠ dtM = 1 μ+ df∗(s) ds s=μ −f∗(μ) μ . (9)
From (2), (8), and (9), we have E[tK|tM ≥ tK] = E[tK and tM ≥ tK] P r[tM ≥ tK] = 1 μ+ df∗(s) ds s=u −f∗(μ) μ × 1 1− f∗(μ) . (10) E[t∗Kand tM ≤ tK] is derived as
E [t∗Kand tM ≤ tK] = ∞ tK=0 μe−μtK× ⎡ ⎣ tK tM=0 (tk− tM)f (tM)dtM ⎤ ⎦ dtK = f ∗(μ) μ . (11)
Therefore, from (5), (8), and (11), we have E [t∗K|tM ≤ tK] = E [t∗K and tM ≤ tK] P r[tM ≤ tK] = f∗(μ) μ × 1 f∗(μ) = 1 μ. (12) Equation (12) says that E[t∗K|tM ≤ tK] is not affected by the
tM distribution.
To further investigate (8) and (10), we assume that tM has
a Gamma distribution, which has been used in telecommuni-cation modeling [5], [7]. The Gamma-distributed tM has mean
1/λ, variance VM, and Laplace transform
f∗(s) = 1 λVMs + 1 1 λ2 VM . Then, from (8) α = f∗(μ) = 1 λμVM+ 1 1 λ2 VM (13) and E[tK|tM ≥ tK] is expressed as
E[tK|tM ≥ tK] = 1 μ+ df∗(s) ds s=μ −f∗(μ) μ × 1 1− f∗(μ) = 1 μ− 1 λ× 1 λμVM+ 1 1 λ2 VM+1 −1 μ× 1 λμVM + 1 1 λ2 VM × ⎡ ⎢ ⎣ 1 1− 1 λμVM+1 1 λ2 VM ⎤ ⎥ ⎦ . (14)
TABLE I
COMPARISON OFANALYTIC ANDSIMULATIONRESULTS. (a) α (EXPONENTIALtM). (b) E[tk|tM ≥ tk] (GAMMAtM
ANDEXPONENTIALT ). (c) E[t∗k|tM ≤ tk]
(GAMMAtMANDEXPONENTIALT )
When tM is exponentially distributed (i.e., VM = 1/λ2), (13)
is rewritten as α = 1 λμVM + 1 1 λ2 VM = λ λ + μ (15) and (14) is expressed as E[tK|tM ≥ tK] = 1 μ− 1 λ× 1 λμVM + 1 1 λ2 VM+1 − 1 μ× 1 λμVM + 1 1 λ2 VM × ⎡ ⎢ ⎣ 1 1− 1 λμVM+1 1 λ2 VM ⎤ ⎥ ⎦ = 1 λ + μ. (16)
Equations (1), (4), (7), (12), (15), and (16) provide the mean value analysis to show the “trends” of the output measures. These equations are also used to validate the simulation exper-iments. Table I shows that the simulation is consistent with the analytic analysis, and all errors are within 1%.
Fig. 5. Effect of μ. (a) Effect of E[T] on α. (b) Effect of E[T ] on E[tK|tM ≥
tK]. (c) Effect of E[T ] on E[t∗K|tM ≤ tK].
IV. NUMERICALEXAMPLES
According to the analytic and the simulation models, we use numerical examples to investigate how the MSK lifetime T affects the performance of the key caching mechanism. Fig. 5 plots the results for exponential tM. Fig. 5(a) plots α against
E[T ]. The figure indicates that α is an increasing function of E[T ]. It is intuitive that if E[T ] is large, then it is more
likely that the MS will return before the MSK lifetime expires. From (1) lim E[T ]→∞α = limT→∞ e−λT+ λT − 1 λT = 1. Since E[T ] = 1/μ, from (15), we have
lim
E[T ]→∞α =E[T ]lim→∞
λ λ + (1/E[T ])
= 1.
This figure also shows that the exponential T outperforms the fixed T in terms of α.
Fig. 5(b) plots the unused key period E[tK|tM ≥ tK] as a
function of E[T ]. The figure shows that the unused key period increases as E[T ] increases. From (4), we have
lim
E[T ]→∞E[tK|tM ≥ tK] = limT→∞
1 λ− T e−λT 1− e−λT = 1 λ and from (16) lim
E[T ]→∞E[tK|tM ≥ tK] =E[T ]→∞lim
1 λ + (1/E[T ]) = 1 λ. Therefore, the maximum unused key period is E[tM] = 1/λ.
When E[T ] is small (e.g., less than 1/λ), the fixed T outper-forms the exponential T . When E[T ] is large, the exponential T yields better performance in terms of the unused key period.
Fig. 5(c) plots the reused key period E[t∗K|tM ≤ tK] as a
function of E[T ]. The figure indicates that the key reused period increases as E[T ] increases. From (7), we have
lim E[T ]→∞E [t ∗ K|tM ≤ tK] = lim T→∞ λT2 2(λT + e−λT − 1)− 1 λ =∞ and from (12) lim E[T ]→∞E [t ∗ K|tM ≤ tK] = lim E[T ]→∞ 1 1/E[T ] =∞. The figure also indicates that the exponential T outperforms the fixed T in terms of the reused key period.
Fig. 6(a) plots the unused key period E[tK|tM ≥ tK] against
E[T ] and VM. When E[T ]≥ 1/λ, the unused key period
increases as VM increases. This phenomenon is explained as
follows. As VM increases, more long and short tM are
ob-served. Since a random observer (an MS movement) tends to observe long tM, short tM will not contribute to E[tK|tM ≥
tK]. Therefore, more long tK are observed as VM increases.
From (14), we have the equation shown at the bottom of the
Fig. 6. Effect of VM. (a) Effect of VM on E[tK|tM ≥ tK]. (b) Effect of
VMon E[t∗K|tM ≤ tK].
page. When E[tK] < 1/λ, E[tK|tM ≥ tK] E[tK], which is
not sensitive to VM.
Fig. 6(b) plots the reused key period E[t∗K|tM ≤ tK] against
E[T ] and VM. For the exponential T , according to (12),
E[t∗K|tM ≤ tK] = E[T ]. This phenomenon is explained as
follows. Since the residual MSK lifetime tK is exponentially
distributed, the arrival of the MS to the old ASN-GW is a random observer to tK. Thus, from the residual life theorem,
t∗K is also exponentially distributed with the mean E[T ]. For the fixed T , E[t∗K|tM ≤ tK] increases as VM increases. Since
we only consider the case when tM ≤ tK, as VM increases,
lim VM→∞ E[tK|tM ≥ tK] = lim VM→∞ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 μ− 1 λ× 1 λμVM+1 1 λ2 VM+1− 1 μ× 1 λμVM+1 1 λ2 VM 1− 1 λμVM+1 1 λ2 VM ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = 1 μ
short tK periods are observed, and long tKwill not contribute
to E[t∗K|tM ≤ tK]. Thus, for the fixed T , the reused key
period increases as VM increases and eventually approaches
E[tK] = T /2.
Fig. 6 shows that the exponential T outperforms the fixed T in terms of the reused key period. On the other hand, for the unused key period, the fixed T outperforms the exponential T in most cases. Another advantage of the exponential T over the fixed T is that the reused key period E[t∗K|tM ≤ tK]
performance is not affected by the variance VM. This stability
property is important for a telecom-grade system. V. CONCLUSION
This paper has proposed a key caching mechanism to speed up the inter-ASN-GW handoff for mobile WiMAX. With this mechanism, when an MS leaves the old ASN-GW, the MS key record (e.g., the MSK) is cached in the old ASN-GW. If the MS returns to the old ASN-GW before the MSK lifetime expires, it can reuse the MSK without executing the IEEE 802.1X authentication. On the other hand, the old ASN-GW consumes extra storage to maintain the MS key records when the MS leaves the old ASN-GW. This paper has investigated how the period T of the MSK lifetime affects the key caching performance by an analytic model and simulation experiments. Three output measures are evaluated: the key reuse probability, the unused key period, and the reused key period. We have shown that the caching mechanism can effectively speed up the inter-ASN-GW handoff. We also observed that the exponential T outperforms the fixed T in most cases. Moreover, for the reused key period, the exponential T is not affected by the variance of the MS residence period in new ASN-GWs and is more suitable for telecommunication systems. As a final remark, the operator uses our study and the number of serving MSs to calculate the storage budget at an ASN-GW. Our study indicates that E[T ] > 10/λ will not improve performance. Therefore, if E[T ] < 10/λ is selected, the extra storage can be computed from Little’s law, i.e., N = xE[T ] < 10x/λ, where N is the extra storage (the number of MS key records), and x is the rate of the MSs leaving the ASN-GW [10].
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Shih-Feng Hsu received the B.S. degree in computer
science from the National Tsing Hua University, Hsinchu, Taiwan, in 2001. He is currently work-ing toward the Ph.D. degree in computer science and information engineering with the Department of Computer Science, National Chiao Tung University, Hsinchu.
His current research interests include personal communication services, mobile computing, and wireless security in wireless local area network, Worldwide Interoperability for Microwave Access, and Universal Mobile Telecommunication Systems.
Yi-Bing Lin (M’95–SM’95–F’03) received the B.S.
degree in electrical engineering from the National Cheng Kung University, Tainan, Taiwan, in 1983 and the Ph.D. degree in computer science from the University of Washington, Seattle, in 1990.
He is the Dean and the Chair Professor of the De-partment of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. He is also with the Institute of Information Science, Academia Sinica, Taipei, Taiwan. He is the author of the books Wire-less and Mobile Network Architecture (Wiley, 2001), Wireless and Mobile All-IP Networks (Wiley, 2005), and Charging for Mobile All-IP Telecommunications (Wiley, 2008).
Prof. Lin is a Fellow of the Association for Computing Machinery, the American Association for the Advancement of Science, and the Institution of Engineering and Technology. He is the recipient of numerous research awards, including the 2005 NSC Distinguished Researcher award and the 2006 Academic Award of the Ministry of Education. He is a Senior Tech-nical Editor of IEEE NETWORK. He serves on the editorial boards of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the IEEE TRANSACTIONS ONVEHICULARTECHNOLOGY. He has served as the General or Program Chair for many prestigious conferences, including the 2002 ACM MobiCom. He has been a Guest Editor for several first-class journals, including the IEEE TRANSACTIONS ONCOMPUTERS.
![Fig. 5. Effect of μ. (a) Effect of E[T] on α. (b) Effect of E[T ] on E[t K |t M ≥](https://thumb-ap.123doks.com/thumbv2/9libinfo/7659847.140313/5.891.83.413.141.841/fig-effect-μ-effect-e-t-α-effect.webp)
![Fig. 6. Effect of V M . (a) Effect of V M on E[t K |t M ≥ t K ]. (b) Effect of](https://thumb-ap.123doks.com/thumbv2/9libinfo/7659847.140313/6.891.480.807.95.730/fig-effect-v-m-effect-v-k-effect.webp)
