IMF-PR: An Improved Morton-Filter-Based Pseudonym-Revocation Scheme in VANETs

Vehicle ad hoc networks (VANETs) are special wireless networks which help vehicles to obtain continuous and stable communication. Pseudonym revocation, as a vital security mechanism, is able to protect legal vehicles in VANETs. However, existing pseudonym-revocation schemes suffer from the issues of low certificate revocation list (CRL) generation and update efficiency, along with high CRL storage and transmission costs. In order to solve the above issues, this paper proposes an improved Morton-filter-based pseudonym-revocation scheme for VANETs (IMF-PR). IMF-PR establishes a new distributed CRL management mechanism to maintain a low CRL distribution transmission delay. In addition, IMF-PR improves the Morton filter to optimize the CRL management mechanism so as to improve CRL generation and update efficiency and reduce the CRL storage overhead. Moreover, CRLs in IMF-PR store illegal vehicle information based on an improved Morton filter data structure to improve the compress ratio and the query efficiency. Performance analysis and simulation experiments showed that IMF-PR can effectively reduce storage by increasing the compression gain and reducing transmission delay. In addition, IMF-PR can also greatly improve the lookup and update throughput on CRLs.


Introduction
The intelligent transportation system (ITS) integrates a variety of advanced equipment and technologies and has gradually become an important part of the next generation of urban transportation, providing convenience for drivers and passengers [1]. Vehicular ad hoc networks (VANETs), as the key parts of ITSs, adopt dedicated short-range communication technology (DSRC) and enable rapid interconnection between vehicle-to-vehicle (V2V) and vehicle-to-roadside infrastructure (V2I) to ensure that drivers and passengers have access to continuous and reliable services and applications [2]. In order to preserve vehicle identity privacy, VANETs mandates that vehicles utilize pseudonyms instead of real identities to anonymize identities and regularly change pseudonyms to prevent tracking from adversaries [3]. However, when vehicles are attacked, an effective pseudonym-revocation scheme should be developed to quickly remove illegal vehicles from VANETs [4]. The following two mechanisms are typically adopted in pseudonym-revocation schemes.
(1) An automatic revocation mechanism [5][6][7]. If an illegal vehicle is identified, the authority only stops issuing new pseudonyms. When the old pseudonym expires, the illegal vehicle will be automatically revoked. An expired automatic revocation mechanism can help the authority to drop the maintenance cost effectively. However, since the legal vehicles are unable to receive the revocation notice timely, the illegal vehicle can continue to misbehave before the old pseudonym expires. (2) A certificate revocation list (CRL)-based pseudonym revocation mechanism [8]. By distributing CRLs containing illegal vehicles' identities, the authority is able to remove illegal vehicles from VANETs in time. According to [9], the CRL is typically distributed under the following three conditions.
In order to solve the above problems, the authors of [14] separated and distributed CRLs to vehicles in the form of data streams to reduce communication costs and CRL distribution delay. However, [14] only focused on CRL distribution efficiency and ignored the optimization of CRL's data structure. Taking advantage of the low computational cost and high verification efficiency of HMAC, [7,15] replaced the CRL's linear data structure with a hash table. By regularly updating the hash chain, the authority ensures that legal vehicles can obtain relevant information of illegal vehicles in a timely manner. However, the above two schemes are not able to solve high storage and transmission overhead issues caused by the CRL growth. Ref. [16] adopts a Merkle hash tree (MHT) to optimize CRL data structure. However, the MHT-based cannot support the deletion operation. Moreover, pseudonym validity was verified by obtaining proofs from Roadside Units (RSUs), which imposes huge computational and communication overheads on RSUs. Ref. [17,18] adopted the Bloom filter (BF) to optimize the CRL's data structure. Likely, the BF-based schemes cannot support the deletion operation as the MHT-based scheme. As a result, the authority must rebuild the CRL before distribution, which leads to low CRL update efficiency. Ref. [19] proposed a pseudonym-revocation scheme based on Cuckoo filter. The scheme supports CRL data-deletion operations with a minimal storage overhead and provides higher data query performance by optimizing the CRL data structure. However, this scheme ignores the maintenance cost of the historical CRLs by authority. As historical CRLs continue to multiply, the storage cost of the authority will increase linearly.
Unlike the Bloom filter (BF), which only supports lookups and insertions but not deletions in its simplest form, the Morton filter (MF) supports lookups, insertions, and deletions. In VANETs, it is necessary to delete expired pseudonyms in CRLs in time to avoid huge overheads caused by CRLs storage and transmission. In addition, MF adopts a compressed block format that permits storing a logically sparse filter compactly in memory. Compared to the stock Cuckoo filter (CF), MFs particularly excel for workloads that use large filters. Due to the large number of vehicles in VANETs, a large number of pseudonyms need to be revoked; therefore, compared with BF and CF, MF is more suitable for VANETs.
Moreover, to improve the update efficiency, an improved Morton-filter-based pseudonymrevocation scheme for VANETs (IMF-PR) is proposed in this paper. IMF-PR optimizes the Morton filter to improve the efficiency of CRL generation and query. Meanwhile, in order to reduce the CRL storage overhead and improve CRL distribution efficiency, IMF-PR improves the CRL data structure and implements the distributed CRL management mechanism. To be specific, the main contributions are described as follows.
(1) Based on the previous research [20], IMF-PR establishes a distributed CRL management mechanism, in which authority is responsible for maintaining CRLs, including long-term pseudonyms of illegal vehicles, and the base station maintains CRLs containing multiple illegal vehicles' temporary pseudonyms that are valid within the scope of base-station management, so as to reduce the computational overhead of authority on CRLs and the communication overhead of distributing CRLs. (2) IMF-PR improved the Morton filter to optimize the CRL management mechanism.
Temporary storage space is used to support the dynamic management of illegal vehicles' information, so as to keep CRL generation and update efficiency and reduce the CRL storage overhead. (3) CRLs in IMF-PR store illegal vehicles' information based on the improved Morton filter's data structure rather than sequential structure to improve the compression ratio of CRLs and lower the transmission cost of CRLs. Furthermore, the entities owning CRLs can determine whether the surrounding vehicles are legitimate by calculating the fingerprint, which improves the query efficiency. (4) The compression gain and transmission delay results demonstrate that, compared with C 2 RL [17] and C 3 RL [19], CRLs in IMF-PR can be more quickly transmitted to the whole VANETs than them. Moreover, the simulation showed that IMF-PR's query throughput and update throughput are superior to those of C 2 RL [17] and C 3 RL [19] in most cases.

VANETs
VANETs, as essential parts of ITSs, are able to support the connection between vehicles and infrastructures. In VANETs, a broad range of mobile communication technologies are integrated into roadside units (RSUs) and vehicles to relieve traffic congestion and improve driving safety. As shown in Figure 1, vehicles with onboard units (OBUs) adopt dedicated short range communication (DSRC) to communicate with RSUs or other OBUs and obtain required services [8]. RSUs provide safety-related services, efficiency-related services, entertainment-related services, and so forth, for surrounding vehicles. For example, to guarantee traffic efficiency and safety, RSUs provide road-congestion management and collision alerts for surrounding vehicles [21]. A road-congestion application can provide the best routes for vehicles, and communication occurs among the vehicles and from vehicles to RSUs. This application's goal is to decrease congestion and improve traffic efficiency. In order to prevent malicious vehicles from obtaining services, it is necessary to verify the validity of pseudonyms of vehicles applying for services. The collision alert's goal is to avoid and decrease the number of accidents. This is an application category sensitive to the delay. To reduce the delay, road conditions use vehicle-to-vehicle communication. Moreover, the pseudonym's validation must be as fast as possible to avoid the illegal user from broadcasting a bogus message. Therefore, in both examples, it is necessary to design a pseudonym-revocation scheme with a low distribution delay and an efficient query. The base station (BS) can communicate with the external network and provide network communication services for vehicles. In addition, a BS can assist authorities in providing CRL generation and distribution services for vehicles. In VANETs, two different types of communication are used to support applications and services: (1) vehicle-toinfrastructure (V2I) communication. V2I refers to the communication between vehicles and infrastructures (e.g., RSUs). When entering the signal coverage range of an RSU, vehicles are able to adopt DSRC technology to request an RSU to obtain the required services. An RSU has external network communication capabilities through interconnection with the surrounding BS and provides necessary services for surrounding vehicles. (2) Vehicle-tovehicle (V2V) communication. V2V refers to the communication between vehicles, which is completed by vehicles independently without the participation of RSUs. Depending on V2V communication, vehicles can obtain the driving status of surrounding vehicles in time to ensure driving safety and improve traffic flow.

Morton Filter
The Morton filter (MF) [22] is a data structure to support data storage and queries with low spatial complexity and high efficiency. The MF enhances the efficiency of data insertion, querying, and deletion by optimizing a cuckoo filter (CF) [23] and improves space complexity through sparse matrix compression. As shown in Figure 2, the MF takes a block as the basic data unit to store the compressed data. Each block consists of the following three parts. Overflow tracking array (OTA). OTA is composed of a bit vector, which tracks the fingerprint overflow status in the block by setting a bit. The fingerprint can be located by querying the value recorded in OTA.
There are three core algorithms in the MF.
(1) Lookup:Given a key K and a MF, we first compute the fingerprint F = H F (K) of K. Then, we compute x = H 1 (K) to determine the bucket index for its bucket and divide x by the buckets per block B to yield the block index b. Next, we calculate l = mod(x, B) to get the bucket index. We use F, b, l to check for the presence of K in the bucket. If not, we recompute x = H 2 (K) and then continue with the rest of the operation. (2) Insert: Given a key K and a MF, we first compute the fingerprint F = H F (K) of K. Then, we compute x = H 1 (K) to determine the bucket index for its bucket and divide x by the buckets per block B to yield the block index b. Next, we calculate l = mod(x, B) to get the bucket index. We use F, b, l to store K in the bucket. If successful, we insert F. Else, we recompute x = H 2 (K), and then continue with the rest of the operation. (3) Delete: Similarly to lookup, given a key K and a MF, we first compute the fingerprint F = H F (K) of K. Then, we compute x = H 1 (K) to determine the bucket index for its bucket and divide x by the buckets per block B to yield the block index b. Next, we calculate l = mod(x, B) to get the bucket index. We use F, b, l to check for the presence of K in the bucket. If successful, we delete F. Otherwise, we recompute x = H 2 (K) and then continue with the rest of the operation.

System Overview
In this section, the IMF-PR system architecture, the improved Morton filter (IMF) data structure, and the IMF-PR-based CRL (IMF-CRL) are elaborated.

IMF-PR System Architecture
As shown in Figure 3, the IMF-PR system architecture includes four types of entities: TA, BS, RSU, and vehicle. TA. According to the illegal vehicle's long-term pseudonym PS TA τ , TA is able to query the real identity of the vehicle and the related long-term pseudonym set [1,n] in CRL TA , and distribute CRL TA to the BS.
BS. When receiving CRL TA from TA, the BS verifies whether CRL TA 's signature is legal. If CRL TA is legal, the BS queries the temporary pseudonym set {PS BS i } i∈ [1,m] issued by the BS according to the long-term pseudonym {PS TA i } i∈ [1,n] . The BS integrates all longterm pseudonyms, temporary pseudonyms, and expiration dates of all illegal vehicles to generate and distribute I MF − CRL to all RSUs within the managed range.
RSU. When receiving I MF − CRL from the BS, RSU verifies the legitimacy of I MF − CRL's signature. If I MF − CRL is legal, RSU refuses to communicate with the illegal vehicles contained in I MF − CRL, then distributes the I MF − CRL to all legal vehicles within the communication range; Vehicle. After receiving I MF − CRL from RSU, the legal vehicle verifies I MF − CRL. Once the verification is successful, the vehicle refuses to communicate with the illegal vehicles recorded in I MF − CRL.

Improved Morton Filter
The improved Morton filter (IMF) data structure that stores the identity information of illegal vehicles in the BS is shown in Figure 4. Three partitions make up the storage space based on the IMF data structure: CRL generation/storage space, data-sharing space, and temporary storage space. If there are new pseudonyms with expiration dates, they are added to the eList and the pList in the temporary storage space. Then, add the corresponding fingerprints to the f pList in the data-sharing space. Finally, update the < FCA, FSA >, and < date, link CRL > in the CRL generation/storage space, which means that the entire IMF-CRL construction process is complete. (3) Temporary storage space is composed of two linked lists: eList and pList, which are responsible for maintaining the information of the illegal vehicles corresponding to the latest I MF − CRL issued. Each node eNode in eList is composed of three triples: < next pn , EXP, and next en >, where next pn stores the address of the pseudonym linked list pList, EXP is responsible for storing and maintaining the expiration date of illegal vehicles pseudonyms, and next en points to the next eNode in the eList. Each eNode is stored in eList in the order of EXP increment. pNode is composed of three triples:< next pn , PS, and link f p >, where next pn points to the next pNode with the same EXP, PS is the pseudonym of the illegal vehicle, link f p stores the address of fingerprint F stored in the f pList in the data-sharing space. Figure 5 shows the I MF − CRL data structure. I MF − CRL substitutes ID TA , identifying the TA's identity in CRL based on 1609 with ID BS and stores the fingerprints of the illegal vehicles based on IMF data structure instead of the sequence list. The notation and explanations for the I MF − CRL are shown in Table 1.

Notation Description version
The version of CRL, which is set to 1 by default.

CRL series
The CRL serial number, indicating whether the CRL is associated with a specific certificate.

ID BS
BS's identity.

issue date
The issuance time of I MF − CRL. next CRL Next issuance time of I MF − CRL with the same CRL series. priorityIn f o Assist legal vehicles with insufficient storage space to identify information that needs to be retained or discarded.

CRL serial
Counter, the value is incremented by 1 when a full I MF − CRL or incremental I MF − CRL is issued. entries The vehicles' fingerprints information. signature CRL's signature.

The Proposed Scheme
The flow diagram of the proposed scheme is shown in Figure 6. The proposed scheme includes IMF-CRL construction, IMF-CRL distribution, and IMF-CRL resolution. In addition, the IMF-CRL construction includes a data-update algorithm and IMF-CRL generation.

IMF-CRL Construction
In the CRL generation period, TA follows the WAVE-CRL construction algorithm to load the long-term pseudonyms and expiration date of illegal vehicles into CRL TA , and then sends CRL TA to all BSs in VANETs via the secure channel.
When receiving CRL TA , the BS first queries the temporary pseudonyms and expiration date issued by the BS in accordance with the entries contained in CRL TA . The BS updates eList and pList in the temporary storage space and f pList in the data-sharing space based on the above illegal-vehicle information. Then, given the storage list f pList and current date date, the BS updates CRL generation/storage space and generates I MF − CRL. Finally, the BS generates the I MF − CRL based on the updated f pList.

Data-Update Algorithm
The data-update algorithm consists of the REPF algorithm (remove the expired pseudonym and fingerprint algorithm) and the INPF algorithm (insert a new pseudonym and fingerprint algorithm).
Algorithm 1 is in charge of deleting the old pseudonyms, expiration dates, and fingerprints of illegal vehicles stored in IMF. As shown in Figure 7, the details of the REPF algorithm follow.
(1) Query the head node (eNode) in eList. If the EXP stored in eNode is expired, then determine the pList pointed to by next pn . For the link f p stored by each pNode in the pList, record the address of the fingerprint F in the pNode.   pList ← popFront(pList) 13: end while 14: eList ← popFront(eList) 15: else 16: break 17: end if 18: return eList, f pList, pList end function Algorithm 2 is in charge of storing new illegal vehicles' pseudonyms, expiration dates, and fingerprints in IMF. As shown in Figure 8, given a new pseudonym PS and expiration date EXP, the BS needs to execute the following operations.
(1) Determine the pList of the new pseudonym PS by traversing the eNode: (a) When there exist eNode with EXP in the eList, record the next pn of the last pNode in the pList pointed to by the next pn in eNode as lastnext pn ; (b) When the eList is empty, create a new eNode, add EXP to eNode, set next en and next pn to null, and eList points to the new eNode. Finally, record the next pn in the eNode as lastnext pn . (c) When there is no eNode with EXP in eList, create a new eNode, add EXP to the eNode, and set next pn to null. Add the eNode in incremental order (next en of the previous node of the eNode points to the node, and the next en of the eNode points to the next node) and record the next pn in the eNode as lastnext pn .

IMF-CRL Distribution
Based on the WAVE − CRL generation protocol, TA generates and transmits CRL TA to all BSs, which contains the long-term pseudonym set, corresponding to the expiration date set of illegal vehicles, and the signature. After receiving the CRL TA , the BS first uses the TA's public key to verify CRL TA . If CRL TA is regarded as legal, the BS queries all temporary pseudonym sets issued by the BS according to the pseudonyms and expiration date stored in CRL TA . Then, the BS generates the certificate revocation list I MF − CRL through the IMF-CRL construction algorithm and transmits the I MF − CRL to RSUs. Next, the RSU verifies I MF − CRL. If I MF − CRL is legal, the RSU stores I MF − CRL locally and sends I MF − CRL to surrounding legal vehicles. Finally, the legal vehicle that has received I MF − CRL verifies I MF − CRL. If the verification is successful, the vehicle updates the local I MF − CRL.

IMF-CRL Resolution
When legal vehicle v and the RSU receive the request from surrounding vehicles (e.g., v ), vehicle v and the RSU checks whether the fingerprint of vehicle v is recorded in I MF − CRL. Given I MF − CRL, and the pseudonym PS and expiration date EXP of v , the process of IMF-CRL resolution is as follows:

Performance Analysis
In this section, IMF-PR, C 2 RL [17], and C 3 RL [19] are compared and analyzed in terms of compression gain and transmission delay. Additionally, the simulation framework Veins was adopted to demonstrate the CRL query throughput and CRL-update throughput. C 2 RL [17] designed a lightweight online certificate status protocol (TinyOCSP) to save on energy. Then, the C 2 RL integrates CRL compression using Bloom filters with TinyOCSP to further reduce the certificate validation overhead. C 3 RL [19] proposed an improved pseudonym certificate revocation scheme, using a Cuckoo filter for compression. In order to optimize deletion efficiency, C 3 RL [19] designed the concept of the Certificate Expiration List (CEL), which can be implemented with a priority queue. The scheme greatly improves the lookup performance on CRLs and reduces the revocation operation costs through deletion.

Compression Gain
Compression gain ζ is defined as the ratio of the size of WAVE-CRL to the size of the CRL in IMF-PR, C 2 RL, and C 3 RL. With the same number of illegal vehicles, the larger the ζ, the lower the space complexity, and the shorter the CRL length.
According to [8], WAVE-CRL consists of three parts: unsigned CRL (L h = 24 bytes), entries comprising pseudonym and expiration date (14 bytes for each entry), and signature (L s = 64 bytes).
In the C 2 RL scheme based on the Bloom filter, given the array length m of the Bloom Filter, the length of C 2 RL is l h + (m/8) + L S bytes. Given the number of entries is n, the average length of each entry in the Bloom filter is C C 2 RL = m/(8 * n) bytes. According to [24], C C 2 RL is defined by the false-positive rate as C C 2 RL = m/n = −ln2log 2 bits. As a result, In the C 3 RL scheme based on the Cuckoo filter, assume there are m buckets in the Cuckoo filter, each bucket is able to hold up to b entries, each entry's fingerprint length is f , and the number of entries is n. The length of C 3 RL is l h + ( f mb/8) + L S bytes; each entry is on average C C 2 RL = f mb/(8 * n) bytes. According to [25], given a loading factor α, C C 3 RL is expressed by false-positive rate as C C 3 RL = m/n = (3 − log 2 )/α bits. Consequently, In IMF-PR, suppose that there are m buckets in the IMF, each bucket holds at most b entries, the fingerprint length of each entry is f , and the number of entries is n. The total length of IMF-PR is l h + ( f log 2 b /8) + n f + L S bytes; each entry is on average C I MF−PR = (mlog 2 (b + 1) + b f )/(8 * n) bytes. According to [22], C I MF−PR defines the false-positive rate as C I MF−PR = (log 2 5 − log 2 )/4α bits. Thus, The compression gains of IMF-PR, C 2 RL, and C 3 RL at the same load factor α = 0.5 are shown in Figure 9. When the number of pseudonyms revoked remains constant, the compression gain rises in tandem with the false-positive rate. As a result, in order to achieve a low false-positive rate, a larger storage capacity needs to be required. Simultaneously, as the false-positive rate increases, the compression gains of C 2 RL and IMF-PR rise quickly, and the growth rate gradually becomes flat with the false-positive rates of 0.1% to 0.2%. Meanwhile, the overall growth rate of C 3 RL is modest. With the same false-positive rate, IMF-PR guarantees the maximum compression gain. In addition, as the number of entries stored in the CRL grows, the compression gains of the three schemes starts to increase dramatically and level off when the number of entries reaches 400 to 800. Meanwhile, the compression gain of C 3 RL increases the slowest and the growth rate of IMF-PR is the highest. When storing the same amount of entries, IMF-PR can provide higher compression gains and shorter the length of the CRL. When the number of entries is 2000 and the false-positive rate is 0.01, the compression gains of C 2 RL and C 3 RL are 2.5% and 75.4% lower than those of IMF-PR, respectively. Figure 10 shows the compression gains of IMF-PR, C 2 RL, and C 3 RL at the same load factor: α = 0.8. Compared with Figure 9, the compression gain of the IMF-PR is significantly increased, and the high compression gain of CRL can be guaranteed while keeping the false-positive rate low. Since the storage space in C 2 RL has nothing to do with the load factor, the results in Figures 9 and 10 are consistent, whereas the compression gain of C 3 RL shows a rather smooth increase. In comparison to Figure 9, IMF-PR and C 3 R achieved higher compression gain with the same storage entry and load factor, while the compression gain of IMF-PR was significantly accelerated and the compression gain of C 3 RL was relatively slow. When the number of entries was 400 to 800, the compression gain of the three schemes flattened out. When the number of entries was 2000 and the false-positive rate was 0.01, the compression gains of C 2 RL and C 3 RL were 56.7% and 73.4% lower than that of IMF-PR, respectively.

Transmission Delay
The transmission delay (TD) is defined as the time from a CRL being distributed to when it is received [26]. In the CRL distribution protocol, the transmission is relatively stable and the delay is small due to the wired transmission among the authority, BSs, and RSUs. This section only discusses the transmission delay from RSU distributing CRL to the vehicle receiving CRL. The TD is defined as where |WAVE < CRL > | denotes the data packet size of CRL defined by the WAVE standard. The RSU that receives the CRL needs to layer the data via WSMP, LLC, MAC, and PHY. The RSU then transmits the WAVE-packets to surrounding vehicles via the physical layer. According to the 1069.2 standard [8], given the length of the CRL L CRL , the length of the WAVE-packet is L WSA =L CRL + 121 bytes. Figure 11 shows the transmission delays of IMF-PR, C 2 RL, and C 3 RL for various bandwidths and entries sizes when α = 0.5, = 0.1%. The transmission delays of the three schemes all show a decreasing trend as bandwidth increases. Since the CRL compression gain of the IMF-PR is the highest when storing the same items, it assures that the VANETs can complete the CRL distribution with the lowest communication cost. Therefore, IMF-PR can complete the distribution of the CRL with lower transmission delay than C 2 RL and C 3 RL. When the number of illegal vehicles included in the CRL is 0, the CRLs of the three schemes only contain 121 bytes, so the three schemes can finish the CRL distribution with extremely little transmission delay. As the entries of illegal vehicle stored in the CRL grow, the transmission delays of the three schemes showed an upward trend. C 3 RL had the most obvious upward trend due to having the lowest compression rate. By owning the highest compression rate, IMF-PR distributes the CRL with the least transmission delay and the slowest rise rate. When the number of entries is 2000 and the bandwidth is 10 Mbps, the transmission delay of IMF-PR is 10.3% and 74.9% lower than those of C 2 RL and C 3 RL, respectively.

Simulation
Based on the Veins framework, we conducted simulation experiments on IMF-PR, C 2 RL, and C 3 RL in terms of CRL query throughput, where the traffic map of Tianhe District in Guangzhou was embedded as the simulation scenario (2000 × 2000 m 2 ). The simulation tool we used is the open-source framework Vehicles in Network Simulation (Veins) [27]. Veins implements the IEEE 802.11p protocol in the physical and MAC layers and manages the data transmission between OMNET++ and SUMO through TraCI [28]. The data-transmission rate and transmission power are defined as the default values of 6 Mbps and 20 mW, respectively. Then, we define the simulation time and number of cars as 500 s and 20-200, respectively [3,29]. We adopted the open-source MF-library at Github and the main configuration Morton3_8 from the Morton filter proposal paper [22], which means a 3-slot bucket with 8-bit fingerprints. The parameters of the simulation are shown in Table 2.  [30]. The higher the query throughput and the faster the query speed, the quicker the RSU, and the vehicle can determine whether the surrounding vehicle is legal or not. Figure 12 demonstrates the CRL query throughput of IMF-PR, C 2 RL, and C 3 RL under different load factors, α. Both the number of vehicles in the simulation and the running time of the simulation affect the load factor. As the number of vehicles in the simulation and the simulation time increased, the load factor also increased gradually. As the load factor increased, the throughputs of C 2 RL and C 3 RL remained relatively stable. However, IMF-PR showed a considerable downward trend, while still maintaining a high query throughput. Furthermore, when the load factor is low, IMF-PR can complete query operations faster than C 2 RL and C 3 RL. As can be seen in the figure, when the load factor ranges from zero to one, the query-throughput advantage of IMF-PR over C 3 RL gradually decreases from 62.5% to 25%, and compared with C 2 RL, the query-throughput advantage of IMF-PR decreases from 84.4% to 68.8%. Moreover, when the load factor is increased from 0 to 0.5, the load factor of IMF-PR decreases by 10.5%, and when the load factor is increased from 0.5 to 1, the load factor of IMF-PR decreases by 42.8%. As a result, in order to achieve a high query throughput, more storage capacity is required to achieve a lower load factor.

Update Throughput
Update throughput is defined as the number of CRL update operations performed by an authority per unit of time [30]. The higher the update throughput, the faster the authority can update the CRL. The updating of the CRL includes deleting illegal vehicles with expired pseudonyms or falsely reported vehicle information in the old CRL and inserting new illegal-vehicle information. C 2 RL does not support deletion operations, instead verifying the update throughput by insert operations. Figure 13 shows the CRL-update throughputs for IMF-PR, C 2 RL, and C 3 RL with a load factor of 0.5. Both the number of vehicles in the simulation and the running time of the simulation affect the number of entries that are inserted and deleted. As the number of vehicles in the simulation and the simulation time increase, the number of entries inserted and deleted also increases gradually. However, the percentage of entries inserted does not increase with the number of vehicles and simulation time. It can be seen in the figure that the throughput of C 2 RL is not affected by the percentage of entries inserted. As the proportion of inserted entries continues to increase, the proportion of deleted entries decreases and the throughput of C 3 RL continues to increase, which means when the pseudonyms used by vehicles in VANETs have a long validity period-that is, C 3 RL does not need to delete expired pseudonyms frequently-so the throughput of C 3 RL is more advantageous. However, in order to protect the privacy of the vehicle, the validity periods of the pseudonyms in VANETs cannot be too long. In IMF-PR, since eList in the temporary storage space needs to be updated during inserting entries, the throughput continues to drop. This is because in IMF-PR, entry insertions can only be done one at a time and entry deletions can be done in batches, so throughput decreases as the proportion of entries inserted increases; in contrast, throughput increases as the percentage of entries deleted increases. As can be seen in Figure 13, IMF-PR's throughput is lower than C 3 RL's when the percentage of entries inserted is higher than 70%. When the insertion percentage is zero, the update throughput of IMF-PR is 37.9% and 51.7% higher than those of C 3 RL and C 2 RL, respectively. When the insertion percentage is one, the update throughput of C 3 RL is 20% higher than that of IMF-PR, and the update throughput of IMF-PR is 30% higher than that of C 2 RL.

Conclusions
In this paper, an improved Morton-filter-based pseudonym-revocation scheme for VANETs (IMF-PR) was proposed to address the issues of high CRL maintenance, distribution, and query costs. By improving the Morton filter, IMF-PR enhances the efficiency of generating and updating CRL, lowers the CRL's storage cost, and enables legal vehicles to quickly query and verify the surrounding vehicles. Moreover, IMF-PR supports the decentralized CRL management mechanism, which not only reduces the CRL maintenance cost of TA, but also improves CRL distribution efficiency. The performance analysis and simulation results showed that the compression ratio of IMF-PR is better than those of C 2 RL and C 3 RL, and the transmission delay of IMF-PR is 10.3% and 74.9% lower than those of C 2 RL and C 3 RL, respectively. In addition, in the worst case, the query throughput of IMF-PR is 25% and 68.8% higher than those of C 3 RL and C 2 RL, respectively. Moreover, IMF-PR's update throughput is better than those of the other two schemes when the percentage of entries inserted is less than 70%. In the future, we will explore a novel anonymous authentication scheme to support secure V2I and V2V communication.