—For avoding the attacks from quantum computing (QC), this study applies the post-quantum cryptography (PQC) methods without hidden subgroups to the security of vehicu- lar communications. Due the mainstream technologies of PQC methods (i.e. lattice-based cryptography methods), the standard digital signature methods including Dilithium and Falcon have been discussed and compared. This study uses a queueing model to analyze the performance of these standard digital signature methods for selection decision-making. Index Terms —Post-quantum cryptography, vehicular commu- nications, queueing model, performance analysis I. I NTRODUCTION In the recent years, quantum computing (QC) has been more and more popular, and the quantum Fourier transform has been proposed to solve hidden subgroup problems by polynomial time computability. Furthermore, some cryptogra- phy methods (e.g. RSA (Rivest-Shamir-Adleman) and Elliptic Curve Cryptography (ECC)) including hidden subgroups may be cracked by QC. Therefore, the post-quantum cryptography (PQC) methods (e.g., lattice-based, multivariate-based, and code-based cryptography methods [1]) without hidden sub- groups are important for defensing the attacks from QC. For the security of vehicular communications, Security Credential Management System (SCMS) [2] and Coopera- tive Intelligent Transportation System Credential Management System (CCMS) have been designed based on Public Key Infrastructure (PKI). However, ECC is the used cryptography method in these systems, and security threats may be occurred by QC. Therefore, the PQC methods could be considered instead of ECC for improving the security levels. In this study, the PQC methods that are applied to SCMS and CCMS will be surveyed and discussed. Furthermore, the lattice-based cryptography methods are the mainstream technologies of PQC methods [1]. Therefore, the performance of standard lattice-based cryptography methods (i.e. Dilithium and Falcon) will be compared. The main contributions of this study are listed as follows. •This study implements standard lattice-based cryptog- raphy methods and measures the signature length and computation time of each method. •This study uses the M/M/1 queueing model to analyze the average response time and the number of waiting packets for each standard lattice-based cryptography.II. M EASUREMENT This study implemented the methods of Dilithium [3] and Falcon [4] by C, Java, or Python languages for the digital sig- nature of packets in SCMS and CCMS. The used machine in- cludes Intel® Core™ i7-10510U and 8G RAM. The measured information including the signature length and computation time of each method is listed in Table I. The results show that Falcon has shorter signature lengths and Dilithium has shorter computation time. The detailed discussions are presented in the following subsections. TABLE I THESIGNATURE LENGTH AND COMPUTATION TIME OF EACH METHOD Method LanguageSecurity LevelSignature Length (bytes)Signature Time (ms)Verifiaction Time (ms) Dilithium C Level 2 2,420 1.183 0.274 Level 3 3,293 1.888 0.438 Level 5 4,595 2.299 0.720 Java Level 2 2,420 0.363 0.064 Level 3 3,293 0.554 0.091 Level 5 4,595 0.636 0.122 Falcon C Level 1 666 6.896 0.190 Level 5 1,280 13.804 0.358 Python Level 1 666 48.718 12.965 Level 5 1,280 66.907 18.466 A. Signature Length and Transmission Time According to the transmission rate of each communication protocol (e.g., IEEE 802.11p, IEEE 802.11bd, LTE-V2X, 5G NR-V2X, Zigbee, and LoRa [5]), the transmission time of signature by each PQC method could be estimated based on the signature lengths in Table I. For instance, the signature length of Dilithium based on Security Level 2 is 2,420 bytes, and the transmission time of the signature is 0.8 ms by the transmission rate of 23.08 Mbps through IEEE 802.11p (shown in Fig. 1). The results show that LTE-V2X and 5G NR- V2X have higher transmission rates for reducing transmission time. Furthermore, Falcon has shorter signature lengths, and the signature by Falcon could be transmitted faster. B. Computation Time and Service Rate For performance analysis, the service rates are considered in a M/M/1 queueing model. The service rates of signature and verification could be estimated based on the computation Fig. 1. The signature transmission time based on different communication techniques. time of signature and verification in Table I. For instance, the computation time lengths of signature and verification by Dilithium based on Security Level 2 in C Language are 1.183 ms and 0.274 ms. Therefore, the service rates of signature and verification by Dilithium based on Security Level 2 in C Language could be estimated as 845.654 and 3,655.784 per second (shown in Table II). The results show that Dilithium in Java Language has short computation time, and the higher service rates could be obtained by Dilithium in Java Language. TABLE II THESERVICE RATES OF SIGNATURE AND VERIFICATION Method LanguageSecurity LevelService Rate for SignatureService Rate for Verification Dilithium C Level 2 845.654 3,655.784 Level 3 529.637 2,280.823 Level 5 434.910 1,388.284 Java Level 2 2,754.821 15,673.981 Level 3 1,805.054 10,940.919 Level 5 1,572.822 8,183.306 Falcon C Level 1 145.004 5,265.375 Level 5 72.443 2,796.264 Python Level 1 20.526 77.129 Level 5 14.946 54.153 III. N UMERICAL ANALYSIS AND DISCUSSIONS A M/M/1 queueing model is adopted according to the service rates in Subsection II.B and the arrival rates of packets from 1 to 14 for numerical analysis. Therefore, the average response time and the number waiting packets of signature by each PQC method could be estimated in Fig. 2 and 3. Furthermore, the average response time and the number waiting packets of verification by each PQC method could be estimated in Fig. 4 and 5. The results show that Dilithium in Java Language has higher performance and may be more suitable for vehicular communications with high-frequency signed transmission. IV. C ONCLUSIONS AND FUTURE WORK This study analyzed the performance of standard lattice- based cryptography methods (i.e. Dilithium and Falcon) for digital signature in vehicular communications. In numerical results, Dilithium in Java Language could obtain less waiting time and lower waiting packet number. Moreover, Falcon could be used for the case with shorter signature lengths. In the future, the more standard PQC methods in the fourth round by National Institute of Standards and Technology (NIST) [1] could be analyzed. Fig. 2. The average response time of each PQC method for signature. Fig. 3. The waiting packet number of each PQC method for signature.