# Improving Throughput of Mobile Sensors via Certificateless Signature Supporting Batch Verification
## 1 Introduction
### Related Work  
The CLPKC was first proposed by [PERSON] and [PERSON] [6] to deal with the key escrow problem in the identity-based encryption (IBE) system [2]. They provided the construction of three schemes including encryption, signature, and key agreement. In  
Figure 1: Framework of certificateless signature with batch verification.  
addition, two basic adversary models in the certificateless cryptosystem were identified, namely, Type I adversary and Type II adversary. Due to the excellent properties of no-certificate, many researchers were attracted and many follow-up work was proposed. [PERSON] and [PERSON] summarized a general secure construction method of the certificateless signature (CLS) scheme [7] and certificateless encryption (CLE) scheme [8]. However, later these constructions were proved to be unsafe by [PERSON] et al. [9] and [PERSON] et al. [10]. In 2005, [PERSON] et al. [11] proved that there was a security risk in the original Al-Riyami and Paterson scheme [6]. [PERSON] et al. [12] re-examined the security model of CLPKC and proposed the concept of a new adversary model called the malicious key generation center (KGC). [PERSON] et al. [13] further subdivided each type opponent into three levels based on their attack capabilities and provided a super secure certificateless signature scheme. Among the known models, security against the super-type adversary achieves the most secure level. Nevertheless, the signature length was slightly long and contained three group elements. In recent years, many new shorter certificateless signature schemes and certificateless aggregate signatures [14; 15] were proposed. There are also some schemes that have been proven to be insecure. For example, [PERSON] [16] analyzed five recent articles and found that they can all be forged by adversaries. Therefore, how to construct secure certificateless signatures still requires a very rigorous approach. For a comparative survey of certificateless signature, ref. [17] is a good reference for the related studies until 2022. Two other related but earlier surveys can be found in [18; 19].  
In addition to solving the key escrow problem, compared to IBCs, another major advantage of certificateless cryptosystem is that they can be implemented without pairing. [PERSON] et al. [20] explored the first certificateless encryption scheme without pairing using the Schnorr signature. However, [PERSON] et al. [21] showed that the scheme in [20] did not consider public key attacks. They fixed the problem using a new scheme with a more stringent security model. The certificateless signature scheme without pairing was finished by [PERSON] et al. [22] in 2010. For the IoT scenario, [PERSON] et al. [23] and [PERSON] et al. [24] designed a certificateless aggregation signature without pairing and [PERSON] et al. [25] proposed a certificateless aggregation signcryption without pairing. Moreover, many certificateless schemes based on other PKI signatures have been studied. Using the RSA signature, [PERSON] et al. [26] also constructed a CLS scheme. Another study point is constructing CLS schemes based on already-published cryptographic standards. In 2022, [PERSON] et al. [27] proposed a CLS scheme (in Chinese) based on the Chinese national cryptographic standard. The scheme is built upon the identity-based standard, namely, the SM9 (SM stands for the Chinese pinyi \"shangmi\", which means a commercial cryptography application) algorithm. As a result, it must rely on the pairing operation. Recently, [PERSON] et al. [28] proposed a new CLS scheme using the SM2 algorithm without pairing. But their scheme requires zero-knowledge proof to verify the user public keys and how to support batch verification remains unknown. For batch verification, the certificateless aggregate signature (CLAS) [15; 29] technique can be considered.