Recently, the deployment of sensors and actuators to collect and disseminate data in various applications such as e-healthcare, vehicular adhoc networks (VANETs) and smart factories has revolutionized several new communication technologies. The Internet of Things (IoT) is one of those emerging communication technologies. These revolutionary applications of IoT in industrial environment are termed as Industry 4.0 and it has vitalized the concept of Industrial IoT (IIoT). Being wireless communication, the authentication and integrity of data are the most important challenges. To mitigate these challenges, several digital signature schemes are proposed in the literature. However, due to identity-based or certificate-less construction, those schemes suffer from inborn key escrow and secret key distribution problems. To resolve such issues, the first certificate-based proxy signature (PFCBPS) scheme without pairing is proposed. The proposed PFCBPS scheme is provably secure in random oracle model (ROM). The performance comparison (in terms of computational costs of different phases and length of resulting delegation and signature) shows that the proposed PFCBPS scheme’s total computational cost is 46.69 ms which is 52.24% of He et al. [8], 61.40% of Debiao et al. [5], 23.33% of Seo et al. [20], 28% of Hu et al. [9] and 36.84% of Verma and Singh [23]. Thus, it is more suitable to IIoT environment than existing competitive schemes.

最近，在各种应用中部署传感器和执行器以收集和分发数据，例如电子医疗，车载自组网（VANET）和智能工厂等，已经彻底改变了几种新的通信技术。物联网（IoT）是新兴的通信技术之一。物联网在工业环境中的这些革命性应用被称为“工业4.0”，它激发了工业物联网（IIoT）的概念。作为无线通信，数据的身份验证和完整性是最重要的挑战。为了减轻这些挑战，文献中提出了几种数字签名方案。但是，由于基于身份或无证书的构造，这些方案存在先天密钥托管和秘密密钥分发问题。为了解决这些问题，提出了不配对的第一个基于证书的代理签名（PFCBPS）方案。所提出的PFCBPS方案在随机预言机模型（ROM）中可证明是安全的。性能比较（就不同阶段的计算成本以及所产生的委派和签名的长度而言）表明，所提出的PFCBPS方案的总计算成本为46.69 ms，占He等人的52.24％。[8]，Debiao等人的61.40％。[5]，Seo等人的23.33％。[20]，胡等人的28％。[9]和Verma和Singh [23]的36.84％。因此，与现有竞争方案相比，它更适合于IIoT环境。性能比较（就不同阶段的计算成本以及所产生的委派和签名的长度而言）表明，所提出的PFCBPS方案的总计算成本为46.69 ms，占He等人的52.24％。[8]，Debiao等人的61.40％。[5]，Seo等人的23.33％。[20]，胡等人的28％。[9]和Verma和Singh [23]的36.84％。因此，与现有竞争方案相比，它更适合于IIoT环境。性能比较（就不同阶段的计算成本以及所产生的委派和签名的长度而言）表明，所提出的PFCBPS方案的总计算成本为46.69 ms，占He等人的52.24％。[8]，Debiao等人的61.40％。[5]，Seo等人的23.33％。[20]，胡等人的28％。[9]和Verma和Singh [23]的36.84％。因此，与现有竞争方案相比，它更适合于IIoT环境。