###### Abstract
In addition, using smart contracts in blockchain technology can automate identity authentication processes, thereby reducing the complexity and cost of manual verification and improving collaboration efficiency. However, existing blockchain-based cross-domain identity authentication schemes in IIoT scenarios lack a sufficient lightweight design. Industrial environments often have stringent latency constraints, with some time-critical applications requiring end-to-end delays in 10 to 100 milliseconds. Current blockchain-based cross-domain identity authentication schemes are mainly categorized into three types: authentication based on blockchain and Public Key Infrastructure (PKI) [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17], authentication based on blockchain and Identity-Based Authentication (IBA) [18; 19; 20; 21; 22; 23; 24; 25; 26; 27], and authentication based on blockchain and Certific-cateless Public Key Cryptography (CL-PKC) [28; 29; 30; 31; 32; 33; 34]. The hierarchical structure of most blockchain-based PKI schemes still poses a challenge. It increases interaction overhead during authentication and indirectly raises latency [7; 12; 14; 15; 16; 17]. Moreover, most existing blockchain-based PKI schemes [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17] rely on computationally intensive operations such as RSA exponentiation, bilinear pairings, and elliptic curve point multiplications. In addition, most blockchain-based IBA [18; 19; 20; 21; 22; 23; 24; 25; 26; 27] and blockchain-based CL-PKC schemes [28; 29; 30; 31; 32; 33; 34] have the inherent privacy risk of using a user's identity as their public key. While their design eliminates the need for digital certificates, it also exposes identity-related information. Once written to the immutable blockchain ledger, such data becomes vulnerable to linkage attacks and long-term privacy compromises. To address these risks, researchers have proposed various privacy-preserving techniques. These include cryptographic obfuscation [19] and zero-knowledge proofs [20; 21; 32; 34]. However, implementing these mechanisms often introduces significant computational overhead and increased protocol complexity. This is particularly problematic in resource-constrained environments such as IIoT. As a result,while blockchain helps alleviate the trust and key escrow issues in traditional IBA by removing the need for a centralized KGC, it simultaneously necessitates more sophisticated and costly mechanisms to preserve user privacy. To address this issue, this paper presents an efficient and lightweight cross-domain authentication scheme that integrates blockchain technology with Chebyshev chaotic mapping. By leveraging blockchain technology to achieve the trusted sharing of device identity information, the scheme significantly reduces the number of interactions required during cross-domain authentication. It simplifies the identity verification process for industrial devices. Furthermore, it employs the semigroup property of Chebyshev polynomials to construct session keys, effectively reducing the authentication process's computational complexity and enhancing the system's overall efficiency and security. Our scheme also assigns IoT devices pseudonyms that are periodically generated using Chebyshev chaotic maps, effectively mitigating traceability risks during cross-domain operations and preserving device privacy. To summarize, our contributions are the following:  
* To address the complexity and single point of failure associated with third-party-dependent cross-domain identity authentication in traditional IIoT systems, we propose a blockchain-based cross-domain authentication scheme tailored for the IIoT. By integrating edge computing, the scheme effectively reduces the number of interactions and latency during device authentication.