## 1 Introduction  
Alliance chain is a type of blockchain that provides decentralization and node manageability, making it widely used in industries including finance, supply chain management, and healthcare [1]. However, as the number of nodes increases, communication congestion and throughput drop become significant scalability obstacles for alliance chain [2]. This situation arises because all transactions within the blockchain structure require nodes to utilize their computational and storage resources [3, 4], resulting in substantial wastage of time and space [5]. The manifestation is the bottleneck of low throughput and high latency caused by the blockchain's difficulty bearing the cost of consensus process and ledger storage [6]. Researchers have suggested various strategies to tackle performance issues and fulfill the practical requirement of enhancing blockchain resources to manage the rising volume of transactions [7]. Existing solutions to achieve blockchain scaling are classified into two categories: off-chain and on-chain [8]. Off-chain solutions adopt a layered system to simplify the blockchain architecture by aggregating the transfer of resources generated by fine-grained payments managed separately in multiple asynchronous subsystems. Only the net result is stored in the blockchain, thus avoiding the high computational cost of traditional blockchain systems [9]. In practical applications, the off-chain payment network requires frequent opening and closing channels to achieve consensus on all completed transactions on the blockchain [10; 11]. This process appreciably impacts the throughput of the blockchain, thus limiting the advantages provided by off-chain solutions [12; 13]. On-chain solutions enhance the functionality and data processing capabilities of the blockchain by improving its protocols and mechanisms [14]. Increasing the block size is typical of on-chain solutions, as it allows for higher transfer limits and reduces costs associated with transfers compared to traditional methods [8]. This approach can have negative implications for block propagation efficiency in terms of time and increase the risk of blockchain forks, leading to a higher probability of orphan blocks and increased maintenance costs [15]. The inefficiency of consensus protocols is the primary cause of blockchain scalability issues [16]. The research community has made significant efforts to address this problem by exploring various innovative consensus methods, yielding remarkable results [17; 18; 19]. There is no need to coordinate or manage various subsystems, and the recording, verifying, and retaining all transactions are characteristics of on-chain solutions. These features make on-chain solutions a mainstream choice for scaling blockchain in environments with varying network bandwidth and computing resources [20]. Sharding protocol is commonly used in distributed databases and cloud infrastructures, which can divide an enormous database into small data fragments and store these fragments on different servers for fast and efficient data management [21]. Elastico [22] pioneered the combination of the sharding protocol with the blockchain in 2016, avoiding the mandatory duplication of communication and computing overhead for each participating node. The sharding protocol has been comprehensively researched and verified in academic papers [23; 24; 25; 26; 27; 28; 29], confirming its effectiveness in enhancing throughput, reducing costs, and preserving decentralization. It has emerged as a prevailing solution for on-chain scaling [30]. During the initial stage of the sharding protocol's development in blockchain, it effectively addressed the resource-intensive challenges public blockchain face [31]. As alliance chain encountered scalability challenges similar to public blockchain in the use process [32], researchers applied sharding protocol to alliance chain, offering valuable assistance in achieving efficient transaction processing [33].