The smart robot is playing an increasingly important role in the social economy, and multi-robot systems will be an important development in robotics. With smart sensing systems, the communications between sensors, actuators, and edge computing systems and robots are prone to be attacked due to the highly dynamic and distributed environment. Since smart robots are often distributed in open environments, as well as due to their limited hardware resources and security protection capabilities, the security requirements of their keys cannot be met with traditional key distribution algorithms. In this paper, we propose a new mechanism of key establishment based on high-order polynomials to ensure the safe key generation and key distribution. Experiments show that the key establishment mechanism proposed in this paper guarantees the security of keys; its storage cost and communication cost are smaller than state-of-the-art mechanisms; and it allows robot components to join and leave the network dynamically, which is more suitable for multi-robot systems. life, and the information security vulnerabilities may lead to leaks, which may cause harm to people in serious cases. Therefore, the safe operation of multi-robot systems requires that the communication among sensors, actuators, and edge computing systems in the system be safe and reliable, and the security of multi-robot networks receives more and more attention from researchers.Cryptographic algorithms are the important means to solve the network security problems. The most critical factor is the security of keys. In multi-node networks, the research on key security has gained significant development. Zhang [6] provided a practical design guide for a secure key generation system that can be changed by a different design. Thai [7] proposed a group key generation mechanism in the physical layer to deliver information through multiple antennas in a wireless network topology. Eschenauer [8] proposed a key pre-distribution algorithm, which performs key pre-distribution by establishing a key pool, but this key pre-distribution algorithm cannot guarantee 100% interworking between nodes. Sencun [9] proposed a localized encryption and authentication protocol (LEAP) to solve the problem of different security requirements when exchanging different types of data between nodes. However, after the initial deployment of the node, the interworking issues when the node and key changes cannot be solved. Jiyong [10] proposed a time-based key management protocol, but the protocol has connectivity issues for node deployment between different time slots. Naor [11] proposed a description schemes for distributing between n servers the evaluation of a function f, which can be used to distribute the operation of a KDC. However, this schemes cannot identify the attacked node in time, and it cannot update the key in time when a component being attacked appears, to avoid the information leakage.Currently, there is much research on key pre-distribution schemes in wireless sensor networks, and so...