## 2 Related Works  
[PERSON] and [PERSON] were the first to present a key management scheme based on random probability (the so-called basic random key distribution scheme, or E-G scheme [17]) for WSNs, which is the foundation of the other key management schemes. The scheme randomly deposits partial keys on the basis of pre-setting all pairwise keys, so it can greatly decrease the node resources cost on the premise of maintaining a certain connectivity in the network. The basic random key pre-distribution scheme concept can be roughly summed up as the following process:  
1. Key pre-distribution. The server (usually in the base station) creates a big key pool \(M\) and each key has a unique _ID_ identifier. Nodes pre-store \(K\) keys randomly selected from the pool and build the key ring. This ensures that two nodes can share at least one of the keys at a certain probability. 2. Shared-key discovery. Each node gets the shared-key by matching the key ring in its own storage with identifier broadcasting. 3. Path key establishment. If the two nodes do not have the shared-key directly, they can develop the path key through the intermediate nodes. The disadvantages of this scheme are obvious, such as the utilization of keys in the key ring is lower, the same key will be established by different nodes, and it will reduce robustness of the system. [PERSON] _et al._[18] proposed an improved random key distribution scheme, called \(q\)-composite scheme, which increases exponentially the difficulty for an adversary to destroy the safety link, but it reduces the network connectivity. [PERSON] _et al._[19] developed a new robust key predistribution scheme by using keys assigned based on the notion of eigenvalues and eigenvectors of a square matrix of a keys pool. [PERSON] _et al._[20] proposed a key predistribution scheme combining the Chinese Remainder Theorem (CRT) with a LU matrix. The scheme achieves smaller storage overhead and better network resilience. Such schemes can be classified as key pre-allocation schemes based on key pools. They use the same key to establish a session key between the nodes, and it can increase the connectivity when the nodes store a certain number of keys, but the network security is usually not better. [PERSON] _et al._ proposed a modified scheme in [21] based on the [PERSON] _et al._ scheme [22], namely a key predistribution scheme based on polynomial pool, and two possible instantiation schemes were presented. In this scheme, the server randomly generates \(S\) bivariate \(t\)-degree polynomials: \(\{f_{i}(x,y)\}\), \(i=1,2,\ldots\) S:  
\[f(x,y)=\sum_{i,j=0}^{t}a_{ij}x^{i}y^{j}=a_{t0}x^{t}+a_{(t-1)1}x^{(t-1)}y+ \cdots+a_{1(t-1)}xy^{(t-1)}+a_{0t}y^{t} \tag{1}\]  
The polynomials have the property \(f(x,y)=f(y,x)\) and each \(a_{ij}\) is different and completely confidential for each node. Prior to the deployment, each node randomly selects \(m\) polynomials, where, \(1\leq m\leq S\), and shares polynomials at a certain probability. After nodes are deployed, if two nodes find there exist shared polynomials, they can calculate the direct session key by exchanging the binomial \(ID\) identifiers and putting the \(ID\) into the binomial, otherwise the two nodes establish a session key with path key agreement. In this scenario, the \(t\)-degree polynomial has a safety threshold \(t\) (\(t\)-degree property), and the key information in other nodes will not be influenced by the captured nodes as long as the number of captured nodes is less than \(t\). The scheme needs to calculate the value of the polynomial during the key establishment, so the computation cost will be increased. However, it will be able to get in return security for the whole network as long as the computational overhead requirement of sensor nodes is satisfied. With increasing network scale, the number of nodes probably captured by an adversary will increase, which makes the polynomial lose its \(t\)-degree property easily.