# Cross-Platform UAV Swarm Key Management in Denied Environments
## 3 Cross-Platform Key Management of UAV Swarms in Denied Environments
### Private Key Establishment in the Regrouping and Arrival Stage
The procedures are shown in Algorithms 2 and 3. #### 3.3.2 Mutual-healing Update

If a UAV in the swarm fails to receive the broadcast message because of strong electromagnetic interferences in the denied environment at the \(t\) task stage, it can request a neighboring UAV (\(UAV\)-\(Neighbor\)) to help recover the swarm key. This process is called mutual-healing, as shown in Figure 7. The algorithm process is shown in Algorithm 4. The \((\varepsilon_{m^{\prime}t},\gamma_{m^{\prime}t})\) randomly selected by the swarm head against forgery attacks is combined with the local hash chain and CRT to complete the mutual-healing update. Figure 7 shows the mutual-healing update process, with the \(m^{\prime}\) swarm as an example:

Figure 5: Network of UAVs in collaboration. Figure 6: Key management in denied environment. ## 4 Scheme Analysis

For the sake of convenient analysis and proof, the symbols for proving are defined, as shown in Table 3.

Figure 7: An example of the mutual-healing update process. \[ \begin{array}{ll}(3)&M_{m^{\prime}t}\
otin\left\{M_{m^{\prime}1},M_{m^{\prime}2}, \cdots,M_{m^{\prime}(t-1)}\right\}\Rightarrow\\ &\left\{\gamma_{m^{\prime}1},\gamma_{m^{\prime}2},\cdots,\gamma_{m^{\prime}(t- 1)}\right\}\Rightarrow\left\{G_{m^{\prime}1_{\cdot}\_key},G_{m^{\prime}2_{ \cdot}\_key},\ldots,G_{m^{\prime}(t-1)_{\cdot}\_key}\right\}\text{.}\ Q.E.D., \text{ the ACP-KMS scheme has forward security, the }\text{UAV}_{j_{i}}\) cannot obtain the key \[\left\{G_{m^{\prime}1_{\cdot}\_key^{\prime}},G_{m^{\prime}2_{\cdot}\_key^{ \prime}},\ldots,G_{m^{\prime}(t-1)_{\cdot}\_key}\right\}\text{ before it joins the swarm.}\end{array} \]

**Lemma 4**. : _The ACPKMS scheme has backward security (BWS). If \(UAV_{j_{i}}\) exits UAVGN at the session stage \(t\), i.e., \(UAV_{j_{i}}\in U_{m^{\prime}1_{\cdot}\_key^{\prime}2_{\cdot}\cdots_{m^{\prime} (t-1)}}\), \(UAV_{j_{i}}\
otin U_{m^{\prime}1_{\cdot}+m^{\prime}(t+1)_{\cdot}\_m^{\prime}(t +2)_{\cdot}\cdots_{m^{\prime}(t+n)}}\). The specific proving process is as follows:_

1. \(UAV_{j_{i}}\
otin U_{m^{\prime}1_{\cdot}+m^{\prime}(t+1)_{\cdot}\_m^{\prime}(t +2)_{\cdot}\cdots_{m^{\prime}(t+n)}}\ UAV_{j_{i}}\in U_{m^{\prime}1_{\cdot}\_m^{ \prime}2_{\cdot}\cdots_{m^{\prime}(t-1)}}\Rightarrow\\ UAV_{j_{i}}\succeq\left\{ID_{j},K_{0},h(\cdot),S_{l}^{NK}\right\}\)_;_
2. _If_ \(UAV_{j_{i}}\vartriangleleft M_{m^{\prime}1},M_{m^{\prime}2},\cdots M_{m^{ \prime}(t-1)}\text{ and }UAV_{j_{i}}\right|\equiv M_{m^{\prime}1},M_{m^{\prime}2}, \cdots M_{m^{\prime}(t-1)}\Rightarrow UAV_{j_{i}}\succeq M_{m^{\prime}1},M_{ m^{\prime}2},\cdots M_{m^{\prime}(t-1)}\)_;_
3. \(M_{m^{\prime}t}\
otin\left\{M_{m^{\prime}t},M_{m^{\prime}(t+1)_{\cdot}\cdot} \cdots M_{m^{\prime}(t+n)}\right\}\Rightarrow\left\{\gamma_{m^{\prime}t}, \gamma_{m^{\prime}(t+1)_{\cdot}\cdot}\cdots\gamma_{m^{\prime}(t+n)}\right\} \Leftrightarrow\\ &\left\{G_{m^{\prime}t_{\cdot}\_key^{\prime}},G_{m^{\prime}(t+1)_{\cdot}\_key ^{\prime}},\cdots G_{m^{\prime}(t+n)_{\cdot}\_key}\right\}\)_;_ \(Q.E.D.\)_, the ACPKMS scheme has backward security, and the UAV cannot obtain the key_ \(\left\{G_{m^{\prime}t_{\cdot}\_key^{\prime}},G_{m^{\prime}(t+1)_{\cdot}\_key^{ \prime}},\cdots G_{m^{\prime}(t+n)_{\cdot}\_key}\right\}\) _after it leaves the swarm._

**Lemma 5**. : _The ACPKMS scheme can resist replay attacks (RRA). During the distribution and update of keys, the task label random number \(\epsilon_{t}\) is introduced. When \(\forall UAV_{j_{i}}\in U_{m^{\prime}t}\) receives the key update message \(M_{m^{\prime}t}\), it checks the freshness of \(\varepsilon_{m^{\prime}t}\). The \(\varepsilon_{m^{\prime}t}\) has already been used in previous sessions, then \(M_{m^{\prime}t}\) is discarded to prevent the attacker (AT) from replaying the broadcast message in previous sessions. Suppose that the \(t\) session broadcast message replayed by the attacker is \(M^{\prime}_{m^{\prime}t}\), the proving process is as follows:_

1.