# Cooperative Jamming-Based Physical-Layer Group Secret and Private Key Generation
## 3 Group Secret and Private Key Generation Scheme
### Key Agreement
#### 3.2.2 The Second Step of Public Discussion  
After the first step, [PERSON], [PERSON], and [PERSON] all obtain the correlated estimates of \(G_{1}\), thereby enhancing common randomness. However, due to the presence of noise, further agreement is still required. [PERSON] transforms \(\mathbf{\tilde{Y}}_{A}\) into \(\mathbf{\tilde{Y}}_{A}^{\Delta}=\left[\tilde{Y}_{A}^{\Delta}(1),\ldots,\tilde {Y}_{A}^{\Delta}(n)\right]^{\prime}\). Similarly, [PERSON] and [PERSON] transform \(\mathbf{\hat{G}}_{1,B}\) and \(\mathbf{\hat{G}}_{1,C}\) into \(\mathbf{\hat{G}}_{1,B}^{\Delta}=\left[\hat{\mathcal{C}}_{1,B}^{\Delta}(1), \ldots,\hat{\mathcal{C}}_{1,B}^{\Delta}(n)\right]^{\prime}\) and \(\mathbf{\hat{G}}_{1,C}^{\Delta}=\left[\hat{\mathcal{C}}_{1,C}^{\Delta}(1), \ldots,\hat{\mathcal{C}}_{1,C}^{\Delta}(n)\right]^{\prime}\), respectively. To generate an SK and a PK with different security permission levels, [PERSON] needs to adopt two different sub-bin partitioning methods according to Slepian-Wolf coding. * [PERSON] randomly and independently partitions the typical \(\mathbf{\tilde{Y}}_{A}^{\Delta}\) sequences into \(2^{nR_{31}}\) bins, with each bin having \(2^{nR_{32}^{SK}}\) sub-bins, where \[R_{31} =\max\left\{H\left(\tilde{Y}_{A}^{\Delta}\middle|\hat{G}_{1,B}^{ \Delta}\right),H\left(\tilde{Y}_{A}^{\Delta}\middle|\hat{G}_{1,C}^{\Delta} \right)\right\}\!,\] (11) \[R_{32}^{SK} =\left[H\left(\tilde{Y}_{A}^{\Delta}\right)-\max\left\{H\left( \tilde{Y}_{A}^{\Delta}\middle|\hat{G}_{1,B}^{\Delta}\right),H\left(\tilde{Y}_{ A}^{\Delta}\middle|\hat{G}_{1,C}^{\Delta}\right)\right\}\right]^{+}\!.\] (12)
* [PERSON] randomly and independently partitions the typical \(\mathbf{\tilde{Y}}_{A}^{\Delta}\) sequences into \(2^{nR_{31}}\) bins, with each bin having \(2^{nR_{32}^{PK}}\) sub-bins, where \[R_{32}^{PK}=\left[H\left(\tilde{Y}_{A}^{\Delta}\middle|\hat{Y}_{R}^{(1),\Delta },\tilde{Y}_{R}^{(2),\Delta}\right)-\max\left\{H\left(\tilde{Y}_{A}^{\Delta} \middle|\hat{G}_{1,B}^{\Delta}\right),H\left(\tilde{Y}_{A}^{\Delta}\middle|\hat{ G}_{1,C}^{\Delta}\right)\right\}\right]^{+}\!.\] (13)  
Hence, each sequence has two indices: bin number and index within the bin. [PERSON] takes the index within the bin of two different partitioning methods as an SK and a PK, respectively, and then sends the bin number as the helper data to [PERSON] and [PERSON]. [PERSON] needs to send \(\max\left\{H\left(\tilde{Y}_{A}^{\Delta}\middle|\hat{\mathbf{G}}_{1,B}^{\Delta }\right),H\left(\tilde{Y}_{A}^{\Delta}\middle|\hat{\mathbf{G}}_{1,C}^{\Delta} \right)\right\}\) bits of information through the public channel. By combining their own observations \(\mathbf{\hat{G}}_{1,B}\), \(\mathbf{\hat{G}}_{1,C}\) and helper data, [PERSON] and [PERSON] can recover \(\mathbf{\tilde{Y}}_{A}\) with the probability being arbitrarily close to \(1\). **Remark 1**. : _From the channel measurement and key agreement stages, it is evident that the estimates of the reference channel \(G_{1}\) play a crucial role in enabling all legitimate nodes to share a group key. During the channel measurement stage, [PERSON] obtains an estimation of \(G_{1}\), as shown in (4). [PERSON], on the other hand, only obtains an estimation of \(G_{2}\), i.e., \(\tilde{Y}_{B}\), as shown in (5). However, following the first step of public discussion in the key agreement stage, [PERSON] gains access to the relay's observation \(\tilde{Y}_{R}^{(1)}\). As illustrated in (8), \(\tilde{Y}_{R}^{(1)}\) encompasses the term \(G_{1}+G_{2}\), allowing [PERSON] to employ a ZF method to estimate \(G_{1}\) based on \(\tilde{Y}_{B}\) and \(\tilde{Y}_{R}^{(1)}\). Similarly, [PERSON] can also obtain an estimation of \(G_{1}\). Subsequently, during the second step of public discussion, all legitimate nodes can leverage the correlated estimates of \(G_{1}\) to generate a common randomness as well as a group key._  
**Remark 2**. : _On the other hand, both the relay and Eve can only acquire superimposed channel observations from both the reference channel \(G_{1}\) and another channel, either \(G_{2}\) or \(G_{3}\).