## 2 System Model
### Uniform Distribution in the Spherical Space
The normalized vector \(\mathbf{X}^{\prime}_{jj}\Big{/}\sqrt{d\sigma_{\mathbf{X}^{\prime}}^{2}}\) with norm \(\left\|\mathbf{X}^{\prime}_{j}\right\|\) and \(\sqrt{d\sigma_{\mathbf{X}^{\prime}}^{2}}\), can be identified on the unit sphere \(\Gamma^{d-1}\)[18; 24], with radius \(r=\left\|\mathbf{X}^{\prime}_{j}\right\|/\sqrt{d\sigma_{\mathbf{X}^{\prime}}^ {2}}\). The mean of \(\left\|\mathbf{X}^{\prime}_{j}\right\|\Big{/}\sqrt{d\sigma_{\mathbf{X}^{\prime }}^{2}}\) is  
\[\mathbb{E}\bigg{[}\left\|\mathbf{X}^{\prime}_{j}\right\|\bigg{/}\sqrt{d\sigma _{\mathbf{X}^{\prime}}^{2}}\bigg{]}=\sigma_{\mathbf{X}^{\prime}}\sqrt{d- \frac{1}{2}}\bigg{/}\sqrt{d\sigma_{\mathbf{X}^{\prime}}^{2}}. \tag{17}\]  
The vector \(\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{\mathbf{X}^{\prime}}^{2}}\) on the unit sphere \(\Gamma^{d-1}\) is identified as  
\[\mathbf{X}^{\prime}_{j}\bigg{/}\sqrt{d\sigma_{\mathbf{X}^{\prime}}^{2}}=r \frac{\mathbf{X}^{\prime}_{j}}{\left\|\mathbf{X}^{\prime}_{j}\right\|}=\frac{ \left\|\mathbf{X}^{\prime}_{j}\right\|/\sqrt{d\sigma_{\mathbf{X}^{\prime}}^{ 2}}r\left\|\mathbf{X}^{\prime}_{j}\right\|}{\left\|\mathbf{X}^{\prime}_{j} \right\|}. \tag{18}\]  
Precisely, the normalized quantity \(\left\|\mathbf{X}^{\prime}_{j}\right\|\Big{/}\sqrt{d\sigma_{\mathbf{X}^{ \prime}}^{2}}\) has variance \(\mathrm{var}\Big{[}\left\|\mathbf{X}^{\prime}_{j}\right\|\Big{/}\sqrt{d\sigma _{\mathbf{X}^{\prime}}^{2}}\Big{]}\leq\frac{\sigma_{\mathbf{X}^{\prime}}^{2} }{2}\bigg{/}d\sigma_{\mathbf{X}^{\prime}}^{2}\). From the spherical symmetry, it follows that if \(d\to\infty\), the normalized random vector \(\mathbf{X}^{\prime  
and it will also converge to one, \(r=\lim\limits_{d\to\infty}\left\|\mathbf{X}^{\prime}_{j}\right/\sqrt{d\sigma_{X^{ \prime}}^{2}}\|=1\). The unit norms of \(\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{X^{\prime}}^{2}}\) play exactly the role of unit fading-coefficients for a logical binary Gaussian channel, since during the transmissions of the messages generated from \(\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{X^{\prime}}^{2}}\) the unit norms \(r=\left\|\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{X^{\prime}}^{2}}\right\|=1\) are also transmitted [11; 21]. To be more exact, the unit norms are only approximated and the distribution of the unit norms also depends on \(d\), and as \(d\to\infty\), it precisely can be described by the Dirac distribution  
\[\mathcal{D}_{d}(x)=\big{(}1/a\sqrt{\pi}\big{)}e^{-\left(x-r\right)^{2}\big{/}a ^{2}}, \tag{21}\]  
where \(a=1\Big{/}\sqrt{d}\) and  
\[r=\lim\limits_{d\to\infty}\frac{\left\|\mathbf{X}^{\prime}_{j}\right\|}{\sqrt {d\sigma_{X^{\prime}}^{2}}}=1. \tag{22}\]  
From \(\mathcal{D}_{d}(x)\) it immediately follows, that the unit norms of the normalized random Gaussian vectors gets closer to \(1\), as \(d\) goes to infinity [18]. As follows from these, for low values of \(d\) the uniform distribution of \(\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{X^{\prime}}^{2}}\) cannot be achieved. In comparison to the multidimensional reconciliation, where the required mathematical operations (the spherical division operator at Alice's side) exist only in \(d=1\), \(2\), \(4\), or \(8\) dimensions [9; 10; 11; 18], the scalar reconciliation process are also existent for arbitrary high dimensions, which makes possible to give a more closer approximation, however it will not refer to the Dirac distribution. Analyzing the situation if the noisy raw data follows Gaussian random distribution with \(\sigma_{X^{\prime}}^{2}>1\), the speed of convergence of the mean \(\mathbb{E}\Big{[}\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{X^{\prime}}^{2}} \Big{]}\) and variance \(\text{var}\Big{[}\mathbf{X}^{\prime}_{j}\Big{/}\sqrt{d\sigma_{X^{\prime}}^{2} }\Big{]}\) will be lower for any \(d\), in comparison if \(\sigma_{X^{\prime}}^{2}=1\) would have hold.