## 4 Theorems and Proofs
\tag{96}\]  
As follows, by applying the procedure [PERSON] can retrieve \(\mathbf{U}_{j}\in\{\mathbf{A},\mathbf{B}\}\) from the noisy \(\mathbf{U}_{j}\) in the vector space \(\mathbf{v}\) of \(\mathbb{R}^{d}\). From the error-corrected \(\mathbf{U}_{j}\) components, [PERSON] finally rebuilds the full key vector \(\mathbf{K}=\Big{(}U_{0},\ldots,U_{(N/d)-1}\Big{)}^{T}\in\mathbb{R}^{N/d}\), which concludes the proof. Proposition 1 demonstrated that there is no need for the use of \(\Gamma^{d-1}\) of \(\mathbb{R}^{d}\) in the error correction, however the corrected noise is not precisely a Gaussian. Theorem 1 reveals that the reconciliation process, in fact, does not require vector operations in \(\mathbb{R}^{d}\), and the noise is a real Gaussian noise in the scalar space \(\mathbb{R}\). **Theorem 1** (Scalar reconciliation of correlated Gaussian variables). : _The Gaussian noise \(\delta_{j}\) on the received scalar \(U^{\prime}_{j}=\sum_{i=0}^{d-1}U^{\prime}_{j,i}\) can be corrected in \(\mathbb{R}\)._  
Proof. : We exploit that the noise on \(U^{\prime}_{j,i}\)-s is \(\delta_{j,i}=\frac{U_{j,i}}{C\big{(}X_{j,i}\big{)}}C\big{(}\Delta_{j,i}\big{)} \in\mathbb{N}\Big{(}0,\sigma_{\delta_{j,i}}^{2}\Big{)}\), while on the sum of the noise of the \(d\) units is a zero-mean Gaussian random variable \(\sum_{i=0}^{d-1}\delta_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{\delta_{j}}^{2} \Big{)}\), that is justified by the CLT and the Lyapunov-condition. [PERSON] will correct the units in the following form:  
\[U^{\prime}_{j}=\sum_{i=0}^{d-1}U^{\prime}_{j,i}=\frac{\sum_{i=0}^{d-1}C\big{(} X^{\prime}_{j,i}\big{)}U_{j,i}}{\sum_{i=0}^{d-1}C\big{(}X_{j,i}\big{)}}=U_{j}+ \delta_{j}\in\mathbb{R}. \tag{97}\]  
First, expresses the secret vector \(\mathbf{U}_{j}\in\mathbb{R}^{d}\) as follows:  
\[\mathbf{U}_{j}=x(\mathbf{A}-\mathbf{B})+\frac{1}{2}(\mathbf{A}+\mathbf{B}), \tag{98}\]  
where \(x\in\{-0.5,0.5\}\in\mathbb{R}\) is a scalar. From this, [PERSON] can also rewrite the noisy \(\mathbf{U}^{\prime}_{j}\) as:  
\[\mathbf{U}^{\prime}_{j}=x(\mathbf{A}-\mathbf{B})+\frac{1}{2}(\mathbf{A}+ \mathbf{B})+\overset{\rightarrow}{\delta}_{j}. \tag{99}\]  
From (99), follows that:  
\[ \begin{array}{ll}U^{\prime}_{j}&=\sum_{i=0}^{d-1}\Big{(}x(A_{i}-B_{i})+ \frac{1}{2}(A_{i}+B_{i})+\delta_{j,i}\Big{)}\\ &=\sum_{i=0}^{d-1}\big{(}x(A_{i}-B_{i})+\frac{1}{2}(A_{i}+B_{i})\Big{)}+\delta _{j}\\ &=\sum_{i=0}^{d-1}U^{\prime}_{j,i}\\ &=U_{j}+\frac{U_{j}}{C\big{(}X_{j}\big{)}}C\big{(}\Delta_{j}\big{)},\end{array}  \tag{100}\]where \(C\big{(}X_{j}\big{)}=\sum_{i=0}^{d-1}C\big{(}X_{j,i}\big{)}\), \(C\big{(}\Delta_{j}\big{)}=\sum_{i=0}^{d-1}C\big{(}\Delta_{j,i}\big{)}\), \(U_{j}=\sum_{i=0}^{d-1}U_{j,i}\) and \(\delta_{j}=\sum_{i=0}^{d-1}\delta_{j,i}\). In fact, [PERSON] does not have to use all elements from (100), because she can apply a simpler process. For this purpose, she draws a new vector, \(\mathbf{d}\):  
\[\mathbf{d}=\frac{\mathbf{A}-\mathbf{B}}{\|\mathbf{A}-\mathbf{B}\|}, \tag{101}\]  
where \(\|\mathbf{A}-\mathbf{B}\|=\sqrt{\sum_{i=0}^{d-1}\big{(}A_{i}-B_{i}\big{)}^{2}}\) is the effective distance of \(\mathbf{A}\) and \(\mathbf{B}\). A useful property of vector \(\mathbf{d}\) drawn in (101), that any independent noise [15] (i.e., independent from the noise on \(\mathbf{U}^{\prime}_{j}\)) could live only in the orthogonal directions to \(\mathbf{d}\), i.e., \((\mathbf{n}_{1},\ldots,\mathbf{n}_{l})\bot\mathbf{d}\). It immediately follows, that the \(\mathbf{n}_{1},\ldots,\mathbf{n}_{l}\) orthogonal directions will have no further importance for [PERSON] in the decoding [15; 16; 17; 18; 19].