## 3 Low-Dimensional Reconciliation
### Notations  
Let \(X\in\mathbb{R}^{N}\) and \(X^{\prime}\in\mathbb{R}^{N}\) the \(N\)-unit length raw data of [PERSON] and [PERSON]. The \(d\)-dimensional vectors \(\mathbf{X}_{j}\in\mathbb{R}^{d}\) and \(\mathbf{X}^{\prime}_{j}\in\mathbb{R}^{d}\), for \(j\ =\ 0,\,j\leq\big{(}N/d\big{)}-1\), of [PERSON] and [PERSON] are defined as:  
\[\mathbf{X}_{j}=\Big{(}X_{j,0},\ldots,X_{j,d-1}\Big{)}^{T}\in\mathbb{N}\Big{(} 0,\sigma_{X}^{2}\Big{)}_{d} \tag{24}\]  
and  
\[\mathbf{X}^{\prime}_{j}=\Big{(}X^{\prime}_{j,0},\ldots,X^{\prime}_{j,d-1}\Big{)} ^{T}\in\mathbb{N}\Big{(}0,\sigma_{X^{\prime}}^{2}\Big{)}_{d}, \tag{25}\]  
where  
\[X_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{X}^{2}\Big{)}\in\mathbb{R} \tag{26}\]  
and  
\[X^{\prime}_{j,i}\in\mathbb{N}\Big{(}0,\sigma_{X^{\prime}}^{2}\Big{)}\in\mathbb{R} \tag{27}\]  
refer to the \(i\)-th unit of the \(j\)-th vector, respectively. [PERSON] and [PERSON] have to share a common secret by using their correlated raw data. For this purpose, they establish a proper code-alphabet \(\mathcal{A}=\{a,b\}\), where \(a\in\mathbb{R}\) and \(b\in\mathbb{R}\) are two public variables (i.e., [PERSON] also has access to it). In the reverse reconciliation, these will be selected uniformly at random in the form of several \(U_{j}\in\{a,b\}\)-s at [PERSON]'s side, with \(\Pr(a)=\Pr(b)=0.5\).  
A secret \(d\)-dimensional key vector \(\mathbf{U}_{j}\) is drawn from a uniform distribution \(\mathcal{U}\) and built up from \(d\) units, \(U_{j,i}\in\mathbb{R}\), as:  
\[\mathbf{U}_{j}\in\mathbb{R}^{d}:\Big{(}U_{j,0},\ldots,U_{j,d-1}\Big{)}^{T}, \text{for }j=0,\ j\leq\big{(}N/d\big{)}-1. \tag{28}\]  
The \(d\) units \(U_{j,i}\in\mathcal{U}\) of \(\mathbf{U}_{j}\) are uniform random variables, and define \(U_{j}\in\mathbb{R}\) as follows:  
\[U_{j}=\sum_{i=0}^{d-1}U_{j,i}\in\mathcal{U}. \tag{29}\]  
The noisy version of (29), \(U^{\prime}_{j}\), is defined as  
\[U^{\prime}_{j}=\sum_{i=0}^{d-1}U^{\prime}_{j,i}. \tag{30}\]From (29) follows, that (28) can be rewritten as \(\mathbf{U}_{j}\in\{\mathbf{A},\mathbf{B}\}\in\mathbb{R}^{d}\), with vectors \(\mathbf{A},\mathbf{B}\) as:  
\[\mathbf{A}:\Big{(}a_{j,0},\ldots,a_{j,d-1}\Big{)}^{T},\Big{\{}\sum_{i=0}^{d-1}a _{j,i}=a\Big{\}},\,\mathbf{B}:\Big{(}b_{j,0},\ldots,b_{j,d-1}\Big{)}^{T},\Big{\{} \sum_{i=0}^{d-1}b_{j,i}=b\Big{\}}. \tag{31}\]  
As follows, [PERSON] granulates the selected \(a\) or \(b\) into \(d\) number of uniformly random variables \(U_{j,i}\), so that the sum of the units will be equal to the selected value.  
The full key \(\mathbf{K}\) is built up as:  
\[\mathbf{K}\in\mathbb{R}^{N/d}:\Big{(}U_{0},\ldots,U_{(N/d)-1}\Big{)}^{T}. \tag{32}\]  
[PERSON] and [PERSON] first agree on \(d\). [PERSON] then sends the \(d\) blocks of  
\[C\big{(}\mathbf{X}^{\prime}_{j}\big{)}\mathbf{U}_{j}=\Big{(}C\big{(}X^{\prime }_{j,0}\big{)}U_{j,0},\ldots,C\Big{(}X^{\prime}_{j,d-1}\Big{)}U_{j,d-1}\Big{)} ^{T}\in\mathbb{R}^{d}, \tag{33}\]  
for \(j=0\), \(j\leq(N/d)-1\), over a classical channel. The scalar quantities \(C\big{(}X_{j}\big{)}\), \(C\big{(}X^{\prime}_{j}\big{)}\), and \(C\big{(}X^{\prime}_{j}\big{)}U_{j}\) are evaluated as  
\[C\big{(}X_{j}\big{)}=\sum_{i=0}^{d-1}C\big{(}X_{j,i}\big{)}\in\mathbb{R},\,C \big{(}X^{\prime}_{j}\big{)}=\sum_{i=0}^{d-1}C\big{(}X^{\prime}_{j,i}\big{)} \in\mathbb{R}, \tag{34}\]  
and  
\[C\big{(}X^{\prime}_{j}\big{)}U_{j}=\sum_{i=0}^{d-1}C\big{(}X^{\prime}_{j,i} \big{)}U_{j,i}\in\mathbb{R}, \tag{35}\]  
respectively.  
[PERSON] receives the \(d\) noisy \(U^{\prime}_{j,i}\) units, and by the addition of the \(d\) units, and via the application of \(C\big{(}X_{j}\big{)}\) she computes \(U^{\prime}_{j}\) as  
\[\begin{array}{ll}U^{\prime}_{j}&=\sum_{i=0}^{d-1}U^{\prime}_{j,i}=C\big{(}X ^{\prime}_{j}\big{)}U_{j}\frac{1}{C(X_{j})}\\ &=\Big{(}\sum_{i=0}^{d-1}C\big{(}X^{\prime}_{j,i}\big{)}\Big{/}\sum_{i=0}^{d-1 }C(X_{j,i})\Big{)}\sum_{i=0}^{d-1}U_{j,i}\,.\end{array} \tag{36}\]  
Thus, [PERSON] has to make an error-correction to remove the noise from \(U^{\prime}_{j}\) to get achieve \(U_{j}\).