## 1 Introduction
This phase basically does the same in the logical layer as the tomography does in the physical layer, and it consists of two main phases: the reconciliation procedure with several error-correction steps, and privacy amplification. The aim of reconciliation is to extract as much valuable information from the correlated raw data as possible and to generate an error-free key between [PERSON] and [PERSON]. The privacy amplification operates on the shared, error-corrected common secret to extract the final key between the parties, and the aim of this phase is to reduce to zero the possible knowledge of an eavesdropper from the elements of the key. The implementation of tomography in the physical layer is a complex problem, and it is intractable in a practical scenario. But, well-characterized solutions can be proposed in the logical layer for the same purpose of giving an analogous, and also more valuable, answer to the reconciliation of correlated Gaussian variables than the physical-layer tomography ever could. The theoretical background that makes the logical layer-based reconciliation possible also allow us to view the noisy physical quantum channel as a binary Gaussian channel in the logical layer [9; 10; 11]. This has the immediate consequence that very efficient binary error-correction tools can be integrated from the world of traditional communication theory into CVQKD--which would not be available for the physical-layer tomography to extract binary information from the correlated Gaussian variables. The raw data shared over the quantum channel is noisy, and this must be corrected to distill the final secret key. Since a large amount of raw data bits have to be shared between the parties, the complexity of the post-processing phase is a critical point in CVQKD protocols, and it has to be in order to be as low as possible. The existing logical layer-based solutions require high-complexity calculations in the high-dimensional spherical space for the reconciliation of Gaussian variables [9; 10; 11]. Since a complex reconciliation is so undesirable, the aim is to find a more efficient solution in the logical layer. A slice method is a different reconciliation approach, which is also used in the current reconciliation steps of CVQKD for short distances, and can be implemented without spherical operations [41]. Basically, the error correction in the reconciliation phase consists of two phases: First, the binary-channel codes (such as LDPC--Low Density Parity Check, turbo codes, polar codes, etc. [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]) that are used for the transmission of the classical bits in the reconciliation phase are corrected. Second, the real Gaussian noise on the received raw-data vector must be corrected, which noise arises from the effect of the quantum channel (i.e., from [PERSON]'s optimal Gaussian attack, which is considered in CVQKD protocols [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]). In this work, we focus on the second phase of reconciliation, which has a crucial role in CVQKD, since this phase makes it possible to correct the errors that are incurred on the quantum channel and to share an error-free key between [PERSON] and [PERSON]. Since the raw data is formulated by continuous real numbers resulted from quadrature measurements at the parties, the reconciliation problem is analogous to the well-known subject of binary-channel coding that operates on binary-channel codes. It also follows that the complicated and difficult to implement physical-layer tomography can be replaced in the logical level by binary error-correction schemes that are easier to implement. According to a critical security requirement of QKD, in the reconciliation phase, only uniform distribution can be transmitted over the classical channel, otherwise the information theoretic security of the protocol cannot be proven [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13].