## 2 System Model  
In comparison to one-way CVQKD protocols, in two-way CVQKD the two uses of the quantum channel lead to superadditive private classical capacity (more precisely, the superadditivity of security threshold leads to a subadditive eavesdropper [1; 2; 3; 4; 5; 6; 7; 8; 14]), which makes it possible to decrease the amount of valuable information leaked to [PERSON]. The subadditive eavesdropper is a consequence of the multiple uses of the quantum channel. The superadditivity of the security threshold can also be expressed in terms of tolerable excess noise and the channel transmission [1]. In the two-way scenario, [PERSON] perturbs the quantum channel \(\mathcal{N}_{1}\), which causes a noise in the transmission that will have an effect on the success of her second attack. From the two attacks, comparatively lower valuable information will be available to [PERSON] so that she would not have made an attack on \(\mathcal{N}_{1}\). The reason for this is that the amount of valuable information transmitted over \(\mathcal{N}_{2}\) is already decreased by the attack of \(\mathcal{N}_{1}\). More attacks add more noise into the transmission, which also decreases the amount of mutual information between [PERSON] and [PERSON]. With the increased number of channel uses we allow [PERSON] to get as much less valuable information as possible. If [PERSON] encodes her information into the noisy state that is received from \(\mathcal{N}_{1}\), and then sends it back to [PERSON] over \(\mathcal{N}_{2}\), then the parties can achieve the desired phenomenon of superadditivity [1; 2; 3; 4]. The amount of valuable information leaked to [PERSON] is also decreased by the multiple uses of the quantum channel. The errors caused by more channel uses can be corrected in the reconciliation phase by traditional error-correction tools. In fact, by utilizing multiple channel uses, we'set a trap' for [PERSON], since again and again she will attack the quantum channel. [PERSON] will also simultaneously decrease the amount of eavesdropped information by her actions. The idea works well, because in the post-processing phase the parties can correct the errors caused by [PERSON], thus, finally, it can be concluded that it was a correct decision to increase the number of channel uses. Of course, if we had perfect amplifiers and ideal devices, then, in theory, it would be possible to completely eliminate [PERSON] from the picture in the asymptotic scenario to make unnecessary the privacy amplification by allowing an infinite amount of channel uses to maximally exploit the superadditivity property (more precisely, the superadditivity of the security-threshold parameter hence the strong subadditivity of [PERSON]). However, in practice it is trivially not possible to circulate over and over the same beam an infinite amount of times, due to the losses and imperfections of the physical devices. Let us review the data components of the protocol that are needed for the appropriate description of the scalar reconciliation for the two-way CVQKD protocol. Our description will be as detailed as desired for further analysis, and will not take into account the particular description of any components of an experimental protocol. The raw data is generated by the use of noisy Gaussian channels \(\mathcal{N}_{1}\) and \(\mathcal{N}_{2}\), and by the parties' internal secrets. The aim of the quantum-level transmission is to generate two nearly identical classical bitstrings between the parties. All of the quantum-level interactions are closed at this point, and the post-processing phase, which uses the raw data of the parties and a classical authenticated channel, is brought to life. The post-processing phase consists of the processes of reconciliation and privacy amplification. The valuable key will be generated in the reconciliation phase by using the raw data and a random secret. It consists of error-correction phases as well.