## 3 The Method Description
When [PERSON] gets no additional information, she is restricted by (3), and her information obtained with the best possible measurement is still below \(NI_{\rm acc}(\mathcal{E}_{\rm E})\). A potential problem may appear due to the information disclosure taking place during privacy amplification procedure, since it makes [PERSON]'s states statistically dependent, and thus her ensemble \(\mathcal{E}_{\rm E}^{N}\) loses product form--see Section 4 for detail. However, in the case when [PERSON] is forced to measure the obtained quantum states before the legitimate users begin privacy amplification routine, the method works well. Instead of disclosing the \(H(\rm X|Y)\) bits during the error correction stage, the legitimate users would consume a part of the pre-distributed key in order to encrypt the classical communication using the one-time pad. Here, \(H(\rm X|Y)=H(\rm XY)-H(\rm Y)\) is the conditional entropy, which characterizes the lack of knowledge about X when the full information about Y is provided [28]. At the same time, after [PERSON]'s measurement, when all the participants operate with classical data, the legitimate users are able to substitute the value \(\chi(\mathcal{E}_{\rm E})\) in the Devetak-Winter equation with \(I_{\rm acc}(\mathcal{E}_{\rm E})\), thus obtaining a higher key generation rate without compromising the security of the whole scheme:  
\[\rm rate\geq\inf_{\rho_{AB}\in T}\Big{(}I(X:Y)-I_{\rm acc}(\mathcal{E}_{\rm E })\Big{)}. \tag{5}\]  
Recall that the set \(\Gamma\) includes all the bipartite states that can be shared between the legitimate users based on the statistics of their measurement results. Let us emphasize that no hardware modification is required for this secret key rate boost. In order to force [PERSON] to measure her states at an early stage and use the bound (4), legitimate users can employ some additional assumptions concerning [PERSON]'s technical abilities. The assumption about the upper-bound on the eavesdropper's quantum memory decoherence time is a natural one typically utilized in a quantum data locking scenario as well as in a QKD scenario with a restricted [PERSON]. This allows us to benefit from postponing the privacy amplification for an amount of time sufficient for the eavesdropper's quantum memory to lose coherence or from encrypting all the classical communication necessary for the stage with an asymmetrical cipher such as AES. In the latter case, the legitimate users are to assume that [PERSON] cannot break a chosen encryption during her quantum memory coherence time. The tactic allows legitimate users to assume that an eavesdropper is to conduct the measurement without any additional knowledge associated with the information from privacy amplification. In this scenario, the size of [PERSON]'s quantum memory is not limited, and her ability to conduct collective measurements is not restricted as well--this significantly distinguishes the approach we propose from the bounded quantum storage model (BQSM), which is built on the assumption concerning the maximal number of quantum states that an eavesdropper can keep in their quantum memory [2; 29; 30]. Nevertheless, our approach makes collective attacks no more efficient than individual ones and thus eliminates the necessity to consider any eavesdropping relying on quantum memory capable of storing more than one quantum state at a time. Moreover, [PERSON] may know all the information concerning bit reconciliation and post-selection procedures, as the availability of the data does not destroy the statistical independence of separate signal states. Thus, we propose a modification of the initial scheme presented in Section 2: the first two stages may remain unchanged, while the subsequent stages are modified in the following way:3. [PERSON] and [PERSON] perform error correction in a standard manner, with the only difference that now they utilize a private channel for the purpose, i.e., all the communication conducted at this stage is encoded by one-time pad cipher using the pre-distributed key.