## 1 Introduction  
Two-way quantum key distribution (QKD) schemes have evolved from a theoretic framework in the context of deterministic schemes to experimental realizations for QKD purposes [1]. The protocol is described by a qubit being sent from one party, say, [PERSON], to another, [PERSON], for her encoding before being sent back to him for a measurement to ascertain [PERSON]'s encoding. Standing in stark contrast to BB84 like prepare and measure schemes where information encoding is simply in the choice of states and bases of qubits to be sent from the encoding party to the receiver [2], the case for two-way QKDs is really in the choice of unitary transformations by an encoding party that would act on the qubitstraveling to and fro between the legitimate parties. This naturally makes use of the quantum channel twice. Another peculiarity of the above-mentioned two-way protocols would be how its security is to be ensured. The protocols' runs are divided into two modes, namely the encoding mode (EM) where message encoded by [PERSON] is followed by measurements by [PERSON] for decoding purposes and the control mode (CM) where [PERSON] would make projective measurements in randomly selected bases such that when the bases coincide with [PERSON]'s then errors can be ascertained. The distribution of the EM and CM would be determined by a factor \(c\), with \(0<c<1\), _i.e._, the probability of [PERSON] randomly choosing an EM. While messages (bits of raw key) are transmitted via the EM, CM has the critical role of estimating the amount of errors in the channels to ascertain [PERSON]'s information gain.  
However, despite its early introduction in 2002 [3], and varying versions that followed, it was only about a decade later that a proper proof for one specific two-way protocol was found for unconditional security in [4] and a proof for a purified version was reported in [5]. In [4], the proof was done for the specific case of non-entangled qubits in one of four states coming from two mutually unbiased bases (this protocol has been referred to as LM05 [6] and we shall refer to it as such hereafter). In [5], a purification of two two-way QKD protocol, namely the Super Dense Coding scheme (SDC) and LM05 was done in order to provide a security proof.  
Unconditional security proofs provide for an information theoretic security picture of QKD. However, a very important fact that sets constraints within the operational context would be the issue of how such proofs are based on asymptotic analysis that holds true only for infinitely long keys; the concern regarding the security nature of realistic keys, which are finitely long, becomes evident. This has spurred a number of studies including [7, 8, 9].  
The central feature of this work is a review of the security analysis done based on protocol purifications in [5] and a natural generalization to the the case for finite sized keys. We begin with a quick description of two most relevant two-way QKD protocols followed by their purifications. After a brief on smooth entropies and its application to deriving finite key rates, we provide finite secure key rates in terms of efficiency for both protocols. These will then be compared to an asymmetrical BB84; essentially a BB84 protocol with one basis preferred over another mainly for a more efficient scenario compared to a standard BB84 where choices of bases are equiprobable [8, 10].