## 1 Introduction  
Cryptography is a well-known technique for achieving communication secrecy, but practical key generation and distribution schemes remain one of the toughest challenges in modern cryptography. The common key distribution protocols [1, 2, 3] rely on symmetric and asymmetric encryption algorithms, which offer computational security based on the complexity of mathematical problems. However, the security of these protocols could potentially be compromised by the advent of powerful computing devices, such as quantum computers [4, 5]. Fortunately, quantum key distribution (QKD) [6] utilizes the properties of quantum mechanics, thereby enabling two legitimate parties (_[PERSON]_ and _[PERSON]_) to generate a secret key based on information-theoretic security principles [7]. Basically, the QKD protocol consists of six steps:  
**(1) Distribution of quantum information:** In the BB84 protocol [6], _[PERSON]_ encodes random bits into the polarization states of single photons, which are then transmitted to _[PERSON]_ over the quantum channel. _[PERSON]_ subsequently randomly selects measurement bases to measure the polarization of a received photon and obtains classical measurement bits. After this step, both parties have a record of binary information, known as the raw key. Upon accumulating a sufficient amount of raw key, they perform the following post-processing steps [8; 9], using the authenticated classical channel to distill secure secret keys. **(2)**: _Sifting_**: _[PERSON]_ and _[PERSON]_ communicate and compare their encoding and measurement bases. Then, any bits in raw key with non-matching bases are discarded, allowing both parties (_[PERSON]_ and _[PERSON]_) to obtain correlated classical bits with the same length, called the sifted key. **(3)**: **Channel error estimation:**_[PERSON]_ and _[PERSON]_ typically estimate the quantum bit error rate (QBER) by using random key sampling. If the estimated QBER exceeds a predetermined threshold value, both parties must abort the QKD protocol to prevent potential security breaches. **(4)**: **Information reconciliation:**_[PERSON]_ and _[PERSON]_ correct the discrepancies between their sifted keys using error-correction algorithms to produce the reconciled key. **(5)**: **Confirmation:**_[PERSON]_ and _[PERSON]_ utilize a universal hash function to verify whether their reconciled keys are identical. If the hash values from _[PERSON]_ and _[PERSON]_ do not match, they can return to the information reconciliation step or abort the QKD protocol. **(6)**: **Privacy amplification:** To eliminate any partial information eavesdropped by _[PERSON]_ through both the quantum and classical channels, _[PERSON]_ and _[PERSON]_ compress their identical keys using universal hashing. The resulting shortened keys, known as the secret keys, are statistically independent of _[PERSON]_'s information and are identical between _[PERSON]_ and _[PERSON]_. Although there have been breakthroughs in QKD, the achievable secret key rates remain inadequate for high-speed industrial applications. One of the primary bottlenecks is the efficiency of classical post-processing algorithms. To address this limitation, this work investigates the effective algorithms for error estimation, information reconciliation, and confirmation steps, aiming to enhance the efficiency of classical post-processing. In QKD post-processing, information reconciliation is typically implemented using interactive protocols such as Cascade [10], which uses a combination of random shuffling and dichotomic search algorithms to identify and correct error positions using subblock parities. Cascade is generally a simple and efficient error-correction protocol. However, its speed is fundamentally limited by its high interactivity. Several modified versions of Cascade have been proposed to improve its speed in parallel operations [11], and to optimize the parameters for optimal implementation [12].