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arXiv:2610.10702

Combining Error Verification and Privacy Amplification in Quantum Key Distribution

Devashish Tupkary, Zhiyao Wang, Ernest Y. -Z. Tan

quant-phSubmitted Oct 7, 2026• Announced Oct 9, 2026• 0 citations
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Abstract

Quantum Key Distribution (QKD) protocols typically involve two separate steps of error-verification and privacy amplification. Both steps involve the application of a hash function sampled from a suitable hash family to classical data. A potentially simpler alternative is to implement a single-step hashing procedure, announce and compare a fixed number of bits of its output for error-verification, and retain the remaining bits as the final output key. This reduces the complexity of classical postprocessing, the amount of classical communication, and the amount of randomness needed in the protocol. While this construction has appeared in the literature, a suitable mathematical justification is lacking. In this work, we show that the security of such protocols, under certain simple conditions, follows from a slightly modified notion of secrecy of the protocol without the final error-verification step. In particular, we show that for most entropy-based proofs, the combined protocol can be implemented with no change to the key rate, and a minor change to the security parameter.

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The European Physical Journal Plus•Jul 9, 2024•12 citations
Quantum Key Distribution (QKD) protocols typically involve two separate steps of error-verification and privacy amplification. Both steps involve the application of a hash function sampled from a suitable hash family to classical data. A potentially simpler alternative is to implement a single-step hashing procedure, announce and compare a fixed number of bits of its output for error-verification, and retain the remaining bits as the final output key. This reduces the complexity of classical postprocessing, the amount of classical communication, and the amount of randomness needed in the protocol. While this construction has appeared in the literature, a suitable mathematical justification is lacking. In this work, we show that the security of such protocols, under certain simple conditions, follows from a slightly modified notion of secrecy of the protocol without the final error-verification step. In particular, we show that for most entropy-based proofs, the combined protocol can be implemented with no change to the key rate, and a minor change to the security parameter.
This paper examines classical privacy amplification using a universal family of hash functions. In quantum key distribution, the adversary's measurement can wait until the choice of hash functions is announced, and so the adversary's information may depend on the choice. Therefore the existing result on classical privacy amplification, which assumes the independence of the choice from the other random variables, is not applicable to this case. This paper provides a security proof of privacy amplification which is valid even when the adversary's information may depend on the choice of hash functions. The compression rate of the proposed privacy amplification can be taken to be the same as that of the existing one with an exponentially small loss in secrecy of a final key.
In a standard quantum key distribution (QKD) scheme such as BB84, two procedures, error correction and privacy amplification, are applied to extract a final secure key from a raw key generated from quantum transmission. To simplify the study of protocols, it is commonly assumed that the two procedures can be decoupled from each other. While such a decoupling assumption may be valid for individual attacks, it is actually unproven in the context of ultimate or unconditional security, which is the Holy Grail of quantum cryptography. In particular, this means that the application of standard efficient two-way error-correction protocols like Cascade is not proven to be unconditionally secure. Here, I provide the first proof of such a decoupling principle in the context of unconditional security. The method requires Alice and Bob to share some initial secret string and use it to encrypt their communications in the error correction stage using one-time-pad encryption. Consequently, I prove the unconditional security of the interactive Cascade protocol proposed by Brassard and Salvail for error correction and modified by one-time-pad encryption of the error syndrome, followed by the random matrix protocol for privacy amplification. This is an efficient protocol in terms of both computational power and key generation rate. My proof uses the entanglement purification approach to security proofs of QKD. The proof applies to all adaptive symmetric methods for error correction, which cover all existing methods proposed for BB84. In terms of the net key generation rate, the new method is as efficient as the standard Shor-Preskill proof.
The amount of secret key that can be obtained from a quantum key distribution run depends on both the physically observed error rates and the mathematical bounds used to certify security. For finite datasets, conservative bounds force users to discard a substantial fraction of the potentially available key. Here we further refine and extend the privacy amplification bounds achievable through the recent leftover hash lemma of Regula and Tomamichel [arXiv:2603.04493] and incorporate them into the security analysis of quantum key distribution based on entropic uncertainty relations. This improves on state-of-the-art key rates in finite-block regimes, certifying more secret key from the same experimental data without changes to the protocol, and outperforming techniques based on entropy accumulation. The results illustrate how sharper mathematical estimates can directly increase the usable output of a quantum communication system.
[[abstract]]An enhanced version of the quantum key distribution (QKD) protocol in quantum cryptography is proposed. We consider that realistic detectors and imperfect devices may introduce noise into the transmission thereby causing a high error rate. The proposed scheme is based on an error rate analysis and a random sampling procedure in statistics to produce a probabilistic bound on the error estimation. Confidence intervals are used to interpret the error estimate with a specified confidence level. Based on the information obtained, Alice and Bob judge what caused the errors and establish confidence levels for the error rate of the remaining untested bits. Hence, they can determine the maximum number of times reconciliation needs to be performed to remove all errors from what remains of a raw key to produce an error-free secret key. As a result, the tedious reconciliation process can be reduced and a neglectful probability of not detecting the existence of remaining errors can be avoided as well. The privacy amplification is then applied: (1) to extract a secret key from a partially secret key, and (2) to enlarge the length of the secret key into the final key. The proposed scheme can work when dealing with errors caused by random noise and eavesdropping.
Privacy amplification is a necessary step in all quantum key distribution protocols, and error correction is needed in each except when signals of many photons are used in the key communication in quantum noise approach. No security analysis of error correcting code information leak to the attacker has ever been provided, while an ad hoc formula is currently employed to account for such leak in the key generation rate. It is also commonly believed that privacy amplification allows the users to at least establish a short key of arbitrarily close to perfect security. In this paper we show how the lack of rigorous error correction analysis makes the otherwise valid privacy amplification results invalid, and that there exists a limit on how close to perfect a generated key can be obtained from privacy amplification. In addition, there is a necessary tradeoff between key rate and security, and the best theoretical values from current theories would not generate enough near-uniform key bits to cover the message authentication key cost in disturbance-information tradeoff protocols of the BB84 variety.
Abstract Since the 80 s when it was first proposed, Quantum Key Distribution (QKD) elicited great interest in the field of cryptography as a unique procedure for key generation that could in principle guarantee unconditionally secure communication “by the laws of Physics”. In the last fifteen years commercial solutions have started appearing on the market, showing that practical implementations of the protocol were not only possible but also competitive in terms of security and achievable secret-key rate. In this work we describe a simulation of the historical QKD protocol E91 on the IBM Quantum platform, making use of the qubit formalism to represent the quantum states received by two communicating nodes. Having implemented also the post-processing steps for the error correction and the privacy amplification, this model can represent a simple stand-alone tool to study the performance not only of one-to-one communication but of more complex systems that rely on QKD for security, one above all QKD networks.