We show that the Mayers-Shor-Preskill approach and Renner's approach to proving the security of quantum key distribution (QKD) are essentially the same. We begin our analysis by considering a special case of QKD called privacy amplification (PA). PA itself is an important building block of cryptography, both classical and quantum. The standard theoretical tool used for its security proof is called the leftover hashing lemma (LHL). We present a direct connection between the LHL and the coding theorem of a certain quantum error correction code. Then we apply this result to proving the equivalence between the two approaches to proving the security of QKD.
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.
We introduce list privacy amplification (LPA), a relaxation of the final step of quantum key distribution (QKD) in which Alice and Bob extract a list of
L candidate keys from a raw string correlated with an eavesdropper Eve, with the guarantee that at least one key is perfectly secret while Eve cannot identify which. This parallels list decoding in error-correcting codes: relaxing unique decoding to list decoding increases the decoding radius; analogously, list extraction increases achievable key length beyond the standard quantum leftover hash lemma (QLHL). Within the abstract cryptography framework, we formalise LPA and prove the \emph{Quantum List Leftover Hash Lemma} (QLLHL): an
L-list of
ℓ-bit keys can be extracted from an
n-bit source with smooth min-entropy
k iff
ℓ≤k+logL−2log(1/ε)−3, yielding a tight additive
logL gain over QLHL. This gain arises because the index of the secure key is chosen after hashing and hidden from Eve, effectively contributing
logL bits of entropy. Applying QLLHL to BB84-type QKD, a list size
L=2αn′ increases the tolerable phase-error threshold from
h−1(1−h(eb)) to
h−1(1−h(eb)+α), exceeding the standard
≈11% bound for any
α>0. We prove tightness via a matching intercept-resend attack, establish composability with Wegman--Carter authentication, and present two constructions: a polynomial inner-product hash over
F2m and a Toeplitz-based variant, running in
O(nL) and
O(nLlogn) time.
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.
By applying the framework of entropic uncertainty relation (EUR) and the Quantum Leftover Hash Lemma (QLHL), we introduce a security-proof method for variable-length side-channel-secure (SCS) quantum key distribution (QKD) against coherent attacks. This method reframes composable security as a statistical fluctuation problem of phase errors, enabling direct proofs against coherent attacks through observables and virtual observables. It yields tight key rates for the SCS protocol and reduces pulse requirements by over two orders of magnitude compared to prior works that employ the post-selection technique. {Utilizing the characteristic of the SCS protocol that it has no untagged-bit bit-flip errors, we prove that the secure key length for the SCS protocol can be determined after Cascade error correction, using its actual parity disclosures. We also explain how the same argument extends to protocols with compatible bit-error-pattern and phase-error measurements.} We further identify sufficient conditions under which the final key length may be determined after error correction in a broader class of QKD protocols. Under the framework of EUR and QLHL, we clarify the applicability of several commonly used concentration bounds to variable-length QKD and the appropriate manner of their implementation. This work enhances the practical value of the SCS protocol and clarifies the security justification of key-rate formulas used in practical variable-length QKD implementations.
Many researchers have contributed to creating Quantum Key Distribution (QKD) since the first protocol BB84 was proposed in 1984.One of the crucial problems in QKD is to guarantee its security with finite-key lengths by Privacy Amplification (PA).However, finite-key analyses show a trade-off between the security of BB84 and the secure key rates.This study analyses two examples to show concrete trade-offs.Furthermore, even though the QKD keys have been perceived to be arbitrarily secure, this study shows a fundamental limitation in the security of the keys by connecting Leftover Hash Lemma and Guessing Secrecy on the QKD keys.