One Key Good, L Keys Better: List Decoding Meets Quantum Privacy Amplification

Quantum Physics [Submitted on 18 Mar 2026] One Key Good, L Keys Better: List Decoding Meets Quantum Privacy Amplification Prateek P. Kulkarni 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 \ell-bit keys can be extracted from an n-bit source with smooth min-entropy k iff \ell \le k + \log L - 2\log(1/\epsilon) - 3, yielding a tight additive \log L gain over QLHL. This gain arises because the index of the secure key is chosen after hashing and hidden from Eve, effectively contributing \log L bits of entropy. Applying QLLHL to BB84-type QKD, a list size L = 2^{\alpha n'} increases the tolerable phase-error threshold from h^{-1}(1 - h(e_b)) to h^{-1}(1 - h(e_b) + \alpha), exceeding the standard \approx 11\% bound for any \alpha > 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 \mathbb{F}_{2^m} and a Toeplitz-based variant, running in O(nL) and O(nL \log n) time. Comments: 18 pages Subjects: Quantum Physics (quant-ph); Cryptography and Security (cs.CR) Cite as: arXiv:2603.18097 [quant-ph]   (or arXiv:2603.18097v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.18097 Focus to learn more Submission history From: Prateek P. Kulkarni [view email] [v1] Wed, 18 Mar 2026 10:37:05 UTC (19 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.CR References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)