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

Prateek P. Kulkarni

Abstract

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 \[ \ell \le k + \log L - 2\log(1/ε) - 3, \] 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 $L = 2^{αn'}$ increases the tolerable phase-error threshold from $h^{-1}(1 - h(e_b))$ to $h^{-1}(1 - h(e_b) + α)$, exceeding the standard $\approx 11\%$ 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 $\mathbb{F}_{2^m}$ and a Toeplitz-based variant, running in $O(nL)$ and $O(nL \log n)$ time.

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

Abstract

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 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 -list of -bit keys can be extracted from an -bit source with smooth min-entropy iff yielding a tight additive gain over QLHL. This gain arises because the index of the secure key is chosen after hashing and hidden from Eve, effectively contributing bits of entropy. Applying QLLHL to BB84-type QKD, a list size increases the tolerable phase-error threshold from to , exceeding the standard bound for any . 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 and a Toeplitz-based variant, running in and time.
Paper Structure (26 sections, 10 theorems, 27 equations, 2 tables, 2 algorithms)

This paper contains 26 sections, 10 theorems, 27 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our question.
  3. Our contributions.
  4. On the practical significance.
  5. Related work.
  6. Paper organisation.
  7. Preliminaries
  8. Notation and Quantum States
  9. Smooth Min-Entropy Against Quantum Adversaries
  10. Two-Universal Hash Families
  11. The Quantum Leftover Hash Lemma (Standard, $L=1$)
  12. Composable Security
  13. List Security for Privacy Amplification
  14. The Quantum List Leftover Hash Lemma
  15. Tightness of the Bound
  16. ...and 11 more sections

Key Result

Lemma 3

Let $\rho_{XE}$ be a CQ state and $\varepsilon, \delta \ge 0$.

Theorems & Definitions (28)

  • Definition 1: Min-entropy Renner2008
  • Definition 2: Smooth min-entropy Renner2008TCR2009
  • Lemma 3: Standard properties Renner2008Tomamichel2015
  • Definition 4: Strong two-universality Carter1979
  • Theorem 5: QLHL Tomamichel2011
  • Definition 6: $L$-list key functionality
  • Remark 7: Reduction to standard at $L=1$
  • Definition 8: List-$\varepsilon$-security
  • Remark 9: Using the list key in practice
  • Theorem 10: Quantum List Leftover Hash Lemma (QLLHL)
  • ...and 18 more