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Universally-composable privacy amplification from causality constraints

Lluis Masanes

arXiv:0807.2158v4quant-phcs.CRcs.ITmath.IT

TL;DR

The paper addresses how to establish security for keys generated from Bell-violating correlations, where earlier schemes had only partial security proofs. It proves universally-composable security using no-signaling constraints, and concludes that secure key distribution can be implemented with untrusted quantum devices.

  • Problem

    Earlier Bell-correlation key-distribution schemes had partial security proofs, leaving a general proof under the strongest security criterion as the central gap.

    Source: p. 1

  • Method

    The proof uses privacy amplification on a function of Alice’s outcomes and relies on multipartite no-signaling rather than the validity of quantum mechanics.

    Source: pp. 1–3

  • Results

    The paper provides a universally-composable security proof for key distribution from Bell-violating correlations under no-signaling assumptions.

    Source: p. 1

  • Takeaways & Limitations

    The result permits secure protocols with untrusted quantum preparation and measurement devices, while requiring trust in classical computers and the random number generator.

    Source: p. 4

    The simple scheme does not work with quantum-generated correlations, although the BHK protocol addresses this problem.

    Source: p. 2

Abstract

from arXiv · show

We consider schemes for secret key distribution which use as a resource correlations that violate Bell inequalities. We provide the first security proof for such schemes, according to the strongest notion of security, the so called universally-composable security. Our security proof does not rely on the validity of quantum mechanics, it solely relies on the impossibility of arbitrarily-fast signaling between separate physical systems. This allows for secret communication in situations where the participants distrust their quantum devices.

APPENDIX

The appendix proves the two stated lemmas, using no-signaling constraints for the first and a random-hashing concentration argument for the second. It bounds the probability that a randomly chosen hash function fails the required property.

Proof of Lemma 1

  • Bob’s inability to signal to Alice, together with the no-signaling condition for Alice’s individual systems, supports the first lemma.The proof treats Bob’s systems as a single system when applying the no-signaling condition.Source: APPENDIX · p. 4

Proof of Lemma 2

  • The proof labels vector entries by (a, b, x, y) and uses bitwise product, bitwise xor, and Hamming weight notation.It expresses the entries of the tensor-product CHSH vector using these quantities.Source: APPENDIX · pp. 4–5
  • For the second lemma, a uniformly random hash makes the indicators for membership in the sets A_k independent, each with probability 2^-Ns.The proof applies a concentration bound to sums built from these indicators and bounds the associated terms using M.Source: APPENDIX · p. 5
  • The concentration argument uses an exponential expansion under a constraint on β, then combines bounds across keys and entries.The appendix states that the probability of failure is bounded across 2^Ns key values and 16^Nr entries.Source: APPENDIX · p. 5
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