Key Distillation and the Secret-Bit Fraction

TL;DR

This paper introduces a new measure called the secret-bit fraction to evaluate the quality of secret key distillation from noisy correlations, establishing conditions for distillability and providing explicit formulas in certain cases.

Contribution

It defines the maximal secret-bit fraction Lambda[P_ABE], proves its properties as a secrecy monotone, and derives conditions and formulas for secret key distillation from single-copy distributions.

Findings

Lambda[P_ABE] is a secrecy monotone.

If Lambda[P_ABE]>1/2, the distribution is distillable.

Explicit formulas for Lambda[P_ABE] when eavesdropper is decoupled and data is binary.

Abstract

We consider distillation of secret bits from partially secret noisy correlations P_ABE, shared between two honest parties and an eavesdropper. The most studied distillation scenario consists of joint operations on a large number of copies of the distribution (P_ABE)^N, assisted with public communication. Here we consider distillation with only one copy of the distribution, and instead of rates, the 'quality' of the distilled secret bits is optimized, where the 'quality' is quantified by the secret-bit fraction of the result. The secret-bit fraction of a binary distribution is the proportion which constitutes a secret bit between Alice and Bob. With local operations and public communication the maximal extractable secret-bit fraction from a distribution P_ABE is found, and is denoted by Lambda[P_ABE]. This quantity is shown to be nonincreasing under local operations and public communication, and nondecreasing under eavesdropper's local operations: it is a secrecy monotone. It is shown that if Lambda[P_ABE]>1/2 then P_ABE is distillable, thus providing a sufficient condition for distillability. A simple expression for Lambda[P_ABE] is found when the eavesdropper is decoupled, and when the honest parties' information is binary and the local operations are reversible. Intriguingly, for general distributions the (optimal) operation requires local degradation of the data.