Exponential decreasing rate of leaked information in universal random privacy amplification

We derive a new upper bound for Eve’s information in secret key generation from a common random number without communication. This bound improves on Bennett et al(1995)’s bound based on the R'enyi entropy of order 2 because the bound obtained here uses the R'enyi entropy of order $1+s$ for $s \in [0,1]$. This bound is applied to a wire-tap channel. Then, we derive an exponential upper bound for Eve’s information. Our exponent is compared with Hayashi(2006)’s exponent. For the additive case, the bound obtained here is better. The result is applied to secret key agreement by public discussion.

💡 Research Summary

The paper addresses the fundamental problem of quantifying the amount of information that an eavesdropper (Eve) can obtain when a secret key is generated from a shared random variable without any interactive communication. The classical bound, due to Bennett, Brassard, and Robert (1995), relies on the Rényi entropy of order 2 (α = 2) and yields an upper bound on Eve’s mutual information that is often loose, especially when the underlying distribution is far from uniform.

The authors propose a more general approach by employing the Rényi entropy of order 1 + s, where the parameter s can be any value in the interval