Direct Proof of Security of Wegman-Carter Authentication with Partially Known Key

TL;DR

This paper provides a direct, proof-based analysis of the Wegman-Carter authentication scheme's security when the key is partially known, demonstrating its UC-security without relying on composability theorems.

Contribution

It offers the first direct security proof for Wegman-Carter authentication with partially known keys, in both information-theoretic and UC frameworks.

Findings

Success probability bounded by +\u00a0||' success probability bound

Trace distance increases to ||' after message-tag pair

Authenticated channel indistinguishability proven without composability theorem

Abstract

Information-theoretically secure (ITS) authentication is needed in Quantum Key Distribution (QKD). In this paper, we study security of an ITS authentication scheme proposed by Wegman & Carter, in the case of partially known authentication key. This scheme uses a new authentication key in each authentication attempt, to select a hash function from an Almost Strongly Universal$_2$ hash function family. The partial knowledge of the attacker is measured as the trace distance between the authentication key distribution and the uniform distribution; this is the usual measure in QKD. We provide direct proofs of security of the scheme, when using partially known key, first in the information-theoretic setting and then in terms of witness indistinguishability as used in the Universal Composability (UC) framework. We find that if the authentication procedure has a failure probability $\epsilon$ and the authentication key has an $\epsilon'$ trace distance to the uniform, then under ITS, the adversary's success probability conditioned on an authentic message-tag pair is only bounded by $\epsilon+|\mT|\epsilon'$, where $|\mT|$ is the size of the set of tags. Furthermore, the trace distance between the authentication key distribution and the uniform increases to $|\mT|\epsilon'$ after having seen an authentic message-tag pair. Despite this, we are able to prove directly that the authenticated channel is indistinguishable from an (ideal) authentic channel (the desired functionality), except with probability less than $\epsilon+\epsilon'$. This proves that the scheme is ($\epsilon+\epsilon'$)-UC-secure, without using the composability theorem.