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

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 Universal2 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 ε and the authentication key has an ε′ trace distance to the uniform, then under ITS, the adversary’s success probability conditioned on an authentic message-tag pair is only bounded by ε + |T |ε′, where |T | is the size of the set of tags. Furthermore, the trace distance between the authentication key distribution and the uniform increases to |T |ε′ 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 ε+ ε′. This proves that the scheme is (ε + ε′)-UC-secure, without using the composability theorem.