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.