  We investigate the construction of weakly-secure index codes for a sender to
send messages to multiple receivers with side information in the presence of an
eavesdropper. We derive a sufficient and necessary condition for the existence
of index codes that are secure against an eavesdropper with access to any
subset of messages of cardinality $t$, for any fixed $t$. In contrast to the
benefits of using random keys in secure network coding, we prove that random
keys do not promote security in three classes of index-coding instances.
