  The Stinson-Wei traceability scheme (known as traceability scheme) was
proposed for broadcast encryption as a generalization of the Chor-Fiat-Naor
traceability scheme (known as traceability code). Cover-free family was
introduced by Kautz and Singleton in the context of binary superimposed code.
In this paper, we find a new relationship between a traceability scheme and a
cover-free family, which strengthens the anti-collusion strength from $t$ to
$t^2$, that is, a $t$-traceability scheme is a $t^2$-cover-free family. Based
on this interesting discovery, we derive new upper bounds for traceability
schemes. By using combinatorial structures, we construct several infinite
families of optimal traceability schemes which attain our new upper bounds. We
also provide a constructive lower bound for traceability schemes, the size of
which has the same order with our general upper bound. Meanwhile, we consider
parent-identifying set system, an anti-collusion key-distributing scheme
requiring weaker conditions than traceability scheme but stronger conditions
than cover-free family. A new upper bound is also given for parent-identifying
set systems.
