  We consider a communication network where there exist wiretappers who can
access a subset of channels, called a wiretap set, which is chosen from a given
collection of wiretap sets. The collection of wiretap sets can be arbitrary.
Secure network coding is applied to prevent the source information from being
leaked to the wiretappers. In secure network coding, the required alphabet size
is an open problem not only of theoretical interest but also of practical
importance, because it is closely related to the implementation of such coding
schemes in terms of computational complexity and storage requirement. In this
paper, we develop a systematic graph-theoretic approach for improving Cai and
Yeung's lower bound on the required alphabet size for the existence of secure
network codes. The new lower bound thus obtained, which depends only on the
network topology and the collection of wiretap sets, can be significantly
smaller than Cai and Yeung's lower bound. A polynomial-time algorithm is
devised for efficient computation of the new lower bound.
