  In the paradigm of network coding, the information-theoretic security problem
is encountered in the presence of a wiretapper, who has capability of accessing
an unknown channel-subset in communication networks. In order to combat this
eavesdropping attack, secure network coding is introduced to prevent
information from being leaked to adversaries. For any construction of secure
linear network codes (SLNCs) over a wiretap network, the based field size is a
very important index, because it largely determines the computational and space
complexities, and further the efficiency of network transmission. Further, it
is also very important for the process of secure network coding from
theoretical research to practical applications. In the present paper, we
proposed new lower bounds on the field size for constructing SLNCs, which shows
that the field size can be reduced considerably without giving up any security
and capacity. In addition, since the obtained lower bounds depend on network
topology, how to efficiently determine them is another very important and
attractive problem for explicit constructions of SLNCs. Motivated by a desire
to develop methods with low complexity and empirical behavior to solve this
problem, we propose efficient approaches and present a series of algorithms for
implementation. Subsequently, the complexity of our algorithms is analyzed,
which shows that they are polynomial-time.
