  In network communications, information transmission often encounters
wiretapping attacks. Secure network coding is introduced to prevent information
from being leaked to adversaries. The investigation of performance bounds on
the numbers of source symbols and random symbols are two fundamental research
problems. For an important case that each wiretap-set with cardinality not
larger than $r$, Cai and Yeung proposed a coding scheme, which is optimal in
the senses of maximizing the number of source symbols and at the same time
minimizing the number of random symbols. In this letter, we further study
achievable lower bound on the number of random key and show that it just
depends on the security constraint, and particularly, is independent to the
information amount for transmission. This implies that when the number of
transmitted source message changes, we can't reduce the number of random key to
keep the same security level. We further give an intuitive interpretation on
our result. In addition, a similar construction of secure linear network codes
is proposed, which achieves this lower bound on the number of random key no
matter how much information is transmitted. At last, we also extend our result
to imperfect security case.
