  Boolean nested canalizing functions (NCFs) have important applications in
molecular regulatory networks, engineering and computer science. In this paper,
we study their certificate complexity. For both Boolean values $b\in\{0,1\}$,
we obtain a formula for $b$-certificate complexity and consequently, we develop
a direct proof of the certificate complexity formula of an NCF. Symmetry is
another interesting property of Boolean functions and we significantly simplify
the proofs of some recent theorems about partial symmetry of NCFs. We also
describe the algebraic normal form of $s$-symmetric NCFs. We obtain the general
formula of the cardinality of the set of $n$-variable $s$-symmetric Boolean
NCFs for $s=1,\dots,n$. In particular, we enumerate the strongly asymmetric
Boolean NCFs.
