  In this paper, we obtain complete characterization for nested canalyzing
functions (NCFs) by obtaining its unique algebraic normal form (polynomial
form). We introduce a new concept, LAYER NUMBER for NCF. Based on this, we
obtain explicit formulas for the the following important parameters: 1) Number
of all the nested canalyzing functions, 2) Number of all the NCFs with given
LAYER NUMBER, 3) Hamming weight of any NCF, 4) The activity number of any
variable of any NCF, 5) The average sensitivity of any NCF. Based on these
formulas, we show the activity number is greater for those variables in out
layer and equal in the same layer. We show the average sensitivity attains
minimal value when the NCF has only one layer. We also prove the average
sensitivity for any NCF (No matter how many variables it has) is between 0 and
2. Hence, theoretically, we show why NCF is stable since a random Boolean
function has average sensitivity $\frac{n}{2}$. Finally we conjecture that the
NCF attain the maximal average sensitivity if it has the maximal LAYER NUMBER
$n-1$. Hence, we guess the uniform upper bound for the average sensitivity of
any NCF can be reduced to 4/3 which is tight.
