  The algebraic degree of Boolean functions (or vectorial Boolean functions) is
an important cryptographic parameter that should be computed by fast
algorithms. They work in two main ways: (1) by computing the algebraic normal
form and then searching the monomial of the highest degree in it, or (2) by
examination the algebraic properties of the true table vector of a given
function. We have already done four basic steps in the study of the first way,
and the second one has been studied by other authors. Here we represent a
method for fast computing (the fastest way we know) the algebraic degree of
Boolean functions. It is a combination of the most efficient components of
these two ways and the corresponding algorithms. The theoretical time
complexities of the method are derived in each of the cases when the Boolean
function is represented in a byte-wise or in a bitwise manner. They are of the
same type $\Theta(n.2^n)$ for a Boolean function of $n$ variables, but they
have big differences between the constants in $\Theta$-notation. The
theoretical and experimental results shown here demonstrate the advantages of
the bitwise approach in computing the algebraic degree - they are dozens of
times faster than the byte-wise approaches.
