  Here we consider an approach for fast computing the algebraic degree of
Boolean functions. It combines fast computing the ANF (known as ANF transform)
and thereafter the algebraic degree by using the weight-lexicographic order
(WLO) of the vectors of the $n$-dimensional Boolean cube. Byte-wise and bitwise
versions of a search based on the WLO and their implementations are discussed.
They are compared with the usual exhaustive search applied in computing the
algebraic degree. For Boolean functions of $n$ variables, the bitwise
implementation of the search by WLO has total time complexity $O(n.2^n)$. When
such a function is given by its truth table vector and its algebraic degree is
computed by the bitwise versions of the algorithms discussed, the total time
complexity is $\Theta((9n-2).2^{n-7})=\Theta(n.2^n)$. All algorithms discussed
have time complexities of the same type, but with big differences in the
constants hidden in the $\Theta$-notation. The experimental results after
numerous tests confirm the theoretical results - the running times of the
bitwise implementation are dozens of times better than the running times of the
byte-wise algorithms.
