  Differentially 4-uniform permutations on $\gf_{2^{2k}}$ with high
nonlinearity are often chosen as Substitution boxes in both block and stream
ciphers. Recently, Qu et al. introduced a class of functions, which are called
preferred functions, to construct a lot of infinite families of such
permutations \cite{QTTL}. In this paper, we propose a particular type of
Boolean functions to characterize the preferred functions. On the one hand,
such Boolean functions can be determined by solving linear equations, and they
give rise to a huge number of differentially 4-uniform permutations over
$\gf_{2^{2k}}$. Hence they may provide more choices for the design of
Substitution boxes. On the other hand, by investigating the number of these
Boolean functions, we show that the number of CCZ-inequivalent differentially
4-uniform permutations over $\gf_{2^{2k}}$ grows exponentially when $k$
increases, which gives a positive answer to an open problem proposed in
\cite{QTTL}.
