  In 2007, Carlet and Ding introduced two parameters, denoted by $Nb_F$ and
$NB_F$, quantifying respectively the balancedness of general functions $F$
between finite Abelian groups and the (global) balancedness of their
derivatives $D_a F(x)=F(x+a)-F(x)$, $a\in G\setminus\{0\}$ (providing an
indicator of the nonlinearity of the functions). These authors studied the
properties and cryptographic significance of these two measures. They provided
for S-boxes inequalities relating the nonlinearity $\mathcal{NL}(F)$ to $NB_F$,
and obtained in particular an upper bound on the nonlinearity which unifies
Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the
Workshop WCC 2009 and in its postproceedings in 2011, a further study of these
parameters was made; in particular, the first parameter was applied to the
functions $F+L$ where $L$ is affine, providing more nonlinearity parameters.
  In 2010, motivated by the study of Costas arrays, two parameters called
ambiguity and deficiency were introduced by Panario \emph{et al.} for
permutations over finite Abelian groups to measure the injectivity and
surjectivity of the derivatives respectively. These authors also studied some
fundamental properties and cryptographic significance of these two measures.
Further studies followed without that the second pair of parameters be compared
to the first one.
  In the present paper, we observe that ambiguity is the same parameter as
$NB_F$, up to additive and multiplicative constants (i.e. up to rescaling). We
make the necessary work of comparison and unification of the results on $NB_F$,
respectively on ambiguity, which have been obtained in the five papers devoted
to these parameters. We generalize some known results to any Abelian groups and
we more importantly derive many new results on these parameters.
