The concept of transparency order is introduced to measure the resistance of
n
,
m
-functions against multi-bit differential power analysis in the Hamming weight model, including the original transparency order (denoted by
TO
), redefined transparency order (denoted by
RTO
), and modified transparency order (denoted by
MTO
). In this paper, we firstly give a relationship between
MTO
and
RTO
and show that
RTO
is less than or equal to
MTO
for any
n
,
m
-functions. We also give a tight upper bound and a tight lower bound on
MTO
for balanced
n
,
m
-functions. Secondly, some relationships between
MTO
and the maximal absolute value of the Walsh transform (or the sum-of-squares indicator, algebraic immunity, and the nonlinearity of its coordinates) for
n
,
m
-functions are obtained, respectively. Finally, we give
MTO
and
RTO
for (4,4) S-boxes which are commonly used in the design of lightweight block ciphers, respectively.
On the Modified Transparency Order of
n
,
m
-Functions
