  Motivated by the problem of constructing bijective maps with low differential
uniformity, we introduce the notion of permutation resemblance of a function,
which looks to measure the distance a given map is from being a permutation. We
prove several results concerning permutation resemblance and show how it can be
used to produce low differentially uniform bijections. We also study the
permutation resemblance of planar functions, which over fields of odd
characteristic are known not to be bijections and to have the optimal
differential uniformity.
