  The linear decomposition attack provides a serious obstacle to direct
applications of noncommutative groups and monoids (or semigroups) in
cryptography. To overcome this issue we propose to look at monoids with only
big representations, in the sense made precise in the paper, and undertake a
systematic study of such monoids. One of our main tools is Green's theory of
cells (Green's relations).
  A large supply of monoids is delivered by monoidal categories. We consider
simple examples of monoidal categories of diagrammatic origin, including the
Temperley-Lieb, the Brauer and partition categories, and discuss lower bounds
for their representations.
