Monoidal categories, representation gap and cryptography

The linear decomposition attack provides a serious obstacle to direct applications of noncommutative groups and monoids 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.