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.

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Given a chain of groups G 0 ≤ G 1 ≤ G 2 ..., we may form the corresponding chain of their representation rings, together with induction and restriction operators. We may let Res l denote the operator which restricts down l steps, and similarly for Ind l . Observe then that Ind l Res l is an operator from any particular representation ring to itself. The central question that this paper addresses is: "What happens if the Ind l Res l operator is a polynomial in the Ind Res operator?". We show that chains of wreath products {H n S n } n∈N have this property, and in particular, the polynomials that appear in the case of symmetric groups are the falling factorial polynomials. An application of this fact gives a remarkable new way to compute characters of wreath products (in particular symmetric groups) using matrix multiplication. We then consider arbitrary chains of groups, and find very rigid constraints that such a chain must satisfy in order for Ind l Res l to be a polynomial in Ind Res. Our rigid constraints justify the intuition that this property is indeed a very rare and special property.