Monoidal categories, representation gap and cryptography
This addresses a security problem in cryptography for applications using noncommutative structures, but it appears incremental as it builds on existing theory like Green's relations.
The paper tackles the linear decomposition attack in cryptography by proposing to use monoids with large representations, and it provides lower bounds for representations of specific monoidal categories like Temperley-Lieb and Brauer categories.
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.