Monoidal categories, representation gap and cryptography

Abstract

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.

Asymptotics in finite monoidal categories

“…More generally, the paper [CEO23b] studies, working in certain tensor categories, the growth rates of summands of categorical dimension prime to the underlying characteristic. The paper [COT23] studies the growth rate of all summands, while [KST22] studies the Schur-Weyl dual question.…”

IntroSurvey of representation theory

“…While writing this paper, a new preprint[KST22] was posted on the arXiv with the quite surprising comment that light leaves might be relevant in cryptography.…”