Monoidal categories, representation gap and cryptography
Mikhail Khovanov, Maithreya Sitaraman, Daniel Tubbenhauer
TL;DR
The paper develops a cryptography-centered framework built on monoidal categories and their associated diagrammatic monoids to resist linear decomposition attacks. It introduces a precise notion of representation gap and faithfulness, and uses Green’s cell theory to organize representations via apexes and $J$-cells, providing systematic tools to bound nontrivial representations and to identify large, potentially secure—i.e., cryptographically robust—monoids. By examining planar and symmetric diagram monoids (notably Temperley–Lieb and Brauer families) and their truncations, the authors derive concrete lower bounds for representation gaps and semisimple gaps, along with faithfulness thresholds, illustrating how to construct finite monoids with exponentially large gaps relative to size. The work emphasizes set-theoretic realizations of diagrammatic categories, leverages cell-subquotients and Gram matrices, and discusses extensions, cohomology constraints, and 2-representation directions as promising future directions, all aimed at providing cryptographically suitable candidates and a rigorous representation-theoretic foundation for their security properties. Overall, the paper offers a rigorous, representation-theoretic, and diagrammatic toolkit to assess and build monoid-based cryptographic platforms with large representation gaps and robust faithfulness, connecting monoid representation theory with cryptographic resilience in finite, two-dimensional categorical settings.
Abstract
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.