Cyclic Diagram Monoids: Structure & Cryptography

• Cyclic diagram monoids are semigroups that extend classical diagram monoids by integrating a finite cyclic group into the loop-count mechanism.
• They exhibit a unique block-normal form and an exponential representation gap, making them valuable for cryptographic key exchange protocols.
• Their framework generalizes to partition, Motzkin, and rook monoids, enhancing applications in algebra, combinatorics, and cryptographic security.

Cyclic diagram monoids are a class of semigroups constructed by introducing internal cyclic components to classical diagram monoids, most notably within the Temperley–Lieb family, but also encompassing the partition, Motzkin, and rook monoids. These structures are characterized by the incorporation of elements of arbitrary period via a cyclic group parameter, enabling high-order elements fundamental for cryptographic applications. The extension modifies the loop-counting aspect of planar diagram categories, endowing the resulting monoids with new algebraic and representation-theoretic features, including exponential growth in the minimal non-trivial representation dimension (representation gap), which is critical for platform resistance in linear-algebraic cryptanalysis (Liu, 20 Nov 2025).

1. Foundations and Definitions

The classical diagram monoids, such as the planar Temperley–Lieb monoid $\mathsf{TL}_n$, consist of non-crossing matchings between two parallel sets of $n$ points (top and bottom), with multiplication defined via concatenation and a scalar parameter $\delta$ associated with each closed loop formed. These monoids and related structures—partition, Motzkin, and rook monoids—are central in diagrammatic semigroup theory and have extensive applications in statistical mechanics, quantum algebra, and combinatorics.

The cyclic construction replaces the unbounded integer parameter tracking the number of loops (typically $\mathbb{Z}$) with a finite cyclic group $\mathbb{Z}_m$: $$m\mathsf{TL}_n := \bigl(C_n^n \times \mathbb{Z}_m\bigr) = \mathrm{End}_{\mathbf{K}}/(o^m=1)$$ Here, $o$ denotes the loop-generator, and $u_1, \ldots, u_{n-1}$ are the usual Temperley–Lieb generators, with the added relation $o^m=1$. This construction yields $m$-cyclic Temperley–Lieb monoids, the principal example of cyclic diagram monoids (Liu, 20 Nov 2025).

Under this paradigm, every element can be expressed uniquely in block form: $$o^r\,u^{[b_1,a_1]}\,u^{[b_2,a_2]}\cdots u^{[b_k,a_k]}$$ for $0 \le r < m$ and strictly increasing $a_i, b_i$, replicating the normal form for classical Temperley–Lieb monoids with the cyclic constraint on loop-generation.

2. Algebraic Structure and Normal Forms

Cyclic diagram monoids are finite due to the cyclicity imposed on the loop-count. The order of $m\mathsf{TL}_n$ is the product of the number of planar matchings (the classical count for $\mathsf{TL}_n$) and the order of the cyclic group ($m$), aligning with the relation $|m\mathsf{TL}_n| = |\mathsf{TL}_n| \cdot m$ (Liu, 20 Nov 2025).

A normal form is available for these elements, using sequences of "hook" elements $u^{[b,a]} = u_b u_{b-1} \cdots u_a$, and a power $r$ of the central loop generator $o$. The multiplication law follows that of the standard monoid, with the difference that loop concatenation increases the exponent of $o$ modulo $m$. In ordinary Temperley–Lieb and related monoids, non-unit elements are typically nilpotent, but the cyclic case yields elements of arbitrarily large period, which is central for cryptographic key exchange protocols.

Examples:

• $2\mathsf{TL}_3$ consists of ten elements (five planar 3-matchings, each with loop parity $0,1$).
• $3\mathsf{TL}_4$ contains forty-two elements (fourteen planar 4-matchings, each with a loop-count modulo 3).

These constructions generalize directly to other planar diagram monoids, for instance, by replacing the loop-count in planar partition, Motzkin, and rook monoids with a cyclic parameter of order $m$.

3. Representation Theory and $H$-Reduction

The representation theory of cyclic diagram monoids leverages Green's preorder and the $H$-reduction technique. For any finite monoid $\mathcal{S}$ over a field $K$, each simple $K\mathcal{S}$-module with apex $J(e)$ (the $\mathcal{J}$-cell containing the idempotent $e$) corresponds bijectively to simple $K\mathcal{H}(e)$-modules, where $\mathcal{H}(e)$ is the maximal subgroup at $e$.

In the $m$-cyclic Temperley–Lieb monoid, $\mathcal{TL}_n$, the $\mathcal{J}$-cells correspond to the number of through-strands in a diagram and the multiplicities of simple modules arise from the $m$-order cyclic structure imposed on each idempotent cell. Over an algebraically closed field containing a primitive $m$th root of unity, the number of simple modules indexed by apex $\mathcal{J}_k$ is exactly $m$ for each valid $k$ (Liu, 20 Nov 2025).

The dimensions of simple modules are governed by the classical Temperley–Lieb theory, with the loop parameter specialized to $\delta = \zeta_m+\zeta_m^{-1}$, where $\zeta_m$ is a fixed primitive $m$th root of unity: $$\dim_K(L(k,r)) = \dim_K(L_k^{\,\mathsf{TL}_n(\zeta_m)})$$ with explicit closed-form expressions from quantum-binomial/Chebyshev recurrence formulas (Liu, 20 Nov 2025).

4. Representation Gap and Exponential Growth

The representation gap, defined as the minimal dimension of a non-trivial simple module of the cyclic diagram monoid, quantifies the algebraic resistance to linear algebraic (matrix-based) attacks, as higher-dimensional representations cannot be efficiently exploited for cryptanalysis in monoid-based cryptographic protocols (Liu, 20 Nov 2025).

For $m\mathsf{TL}_n$ under appropriate field conditions,

$$\mathrm{gap}_K(m\mathsf{TL}_n^{k\le l}) \in \Theta\bigl(2^n\,n^{-5/2}\bigr)$$

This exponential lower bound is achieved by truncating the monoid to exclude one-dimensional simples (retaining cells $\mathcal{J}_l, \dots, \mathcal{J}_k$ for suitable $k, l$), ensuring all non-trivial simples exhibit exponential growth in dimension as $n$ increases.

The ratio of this gap to the square root of the monoid cardinality, termed the gap-ratio, decays only polynomially: $$\mathrm{gap}_K(\mathcal{S})/\sqrt{|\mathcal{S}|} \in \Omega(n^{-7/4})$$ indicating a robust resistance profile relative to monoid size.

5. Cryptographic Applications and Security Rationale

Cyclic diagram monoids address several desiderata for monoid-based cryptographic protocols:

• Existence of elements of high order (the loop generator $o$ of order $m$).
• Efficient sampling and word-computation (block-normal form yields an $O(n^2)$ word problem).
• Expansive representation gap, preventing low-dimensional linearization attacks.

In key exchange protocols, the one-way function $t \mapsto o\cdot u_{i_1}\cdots u_{i_r}$ utilizes high order and structural complexity. An attacker lacking access to non-trivial small-dimensional representations cannot feasibly reduce the combinatorial problem to tractable matrix algebra (Liu, 20 Nov 2025). The block-normal form enables rapid encoding and equality checking of elements, facilitating practical deployments.

A plausible implication is the suitability of cyclic diagram monoids as platforms for post-quantum, non-commutative cryptography, combining computational tractability, high periodicity, and provably large minimal representation dimensions.

6. Connections to Graph Endomorphism Monoids

There is an intrinsic relation between cyclic diagram monoids and the algebraic theory of graph endomorphism monoids, particularly the monoids of endomorphisms and weak endomorphisms of cycle graphs $C_n$. For $n \ge 3$, the endomorphism monoid $\mathrm{End}(C_n)$ and the weak endomorphism monoid $w\mathrm{End}(C_n)$ are formed by full transformations on the vertex set $Q_n = \{1, 2, \ldots, n\}$ respecting the cycle edge structure. These monoids are natural submonoids of the full transformation monoid and are governed by similar combinatorial and algebraic phenomena, such as enumeration (with explicit formulas for their size), minimal generating sets, regularity, and analysis via Green's relations (Dimitrova et al., 2023).

In the context of cyclic diagram monoids, analogous structural tools—partitioning into classes via kernel and image, explicit algorithmic generation, and semigroup-theoretic invariants—inform both the combinatorial and representation-theoretic approaches. This suggests deeper categorical and algebraic links between the diagrammatic and graph-endomorphism semigroup families.

7. Further Generalizations and Open Problems

The cyclic construction applies similarly to a variety of planar diagram monoids beyond Temperley–Lieb, including the planar partition, Motzkin, and rook monoids. Each admits a cyclic version by modulating associated diagrammatic parameters via finite cyclic groups. The systematic control of normal forms, representation classification via $H$-reduction, and the exponential representation gap extends to these structures, broadening the landscape of monoids with cryptographically valuable properties (Liu, 20 Nov 2025).

Ongoing research focuses on analyzing the detailed algebraic and computational properties of these cyclic variants, characterizing their automorphism groups, exploring further cryptographic primitives, and investigating connections with cellular algebras and diagrammatic category theory.

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References:

• "Representations of Cyclic Diagram Monoids" (Liu, 20 Nov 2025)
• "On monoids of endomorphisms of a cycle graph" (Dimitrova et al., 2023)