Token Reordering Mechanism

• Token reordering mechanisms are algorithmic bijections that permute tokens to optimize metrics like error minimization, hardware acceleration, and security across NLP, vision tasks, and blockchain systems.
• They employ methods such as traveling salesman heuristics and dimension reordering to cluster semantically or spatially related tokens, improving inference coherence and computational throughput.
• Applications span from stabilizing encrypted language models and syntactic parsing improvements to ensuring fairness in blockchain protocols, while addressing trade-offs in complexity and scalability.

A token reordering mechanism is any algorithmic or protocol-driven transformation that permutes the order of discrete tokens—where a "token" is a problem-specific atomic unit such as a word, a model input, a transaction, or a vector—to attain specific objectives. Token reordering has emerged as a versatile, domain-agnostic tool for reducing approximation error, exposing latent structure, unlocking hardware acceleration, or enforcing security constraints across cryptosystems, neural sequence models, language and vision tasks, and distributed ledgers.

1. Mathematical Formalization and Structural Role

A token reordering mechanism is fundamentally a bijective map $R : \{1, ..., N\} \to \{1, ..., N\}$ applied to a sequence of tokens. The effect is to produce a permuted sequence $T' = T[R(1)], ..., T[R(N)]$ while retaining full invertibility. The design of $R$ is problem-dependent:

• Metric-Space Reordering: Here, each token $i$ is associated with a point $P_i \in \mathbb{R}^d$ (e.g., LLM embeddings), and $R$ is selected to optimize a global property over $P_1, ..., P_N$ (such as minimizing cumulative adjacent token distances under a chosen metric) (Rho et al., 14 Oct 2025).
• Permutation by Structural Constraints: In certain parsing tasks, $R$ is learned to transform a sequence with potentially complex, non-local dependencies into a canonical order amenable to tractable inference or downstream algorithms (Fernández-González et al., 2021).
• Protocol-Governed Sorting: In cryptoeconomic settings, "tokens" as unique values (e.g., sequence numbers) are attached to operations (like transactions), and a deterministic reordering (often ascending sort) is enforced to achieve fairness or security (Vedula et al., 2023, Churiwala et al., 2022).

The key mathematical guarantee is the bijectivity of $R$, which ensures both reversibility and preservation of information.

2. Token Reordering in Homomorphically Encrypted LLMs

Homomorphic encryption (HE), specifically under the CKKS scheme, prohibits non-arithmetic control flow, making next-token prediction in LLMs fundamentally challenging. All comparison-based logic (e.g., argmax, sampling) must be replaced by polynomial approximations, which leads to the construction of approximate one-hot selectors $\tilde I$. Due to approximation error, $\tilde I$ is non-sparse, with mass spread over multiple tokens. If semantically dissimilar tokens are adjacent in embedding space, this error causes rapid text degeneration.

The traveling salesman-based token reordering mechanism addresses this by finding a permutation $\pi^*$ that minimizes the total adjacent cosine distance,

$$\pi^* = \arg\min_{\pi}\sum_{k=1}^{n-1} d_{\mathrm{cos}}(\bar W_{\pi(k)}, \bar W_{\pi(k+1)})$$

where $\bar W_i$ are unit-normalized embeddings (Rho et al., 14 Oct 2025). The effect is to cluster semantically related tokens in index space, ensuring that any linear combination induced by $\tilde I$ during encrypted inference yields coherent results.

The TSP is solved using the nearest-neighbor heuristic (complexity $O(n^2)$), applied once in preprocessing. Subsequent decoding under encryption operates on this reordered vocabulary, combined with post-processing of $\tilde I$ using a sharpening polynomial $PP(x) = 3x^2 - 2x^3$ to further suppress approximation noise. Theoretical bounds and experiments demonstrate a drastic reduction in text corruption and negligible degradation relative to unencrypted baselines, with corruption scores dropping by up to 50% under full ablation (Rho et al., 14 Oct 2025).

3. Pattern-Aware Token Reordering in Vision Transformers

Pattern-aware token reordering addresses inefficiencies in sparse and quantized attention for high-dimensional visual generation tasks (Zhao et al., 19 Jun 2025). In models processing video or images, attention maps exhibit dispersed, diagonal, or irregular patterns ill-suited to blockwise hardware optimization. The PARO mechanism defines a parameterized permutation $R$—selected from a finite set of frame, height, and width axes—that groups tokens with similar spatial-temporal relationships contiguously. For example,

$$R: i(f,h,w) = f\cdot (H\cdot W) + h\cdot W + w \longrightarrow h\cdot (W\cdot F) + w\cdot F + f$$

where $F$, $H$, $W$ are frame, height, and width dimensions, respectively.

Algorithmically, all 6 possible dimension orderings are evaluated using an efficiency criterion that combines block-level sparsity (the prevalence of near-zero blocks) and quantization-friendliness (local dynamic range). The best permutation is selected offline; tokens, queries, keys, and values are then permuted in memory, after which block-sparse and quantized attention can be performed with near-optimal hardware utilization (Zhao et al., 19 Jun 2025).

Empirically, this approach yields $\sim2\times$ speedups (up to $\sim9\times$ end-to-end with INT4) at negligible quality loss—e.g., PSNR changes near zero and FID deviations indistinguishable from FP16 baselines even when operating at 20–30% density (Zhao et al., 19 Jun 2025).

4. Token Reordering in Syntactic Parsing

Discontinuous constituent parsing, with non-projective dependencies, is reducible to continuous parsing via explicit token reordering. Given a sentence $w$ of length $n$ and its (generally unknown) discontinuous constituent tree $t$, a permutation $f$ is defined by an in-order traversal of $t$, rearranging $w$ into $w'$ such that $t$ on $w$ becomes projective on $w'$. The task reduces to learning $f$ from raw input.

A pointer network (BiLSTM-CNN encoder, LSTM decoder) predicts $f$, and its inverse $f^{-1}$ ensures perfect recoverability. Parsing proceeds as:

1. Pointer network computes $f$;
2. Permute $w$ into $w'$;
3. Feed $w'$ to a standard continuous parser;
4. Invert $f$ on produced terminals to yield the discontinuous parse (Fernández-González et al., 2021).

This approach attains state-of-the-art accuracy (90–95 F₁ on all constituents) and executes $\sim$10× faster than bespoke discontinuous parsers, with pointer reordering itself running at over 500 sentences per second (Fernández-González et al., 2021).

5. Token-Based Reordering in Blockchain Protocols

In distributed ledgers, token reordering mechanisms enforce transactional fairness and mitigate maximal-extractable-value (MEV) attacks. Masquerade (Vedula et al., 2023) introduces non-fungible tokens (short integer values) attached to user transactions. Tokens are purchased with an on-chain fee, then attached to subsequent transactions; within each block, all tokenized transactions are placed first in ascending token-number order, immune to builder reordering. Non-tokenized transactions follow, sorted by fees. The protocol guarantees per-block prefix-determinism for tokenized operations, reducing adversarial MEV exploits from 100% to around 30%, with honest users retaining nearly all intended profits after 10,000 rounds (Vedula et al., 2023).

The CoMMA protocol (Churiwala et al., 2022) uses cryptographically signed interaction tokens bound to user commitments. Transactions can only be executed on-chain by holders of these tokens, and token redemption must occur strictly by increasing index. Hash-based commitments hide operation intent; digital signatures from a counterparty prevent token forgery. Any attempt to reorder or front-run transactions fails, since the on-chain contract reverts redemption attempts out of sequence or for invalid tokens. The protocol imposes constant, modest gas overhead for reservation and execution ($\sim41\,000$ gas to reserve, $\sim25\,000$ gas to redeem), and robustly eliminates all MEV-linked reorderings (Churiwala et al., 2022).

6. Theoretical Analysis and Design Trade-Offs

Token reordering mechanisms are subject to computational, statistical, and protocol-level trade-offs:

• Algorithmic Complexity: TSP-inspired reorderings are NP-hard but tractable with heuristics in practical vocabularies ($O(n^2)$). Structural reorderings in tensors (as in vision models) are typically limited to a small set of axis permutations, enabling exhaustive search.
• Error Propagation: Theoretical bounds (e.g., error in $\tilde I$ under polynomial approximations) are explicitly improved by reordering that localizes approximation support within semantically or structurally contiguous intervals (Rho et al., 14 Oct 2025).
• Storage and Overhead: On-chain token reservation incurs linear storage growth, which can be mitigated by expiry or cleanup protocols (Churiwala et al., 2022).
• Flexibility: Fixed block-aligned reorderings suit static, regular tasks; dynamic or learned reorderings may address domain shift or adaptivity, but introduce implementational complexity.

These mechanisms are extensible to quantized models, approximate lookup structures, federated privacy settings, and multi-modal transformer architectures (Rho et al., 14 Oct 2025, Zhao et al., 19 Jun 2025).

7. Applications, Limitations, and Outlook

Token reordering mechanisms offer general-purpose tools for:

• Stabilizing encrypted inference in privacy-preserving generative models
• Enabling efficient, lossless blockwise sparsification and quantization in high-dimensional neural architectures
• Bridging discontinuous and continuous representations in syntactic parsing
• Enforcing cryptoeconomic fairness in distributed systems

Current limitations include the scalability of TSP solvers, generalization to arbitrary token graphs (beyond axis permutations), and protocol-induced latency or storage bloat in blockchain settings. Extending from static to dynamic or adaptive reordering paradigms, incorporating higher-order constraints or clustering, remains an open area of research (Rho et al., 14 Oct 2025, Zhao et al., 19 Jun 2025).

Token reordering thus functions as a foundational, cross-disciplinary method for structure-exposing, error-minimizing, and adversary-robust computation in machine learning and secure systems.