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arxiv: 2606.04459 · v1 · pith:C6VJOE3Snew · submitted 2026-06-03 · 💻 cs.CR · cs.AI· cs.CC· cs.CL

Token Rankings are Unforgeable Language Model Signatures

Matthew Finlayson , Andreas Grivas , Xiang Ren , Swabha Swayamdipta This is my paper

Pith reviewed 2026-06-28 06:23 UTC · model grok-4.3

classification 💻 cs.CR cs.AIcs.CCcs.CL
keywords token rankingslanguage model signaturesunforgeable signaturesmodel stealingNP-hardnessAPI securitytop-k rankingslogit geometry
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The pith

Token rankings from language models form unique signatures that cannot be forged in polynomial time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Language models impose geometric constraints on their logit outputs that produce a distinct set of feasible top-k token rankings for each model. The paper demonstrates that these rankings identify the originating model even when an API reveals only orderings and not probability values. Because recovering a model that matches a given set of feasible rankings is NP-hard, the signature resists efficient forgery. This property enables APIs to release rankings as a verifiable identifier while limiting exposure enough to block parameter stealing when k is kept small. The k threshold needed to reveal the signature is typically lower than the threshold required to stop stealing attacks.

Core claim

Every language model possesses a unique collection of feasible top-k rankings for large enough k. Finding any other model that shares exactly the same collection is NP-hard, establishing the ranking set as the first known polynomially unforgeable signature. Rankings still permit coarse approximation of the final-layer weights, yet restricting the exposed top-k to a sufficiently small value prevents useful stealing while preserving the signature.

What carries the argument

The set of feasible top-k rankings induced by a model's logit geometry, which uniquely identifies the model and renders forgery NP-hard.

If this is right

  • APIs can release token rankings to prove model identity without enabling forgery.
  • Parameter stealing from rankings remains possible but is blocked by limiting the exposed k.
  • Signature revelation requires smaller k than the k needed to prevent stealing.
  • Rankings supply a more restrictive interface than full logits while retaining identifiability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric hardness might apply to other partial output interfaces such as top-k with ties or sampled tokens.
  • APIs could combine ranking signatures with other lightweight checks to strengthen model provenance without full logit exposure.
  • Determining the smallest k that guarantees uniqueness for a given model family would be a direct next measurement.

Load-bearing premise

A model's parameters create geometric constraints on logits that yield a set of feasible rankings both unique to that model and whose matching problem is NP-hard.

What would settle it

Finding two different models that share identical feasible top-k ranking sets for all sufficiently large k, or exhibiting a polynomial-time algorithm that constructs a model matching an observed set of rankings.

Figures

Figures reproduced from arXiv: 2606.04459 by Andreas Grivas, Matthew Finlayson, Swabha Swayamdipta, Xiang Ren.

Figure 1
Figure 1. Figure 1: We consider the setting where an language model API reveals token rankings, but not [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of feasibility boundaries (smallest [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The realization of the simple allow￾able sequence (1). Projecting the points onto a vector gives a token ordering. Rotating the vector gives a sequence of token rankings [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Checking assumptions for parameter stealing. Left: [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Approximate parameter recovery from rankings on Pythia 70M as the number of training [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Exposing the signature, but not the parameters. Top: as top- [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The probability that a randomly selected ranking can be output by a language model with [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Probability that a random top-𝑘 ranking is feasible for random 10,000×100 matrices sampled from the standard normal distribution N (0, 1). The shaded region indicates the 95% confidence interval, estimated from 30 sampled matrices for each top-𝑘 size. The data show a phase-change between 𝑘 = 9 and 𝑘 = 16, where the matrices go from high to low coverage of top-𝑘 argsorts. By definition, the Pearson correlat… view at source ↗
read the original abstract

Language model parameters are known to impose unique (to each model) geometric constraints on their logit outputs, which serves as a signature that identifies the model, but also leaks the model's final layer parameters when an API distributes logits. We investigate more restrictive APIs that expose token rankings (i.e., their ordering by probability, but not the probability values) and find that rankings also constitute a signature: every model has a unique set of feasible top-$k$ rankings for sufficiently large $k$. Furthermore, the ranking signature is the first known (polynomially) unforgeable signature, since finding a model with the same set of feasible rankings is NP-hard. On the security front, we find that token rankings are already sufficient to approximately steal the final layer of the model, similar to logits, though the approximation is too coarse to forge the signature, and can be effectively countered by restricting the API to top-$k$ tokens with sufficiently small $k$. Since the top-$k$ required to present the model signature is generally smaller than the $k$ required to prevent stealing, it is possible for an API to present an unforgeable signature without leaking model parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that language model APIs exposing only token rankings (orderings by probability, without values) still yield model-unique signatures: each model induces a distinct set of feasible top-k rankings for sufficiently large k, owing to geometric constraints on its logit outputs. It further asserts that this signature is polynomially unforgeable because recovering any model whose feasible ranking set matches a given one is NP-hard. The work also shows that rankings suffice for coarse approximation of the final layer (similar to logit leakage) but that the approximation cannot forge the signature, and that choosing a small enough k prevents stealing while still revealing the signature.

Significance. If the uniqueness and NP-hardness results hold, the contribution would be notable for providing the first polynomially unforgeable signature derived from model outputs, enabling API designs that authenticate models without parameter leakage. The separation between the k needed for signature visibility and the k needed to block stealing is a concrete, actionable observation for API security.

major comments (2)
  1. [Abstract, §3] Abstract and §3 (Uniqueness argument): the claim that geometric constraints on logits produce a unique set of feasible top-k rankings for large k is central but stated without an explicit characterization of the feasible set or a proof that distinct models cannot share the same set; a concrete example or theorem establishing injectivity is required.
  2. [Abstract, §4] Abstract and §4 (NP-hardness): the assertion that finding a model with the same feasible ranking set is NP-hard is load-bearing for the unforgeability claim, yet no reduction, hardness assumption, or proof sketch appears; the manuscript must supply the reduction from a known NP-hard problem and verify that it applies to the ranking-set matching problem as defined.
minor comments (2)
  1. [§2] Define 'feasible top-k ranking' formally at first use and clarify how k is chosen relative to vocabulary size.
  2. [§5] The experimental section on approximate stealing should report the precise k values used and the resulting approximation error to allow direct comparison with the signature k.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. The points raised correctly identify that the uniqueness and NP-hardness claims would be strengthened by explicit formal statements. We will revise the manuscript to incorporate these elements.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (Uniqueness argument): the claim that geometric constraints on logits produce a unique set of feasible top-k rankings for large k is central but stated without an explicit characterization of the feasible set or a proof that distinct models cannot share the same set; a concrete example or theorem establishing injectivity is required.

    Authors: We agree that an explicit characterization and injectivity proof would improve clarity. In the revised manuscript we will add a theorem that formally defines the feasible set of top-k rankings induced by the geometric constraints on logit vectors and proves that this set is distinct for distinct models. A concrete numerical example illustrating non-overlapping feasible sets for two different models will also be included. revision: yes

  2. Referee: [Abstract, §4] Abstract and §4 (NP-hardness): the assertion that finding a model with the same feasible ranking set is NP-hard is load-bearing for the unforgeability claim, yet no reduction, hardness assumption, or proof sketch appears; the manuscript must supply the reduction from a known NP-hard problem and verify that it applies to the ranking-set matching problem as defined.

    Authors: We acknowledge that a formal reduction is required to substantiate the NP-hardness claim. The revised version will contain an explicit polynomial-time reduction from a canonical NP-hard problem (e.g., 3-SAT) to the decision problem of whether there exists a model whose feasible ranking set matches a given target set, together with a verification that the reduction preserves the ranking-set equivalence relation used in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the stated geometric constraints that model parameters impose on logit outputs (a premise presented as known) and extends this to show that top-k rankings induce unique feasible sets for large k. The NP-hardness of recovering a model matching a given ranking set is invoked directly as the basis for unforgeability. No equations or steps reduce a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming within the provided text. The central claims remain independent of the target result and rest on external complexity support rather than internal construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that logit geometry induces unique ranking sets and on the standard complexity assumption that the matching problem is NP-hard; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Model parameters impose geometric constraints on logit outputs that determine a unique collection of feasible top-k rankings for sufficiently large k.
    Stated directly in the abstract as the basis for the signature property.
  • standard math Finding another model with an identical set of feasible rankings is NP-hard.
    Invoked to establish polynomial unforgeability.

pith-pipeline@v0.9.1-grok · 5749 in / 1277 out tokens · 34613 ms · 2026-06-28T06:23:31.519606+00:00 · methodology

discussion (0)

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