Forgery-Resistant Ellipse Signatures

Updated 22 October 2025

Forgery-resistant ellipse signatures are cryptographic constructs leveraging elliptic curve groups or high-dimensional ellipsoids to guarantee unforgeability in digital and model authentication protocols.
They employ precise geometric and algebraic constraints—such as the elliptic curve discrete logarithm problem and affine transformations—to effectively prevent signature forgery.
Innovative implementations integrate public verifiability, blindness, hierarchical revocation, and ultra-lightweight protocols to balance security with computational efficiency.

Forgery-resistant ellipse signatures are cryptographic and geometric constructions whose security and verifiability arise from deep algebraic or geometric constraints—most commonly, the structure of the group of points on an elliptic curve or, in the case of LLMs, the embedding of model outputs onto a high-dimensional ellipsoid surface. These mechanisms underlie a variety of digital signature, blind signature, signcryption, and model-output authentication protocols, in which the resistance to forgery is either information-theoretic or computational, and the signature can be structured for direct public verifiability, hierarchical revocation, or API-level output authentication.

1. Geometric and Mathematical Foundations

Forgery-resistant ellipse signatures derive their core security from precise geometric or algebraic invariants:

Elliptic Curve Signatures: The group structure of points on an elliptic curve $E/\mathbb{F}_p$ defined by $y^2 = x^3 + ax + b$ with a large prime order subgroup means that, for a base point $G$ , all cryptographic operations (key generation, signing, verification) are performed as scalar multiplications/additions within the group, subject to the intractability of the elliptic curve discrete logarithm problem (ECDLP) (Toorani et al., 2010, Toorani et al., 2010, Abid et al., 2013, Hosseini et al., 2013, Bi et al., 2018, Yavuz et al., 2019, Cheng et al., 5 Sep 2025).
Geometric Ellipse Signatures in LLMs: Output log-probability vectors of modern LLMs, after normalization and affine transformation, are constrained to a high-dimensional ellipsoid of the form $\sum_{i,j} Q_{ij} x_i x_j + \sum_i P_i x_i = 1$ , with $Q$ and $P$ determined by the model’s secret parameters. This “ellipse signature” is naturally induced by the architecture itself and serves as a robust, self-contained signature (Finlayson et al., 15 Oct 2025).

These constraints not only define the valid set of signatures but make forging a valid tuple computationally or information-theoretically infeasible.

2. Cryptographic Constructions and Security Mechanisms

Forgery resistance is achieved differently across constructions, but common themes include:

Signcryption and Digital Signature Protocols: These schemes often use ephemeral randomness and couple the signature to both long-term secret keys and session-specific values. For example, the signature $s = t w_A - r \pmod{n}$ with verification equation $sG + R = t W_A$ ensures that forging a signature requires solving for the signer’s private key or recovering ephemeral randomness, both infeasible under ECDLP (Toorani et al., 2010, Toorani et al., 2010).
Blind Signatures (GOST-like, etc.): By embedding multiple independent randomizers (as in blinding parameters $t_1, t_2, t_3$ ) and elliptic curve operations, the signatures decouple the cryptographic evidence from the issuing event, supporting unlinkability, yet maintain unforgeability rooted in ECDLP (Hosseini et al., 2013).
High-Rank Elliptic Curves: By leveraging curves with rank $r>1$ and constructing secret keys as vectors subject to linear constraints (hyperplanes), one achieves signatures whose forgery requires solving for nontrivial relations among $r$ independent generators—a much stronger assumption and a nonstandard, but higher-dimensional, hardness basis (Cheng et al., 5 Sep 2025).
LLM Output Ellipse Signatures: The geometric constraint (logits on a unique, parameter-derived ellipsoid) denies adversaries the ability to produce authentic-looking outputs unless the secret model parameters are known, since producing “on-ellipse” vectors requires knowledge of the affine map and pre-normalized hidden states (Finlayson et al., 15 Oct 2025).

In all domains, the cryptographic hardness or geometric uniqueness underpins resistance to unauthorized output reproduction.

3. Public Verifiability, Blindness, and Advanced Features

Forgery-resistant ellipse signatures often integrate additional security and utility attributes:

Direct Public Verifiability: Certain schemes (notably the signcryption designs in (Toorani et al., 2010, Toorani et al., 2010)) allow any observer to check a signature’s validity using only public data, without access to secret keys, e.g., via verification equations such as $sG + R = t W_A$ .
Blindness: Blind signature protocols introduce extra blinding parameters that hide the link between signature issuance and the final signature, supporting privacy-preserving protocols like electronic voting and electronic cash (Hosseini et al., 2013).
Hierarchical Revocation: High-rank signatures implement group and individual revocation at the algebraic level, so an entire class (represented by a hyperplane in the secret key space) can be revoked efficiently; non-membership is then proven via zero-knowledge proofs (Cheng et al., 5 Sep 2025).
Forgery-Resistance in Model Output: For LLMs, “self-contained” ellipse signatures enable independent, per-output verification, and compact redundancy (every logprob vector can serve as evidence) (Finlayson et al., 15 Oct 2025).

These features are critical for applications requiring auditability, privacy, or fine-grained control over key and signature validity.

4. Implementation Strategies and Performance

The implementation of forgery-resistant ellipse signature schemes deals with both mathematical efficiency and operational constraints:

Efficiency: Elliptic curve–based schemes are favored for resource-constrained environments due to smaller key and signature sizes at given security levels (e.g., 160-bit EC keys ≈ 1024-bit RSA (Toorani et al., 2010, Abid et al., 2013)). They also avoid modular inversion (as in some ElGamal variants), yielding further speed and simplicity advantages (Abid et al., 2013).
Ultra-Lightweight Protocols: Meta-schemes like SEMECS offload all elliptic curve work to the key generation phase, requiring only hashing and modular arithmetic for signing. On constrained platforms (8-bit MCUs), SEMECS achieves up to 19x less battery consumption than established EC signature protocols (e.g., SchnorrQ), maintaining optimal key and signature size (32 bytes for 128-bit security) (Yavuz et al., 2019).
Multiple Curves for Diversity: MECDSA signs on $t$ independent curves and aggregates the results, mitigating single-curve backdoors. Aggregating the $r$ components yields a shorter signature compared to naive concatenation, and choosing $t=2$ is recommended for an optimal balance (Bi et al., 2018).
Geometric Model Verification: Extraction of the ellipsoid constraint from blackbox model APIs is computationally expensive: sample complexity scales as $O(d^2)$ and processing as $O(d^6)$ for $d$ -dimensional embeddings, making forgery impractical for production-scale models (Finlayson et al., 15 Oct 2025).

Implementation designs must account for the trade-offs between precomputation, signature/key size, computational complexity, and resistance to both algebraic and side-channel attacks.

5. Vulnerabilities, Mitigations, and Rigorous API Design

Despite inherent cryptographic strength, implementation vulnerabilities can undermine forgery resistance:

API Vulnerabilities (EdDSA case): The double public key signing function oracle attack shows that if an EdDSA signing API accepts arbitrary (attacker-chosen) public keys, an adversary can recover the private key with only two signature queries. This breaks unforgeability and enables unconstrained forgery; secure designs require either coupling public/private key storage or re-deriving the public key within the signing function to prevent key mismatch (Grierson et al., 2023).
Parameter Validation and Revocation: Robust schemes demand rigorous validation of all public/ephemeral curve points to avoid invalid-curve attacks (Toorani et al., 2010). Certificate management and revocation must be explicitly supported, especially in high-availability or group-revocation scenarios (Cheng et al., 5 Sep 2025).
Trade-offs in Feature Selection: For schemes supporting partial forward secrecy via one-pass protocols, there is a necessary balance between efficiency (single transmission) and the full robustness of mutual-ephemeral schemes (Toorani et al., 2010).

Thus, practical security depends on both theoretical soundness and careful attention to deployment, parameterization, and software design.

6. Broader Applications and Future Directions

Forgery-resistant ellipse signatures have broad applicability and open directions for research:

Applications: Secure mobile communications, IoT authentication, electronic voting, e-cash, digital document signing, accountability in LLM deployment, and hierarchical access management (Toorani et al., 2010, Toorani et al., 2010, Hosseini et al., 2013, Yavuz et al., 2019, Finlayson et al., 15 Oct 2025).
Extensions to Hierarchical and Attribute-Based Constraints: The use of high-rank curves and hyperplane constraint machinery suggests potential adaptation to advanced encryption methods, such as attribute-based or threshold cryptosystems (Cheng et al., 5 Sep 2025).
Model Output Fingerprinting: Ellipse signatures in LLMs enable per-output forensics, regulatory compliance checks, and legal authentication without watermarking or accessing full model weights (Finlayson et al., 15 Oct 2025).
Research on Stronger/Alternate Constraints: There is ongoing interest in developing even stronger geometric or cryptographic constraints that are harder to remove and with reduced extraction cost for large models. Hybrid approaches drawing from both algebraic and geometric invariants may offer further advancements (Finlayson et al., 15 Oct 2025).
Machine Learning–Based Forgery Detection: Deep learning architectures (e.g., SigScatNet with Siamese + scattering wavelets) show promise in automating signature forgery detection at the document/image level, and could be tailored for specialized signature structures, including ellipse-shaped handwritten marks (Chokshi et al., 2023).

These directions illustrate the role of the ellipse—both literal and metaphorical—as a robust, unforgeable signature in modern cryptosystems and model authentication frameworks.

Table: Summary of Key Forgery-Resistant Ellipse Signature Constructions

Scheme / Context	Forgery-Resistance Basis	Notable Features
Signcryption on ECC (Toorani et al., 2010)	ECDLP, ephemeral randomness	Public verifiability; efficient; forward secrecy
Blind signature (ECC GOST style) (Hosseini et al., 2013)	ECDLP, blinding randomness	Unlinkability, privacy, efficiency
High-rank curve PKS (Cheng et al., 5 Sep 2025)	Linear independence in high-rank EC group	Hierarchical revocation; ZK-based proof
MECDSA multi-curve (Bi et al., 2018)	ECDLP on multiple, user-defined curves	Backdoor resistance; short aggregated signature
Model output ellipse (Finlayson et al., 15 Oct 2025)	Geometric (hidden parameter ellipsoid)	Per-output, self-contained verification
SEMECS lightweight (Yavuz et al., 2019)	Hash commitment to precomputed EC secrets	No EC op at signing; energy efficient

Summary: Forgery-resistant ellipse signatures exploit the mathematical structure of elliptic curves or geometric transformations in prediction models to provide computationally or information-theoretically robust resistance to forgery. Modern protocols combine algebraic security, efficient key/signature sizes, and, increasingly, geometric constraints for machine learning applications. Ongoing research focuses on improving efficiency, mitigating API-level risks, and developing even stronger or more flexible signature architectures.