Quantum Tokens for Digital Signatures

• Quantum tokens for digital signatures are quantum cryptographic primitives that encapsulate signing and verification capabilities using quantum states and inherent properties like no-cloning and measurement disturbance.
• They employ models such as BB84-type states, qandy tokens, and hidden subspaces to achieve unforgeability and non-repudiation through precise error thresholds and symmetrization protocols.
• Practical implementations range from direct quantum state transmission to QKD-generated classical keys, balancing rigorous security with current experimental feasibility.

Quantum tokens for digital signatures represent a class of primitives in quantum cryptography wherein the power to sign or verify a digital signature is encapsulated in quantum states—often produced, transmitted, and measured using protocols that exploit fundamental quantum properties such as no-cloning, basis uncertainty, and monogamy of entanglement. These tokens, which may take the form of individual qubits (BB84 states), multi-particle hidden subspaces, or classical keys generated via quantum key distribution (QKD), underpin digital signature schemes with rigorous information-theoretic security not achievable in the classical domain.

1. Foundational Concepts and Quantum Token Models

Quantum tokens exploit quantum information-theoretic phenomena to guarantee unforgeability and non-repudiation in signature schemes. Three principal models appear in the literature:

• BB84-type state tokens: Sequences of qubits prepared in bases such that only recipients with correct measurement choices obtain partial but verifiable information; utilized in QDS schemes such as (Wallden et al., 2014, Amiri et al., 2015). Distribution, symmetrization among recipients, and random-basis measurements enforce transferability and non-repudiation.
• Superpositionless tokens (Qandies): The "qandy model" (Mor et al., 2021) provides quantum tokens that encode one definite property (e.g., color or taste), with measurement in a conjugate basis yielding uniform randomness. This model abstracts the essence of complementarity and no-cloning without reference to quantum superposition, providing a classical combinatorial language for QDS.
• Quantum signing tokens based on hidden subspaces: Tokenized digital signature schemes (Ben-David et al., 2016) generalize quantum money to signatures. The secret key defines a subspace; quantum tokens are single copies of the uniform superposition over this subspace. The signing operation consumes the token, and security relies on no efficient way to projectively obtain membership in both the subspace and its dual.

In all frameworks, quantum no-cloning prevents duplicating signature tokens, and the state disturbance induced by measurement thwarts adversarial attempts at undetectable forgeries.

2. Protocol Design: Generation, Distribution, and Signing

QDS protocols with quantum tokens generally proceed in discrete phases: token/key distribution, signing, and verification/arbitration.

QDS with Qandy Tokens (Protocol P1, (Mor et al., 2021))

1. Private-Key Generation: For each message bit, Alice picks a random $n$-tuple $X_b \in \{R, G, C, V\}^n$ representing qandies encoding either color (R/G) or taste (C/V).
2. Token (Public-Key) Distribution: Two identical $n$-quandy strings $Q_b$ are prepared and sent to Bob and Charlie.
3. Symmetrization and Testing: Each recipient forwards half the unmeasured tokens to the other, then measures all remaining/received qandies in random bases. A small random set is revealed for error testing; abort occurs if measured error exceeds a threshold $s_a$.
4. Signing: Alice reveals $(b, X_b)$.
5. Verification: Each recipient checks for mismatches between their recorded measurements and Alice’s declaration. Acceptance occurs if mismatches are below thresholds $s_a$ (Bob) or $s_v$ (Charlie).
6. Arbitration: Disputes are settled by re-examination of the measurement outcomes relative to $X_b$, with threshold $s_v$ for acceptance.

QDS via QKD-Generated Tokens (Protocols P2, AWKA16, (Mor et al., 2021, Qin et al., 2024, Grasselli et al., 7 Aug 2025, Tarable et al., 2024))

• Instead of distributing quantum tokens directly, underlying QKD protocols generate correlated random keys between Alice and each recipient, which are used as "quantum tokens" in the classical signature phase. Advanced variants (e.g., raw-key QDS using likely bit strings (Qin et al., 2024)) leverage error patterns to maximize rates and minimize key consumption, sometimes involving Carter-Wegman MACs for practical signature size (Tarable et al., 2024).

Tokenized Signature Construction (Ben-David et al., 2016)

• KeyGen: Create a classical description of a hidden subspace; publish an obfuscated verifier.
• TokenGen: Prepare quantum subspace states as one-use tokens.
• Sign_token: Consume the token to obtain either a subspace or orthogonal subspace witness, used as a signature.
• Verify: Public verification by subspace-membership oracles.

3. Security Analysis: Forgery, Repudiation, and Robustness

Quantum token-based QDS achieves information-theoretic (unconditional) security by relying on explicit exponential bounds:

• Forgery: For a forging adversary to pass verification, they must guess sufficient information about a token (e.g., a qandy, BB84 state, or subspace witness). The minimum mismatch rate for an optimal forger is a function of the protocol (e.g., $p_f = 1/4$ in the qandy model). Forgery probability decays exponentially in $n$ as $\exp[-c (p_f - s_v)^2 n]$ for a protocol parameter $c$ (Mor et al., 2021, Wallden et al., 2014, Amiri et al., 2015).
• Repudiation: A dishonest signer attempting to make one recipient accept and another reject cannot control the split of information symmetrically. The probability of successful repudiation follows similar exponential decay, $\exp[-c (s_v - s_a)^2 n]$ (Mor et al., 2021).
• Robustness/Honest-Abort: The probability that honest parties abort (due to noise/channels) is exponentially small as long as error rates remain well below the accepting threshold (Mor et al., 2021, Amiri et al., 2015).

The recommended approach is to select thresholds and key sizes so that the minimal gap between honest error, acceptance, arbitration, and the theoretical forger's limit are equalized—for maximal robustness and balanced security.

Sample Parameters

Parameter	Symbol	Value	Role
# of qandies	$n$	$10^6$	Security parameter
Channel error	$p_e$	0.01	Measured in TEST
Acceptance	$s_a$	0.10	Accept if ≤ $s_a n$ mismatches
Arbitration	$s_v$	0.17	Arbitration if ≤ $s_v n$ mismatches
Forger bound	$p_f$	0.25	Min-error mismatch probability

These settings give forging and repudiation probabilities dropping well below $10^{-9}$, compatible with cryptographic requirements (Mor et al., 2021).

4. Extensions: Superpositionless, Multi-party, and Practicality

Superpositionless QDS and Qandy Model

The qandy model demonstrates that key quantum cryptographic features—complementarity, no-cloning, and measurement-incompatibility—can be expressed without explicit use of superpositions or complex amplitudes. Although the model lacks entanglement and genuinely quantum interference, it provides pedagogically powerful intuition and operational equivalence for single-particle signature primitives (Mor et al., 2021).

Resource-optimized and Multiparty Schemes

Recent work generalizes quantum token mechanisms to practical large-scale and multiparty environments. For instance, QDS over QKD networks supports scalable signature distribution with rigorous bounds on malicious coalition impact and explicit key consumption calculations (Kiktenko et al., 2021, Grasselli et al., 7 Aug 2025). In classical-quantum hybrid models, QKD-derived ephemeral keys serve as tokens for digital signatures, facilitating efficient, information-theoretic security in both transferability and verification, often with signature lengths logarithmic or linear in the document size (Tarable et al., 2024, Grasselli et al., 7 Aug 2025).

Tokenized Signature Schemes and Advanced Features

Quantum tokenized signature schemes (e.g., those based on hidden subspace states) naturally provide one-time signatures, public key verification, revocability, testability, and even functionality akin to quantum money. Their security reductions are closely tied to quantum money lower bounds and the Aaronson–Christiano hidden subspace model (Ben-David et al., 2016). Deeper functionality (e.g., revocable keys, everlasting security, and transfer protocols) can also be constructed, with rigorous proofs (oracle/hardness-limited) supporting their claimed properties.

5. Comparative Analysis: Qandy, BB84, and Advanced Quantum Tokens

Model	Token Type	Need for Superposition/Entanglement	Secure against Repudiation & Forgery?	Resource/Implementation Complexity
Qandy	Color/Taste tokens	No superposition	Yes (proved exponential)	Minimal; combinatorial rules
BB84-based	BB84 qubit strings	Single-qubit superpositions	Yes (even against coherent attacks)	Equivalent to standard BB84 QKD
Tokenized	Subspace states	High-dimensional superposition	Yes (quantum money bounds)	Requires obfuscated subspace oracles
QKD key	Random bit strings	Protocol-level quantum advantage	Yes (by QKD security)	QKD deployment prerequisite

Key distinctions:

• Qandy and BB84-based QDS schemes formalize quantum tokens without entanglement, suitable for basic unforgeable and non-repudiable signatures.
• Tokenized signatures support extensible functionality (multi-use, revocability) and tighter uncloneability but entail substantial experimental and theoretical overhead.
• QKD-generated tokens enable scalable and resource-efficient implementations, as analyzed in recent practical and multiparty schemes.

Not all quantum token schemes support entanglement or phase-dependent multi-particle operations; such primitives are necessary for quantum money protocols and more advanced cryptographic constructs (Ben-David et al., 2016).

6. Practicalities, Limitations, and Future Directions

Implementation feasibility is high for protocols based on QKD/BB84, as all required quantum operations and measurements are within capabilities of current photonic and linear-optical technology (Yin et al., 2015, Dunjko et al., 2013, Grasselli et al., 7 Aug 2025). Superpositionless and classical-quantum hybrid protocols (Qandy, GQaDS) reduce implementation barrier further, working entirely with classical combinatorial logic or ephemeral QKD keys, but are limited in representing full quantum primitives (e.g., lacking entanglement or interference). Tokenized schemes and those relying on hidden subspaces or group actions present experimental challenges but enable richer cryptographic features (revocation, testability, memory-dependent signatures) with theoretical constructions based on LWE, OWF, or group-action assumptions (Morimae et al., 2023, Ben-David et al., 2016).

Limitations include:

• Restricted Functionality: Qandy/qubit-only schemes cannot represent protocols needing phase coherence or entanglement. Protocols are typically limited to classical bit-string messages or signature rates set by underlying channel noise and quantum detection.
• Resource Consumption: Quantum-state-based protocols have signature/key lengths scaling with security parameter; newer approaches optimize signature length (e.g., Carter-Wegman MACs, likely bit strings).
• Authentication: Many efficient QDS schemes rely on information-theoretically secure authenticated classical channels; practical considerations (key recycling, channel failures) must be accounted for in deployment (Grasselli et al., 7 Aug 2025, Yin et al., 2015).
• Revocation and Testability: Recent constructions introduce quantum-key/staterevocation and testability, but public revocation remains an open problem (Morimae et al., 2023).

Ongoing research targets optimal parameterization for large-scale deployment, integration with classical post-quantum signatures, public revocation verification, and reductions in key/signature size by exploiting more efficient universal hash families and new quantum primitives.

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In sum, quantum tokens for digital signatures comprise a versatile set of primitives, models, and protocols at the intersection of quantum information theory and digital cryptography. By leveraging single-copy quantum state transmission, measurement-incompatibility, and quantum key distribution, these schemes achieve unforgeability and nonrepudiation with exponential security under minimal trust assumptions and feasible physical resources. Ongoing developments continue to extend both the theoretical foundation and practical implementation of quantum token-based digital signatures (Mor et al., 2021, Wallden et al., 2014, Ben-David et al., 2016, Qin et al., 2024, Grasselli et al., 7 Aug 2025).