Dynamic Threshold Signature Scheme

• Dynamic threshold signature schemes are cryptographic protocols allowing any subgroup meeting a dynamically set threshold to jointly produce a standard digital signature.
• They leverage secret sharing and distributed key generation to adaptively adjust signing parties and thresholds in response to system state and policy requirements.
• These schemes ensure robust security against adaptive adversaries while supporting recovery, dynamic participation, and seamless integration with existing verification methods.

A dynamic threshold signature scheme is a cryptographic protocol enabling a set of $n$ parties to collaboratively produce a valid digital signature such that any subgroup meeting a dynamically chosen threshold $t$ can sign, while any subset with fewer than $t$ members cannot. Unlike static threshold schemes, dynamic variants allow the threshold or the composition of the signing group to change over time, supporting scenarios with fluctuating participant availability, recovery contingencies, dynamic accountability, and policy-driven adaptation. The core requirements are that a produced signature is verifiable by the standard centralized verification algorithm of the base signature scheme, and that the scheme provides provable security against strong adversarial models, often including adaptive and malicious adversaries.

1. Mathematical Structure and Dynamic Adaptation

Dynamic threshold signature schemes extend the classical $(t, n)$ threshold setting by enabling threshold parameters and the signing set to be selected adaptively—in response to system state, policy, time, or application requirements. The key mathematical building block is secret sharing, often via Shamir’s or generalized linear secret sharing, to distribute the private signing key among $n$ participants:

$x = \sum_{i \in R} \lambda_i x_i \pmod{q}$

where $x_i$ is participant $i$’s share and $\lambda_i$ is a Lagrange coefficient depending on the current authorized signing set $R$ (possibly changing per session).

Dynamic functionality is realized through:

• Interactive distributed key generation (DKG), accommodating parties joining or leaving without global refresh.
• Dynamic aggregation of signature shares using context-specific Lagrange interpolation (e.g., recalculated for each active subset).
• On-the-fly adaptation of the threshold $t$ by adjusting which subsets can sign or by modifying verification scripts in programmable environments (e.g., Bitcoin Taproot).
• Dynamic transition over time, e.g., via time-locked scripts that relax the threshold as key retention probability decays.

An exemplary dynamic scheme in the Bitcoin ecosystem models user and adversary access probabilities as decaying exponentials, parameterized as $p(\tau) e^{-\lambda t}$ for the user and $q(\tau) e^{-\gamma t}$ for the attacker, with optimal threshold $\tau$ and lock times chosen to minimize total expected loss integrated over the contract lifetime (Ray et al., 29 Sep 2025).

2. Protocol Workflows and Core Primitives

Most practical dynamic threshold signature schemes (DSS) implement the following workflow:

• Distributed Key Generation: Key shares are assigned using MPC and/or verifiable secret sharing (VSS) without a trusted dealer. In some constructions, a recovery party is provisioned with encrypted material to support offline recovery, as in threshold ECDSA or EdDSA schemes (Battagliola et al., 2020, Battagliola et al., 2020).
• Signature Generation: For each signing session and threshold value, signers interact to produce and securely aggregate ephemeral nonces and signature shares. For ECDSA and EdDSA, this includes deterministic or pseudo-random MPC-compatible nonce generation (e.g., via “Purify”-style pseudo-random functions), MtA/MtAwc conversions, and sub-protocols to keep nonces and signature shares hidden unless the threshold is met.
• Verification: The resulting signature is valid under the original centralized verification function, thereby providing indistinguishability from standard signatures. No verification infrastructure change is required.
• Recovery and Resilience: If a subset of signers becomes unavailable, recovery parties—provisioned during key generation—can reconstruct missing shares using encrypted recovery material, ensuring continued operability.
• Dynamic Participation: Some schemes anticipate user churn by pre-allocating group structures or public parameters such that new parties can receive keys and join the protocol without key redistribution (Roman'kov, 2022).

3. Security Models and Proof Techniques

Dynamic threshold signature schemes achieve provable security by:

• Reducing the unforgeability of the distributed protocol to that of the underlying standard signature scheme, typically using simulation-based, game-hopping arguments.
• Ensuring robustness under adaptive corruptions, and for some protocols, even under “rushing” adversaries who can decide whom to corrupt based on in-flight protocol messages (Battagliola et al., 2020).
• Leveraging standard assumptions such as the strong RSA (for Paillier-based MPC), DDH, the semantic security of asymmetric encryption, properties of non-malleable commitments, and soundness/zero-knowledge of the accompanying ZKPs.
• Constructing protocols that provide “transparency” (produced signatures are indistinguishable from standard centralized signatures) and “accountability” (misbehaving parties can be detected/attributed via publicly verifiable NIZK proofs or notarized trails in consortium blockchains) (Li et al., 2023).

Security proofs must handle hybrid worlds involving semi-correct executions (proper commitment and decommitment of nonces), rewinding strategies for extracting secrets in the presence of malicious adversaries, and the complexity of managing encrypted recovery data or dynamic notary quorums.

4. Implementation Techniques and Dynamic Functionality

Dynamic adaptation within threshold signature protocols can arise by:

• Allowing the threshold parameter $t$ or the authorized signing set $R$ to change based on application policy or temporal logic (e.g., time-locked Bitcoin scripts that drop the threshold over time, $t_1 > t_2 > \cdots$ with programmable transition points $T_1, T_2, \ldots$) (Ray et al., 29 Sep 2025).
• Designing distributed key assignment so that new users can join (or leave) without global recomputation of existing keys, by leveraging decomposable group structures, hidden multipliers, or coprime subgroup orders (Roman'kov, 2022).
• Integrating programmable accountability mechanisms (e.g., DeTAPS) where the set of notaries or the threshold for revealing signer identities is adjusted at runtime, based on system events or regulatory triggers (Li et al., 2023).
• Supporting recovery via offline parties provided with encrypted key fragments, allowing signing to continue if online parties are lost or compromised (Battagliola et al., 2020, Battagliola et al., 2020).

Performance evaluations in practical deployments demonstrate that while dynamic schemes increase protocol round complexity (e.g., 4–7 rounds for ECDSA-based variants), pre-signing optimizations and efficient DKG mitigate latency, keeping online signing phases within milliseconds to seconds for typical deployment sizes (Faneela et al., 12 Mar 2025, Abdelrahman et al., 29 Jun 2025, Li et al., 2023).

5. Comparisons, Applications, and Practical Significance

Dynamic threshold signature schemes are compared along axes such as:

• Adaptability: Ability to change $t$ on demand; accommodate offline or dynamically joining/leaving signers; and reallocate authority in response to operational needs or key loss rates.
• Transparency: Full compatibility with existing signature verification infrastructure, essential for backward-compatible upgrades (e.g., integrating threshold ECDSA into CBDCs or established cryptocurrency networks) (Abdelrahman et al., 29 Jun 2025).
• Privacy and Accountability: Protocols such as DeTAPS enable public verifiability and privacy-preserving accountability for notary actions, hiding both threshold and signer identities using TEEs and NIZK proofs, in contrast to static schemes which typically expose $t$ or signing set membership (Li et al., 2023).
• Security/Performance Trade-offs: Higher thresholds improve attack resistance but increase risk of self-lockout. Controlled threshold degradation over time (dynamic contracts) achieves improved usability without sacrificing initial security (Ray et al., 29 Sep 2025). Engineering interventions (hardware wallets, fire-resistant storage) can shift optimal parameters (formally, adjusting curvature parameters $a$ and $b$ in the threshold optimization problem).

Dynamic threshold signature schemes see real-world deployment in:

• Cryptocurrency protocols (Bitcoin Taproot, Ethereum multi-signature solutions).
• Central bank digital currencies (CBDCs), often using threshold ECDSA to bolster key management (Abdelrahman et al., 29 Jun 2025).
• Blockchain accountability and access control (dynamic notary quorums, asset freezing).
• Privacy-critical petitioning or reporting where threshold-triggered decryption and anonymity are crucial (Breuer, 2 Aug 2024).

6. Future Directions and Open Challenges

Dynamic threshold signature research is currently addressing:

• The unification of classical and post-quantum settings through composable MPC gadgets and robust dynamic DKG protocols accommodating both DLP/ECDLP and lattice-based schemes (Sedghighadikolaei et al., 2023).
• Efficiency improvements, including asynchronous protocols (e.g., FROST-style) that reduce round complexity, smaller share sizes, and scalable message exchange patterns—vital for IoT and high-throughput applications (Faneela et al., 12 Mar 2025).
• Enhanced robustness and accountability in distributed systems, supporting dynamic membership, recovery from adversarial disruptions, and mechanisms for traceable misbehavior.
• Integration with programmable assets and on-chain logic, leveraging scripting languages and advanced cryptographic primitives to encode temporal and conditional threshold transitions, particularly as post-quantum primitives mature and standardization completes.

7. Representative Formulas and Scheme Summaries

Key mathematical constructs underlying dynamic threshold signature schemes include:

• Dynamic key recombination:

$$x = \sum_{i \in R} \lambda_i x_i \pmod{q}, \quad \lambda_i = \prod_{j \in R, j \ne i} \frac{-j}{i-j} \pmod{q}$$

• Threshold optimization in dynamic contracts:

$$\tau^\ast = \sqrt{\frac{b-a}{ab}}, \qquad V/2 \cdot \int_0^\infty p(\tau) e^{-\lambda t} q(\tau) e^{-\gamma t} dt + V \int_0^\infty \left[1 - p(\tau) e^{-\lambda t}\right] dt$$

• Aggregated ECDSA signing:

$s = \sum_{i=1}^{n} s_i$

• Signature verification (standard form, unchanged):

$U = u_1 B + u_2 \mathcal{Y}, \quad \text{accept if the $x$-coordinate of$U$modulo$q$equals$r$}$

These constructions ensure the adaptability, security, and flexibility at the heart of modern dynamic threshold signature schemes, underpinned by rigorous cryptographic proofs and validated through real-world deployments.