Shared Secrets: Securing Digital Signatures for the Post-Quantum Era

Author: Denis Avetisyan

A new approach distributes the risk and responsibility of cryptographic key management, bolstering security for next-generation digital signatures.

This paper details a FIPS 204-compatible threshold signature scheme for ML-DSA, utilizing masking techniques and distributed key generation to achieve UC security.

Existing threshold signature schemes for post-quantum cryptography often struggle to balance functionality, efficiency, and compatibility with established standards. This paper, ‘FIPS 204-Compatible Threshold ML-DSA via Masked Lagrange Reconstruction’, introduces a practical solution enabling threshold deployment of ML-DSA, a FIPS 204-compliant signature scheme, with arbitrary thresholds and standard signature sizes. We achieve this through masked Lagrange reconstruction, addressing challenges unique to ML-DSA-including rejection sampling, the $r_0$ -check, and nonce distribution-and demonstrate its viability across three deployment profiles with rigorous security proofs. Will this approach pave the way for widespread adoption of distributed key management in secure, post-quantum systems?

The Centrality of Trust: A Systemic Vulnerability

Conventional digital signatures, while widely implemented, inherently depend on a centralized authority to validate transactions and maintain trust. This reliance introduces a critical vulnerability: a single point of failure. Should this authority be compromised – through hacking, malicious insider activity, or even natural disaster – the entire system’s integrity is threatened. Furthermore, this centralized model struggles with scalability. As the number of users and transactions increases, the authority becomes a bottleneck, slowing down processing times and increasing costs. The need to verify every transaction through a single entity limits the system’s capacity to handle large-scale applications, hindering its effectiveness in rapidly growing digital environments and motivating exploration into decentralized alternatives.

Secure multi-party computation (MPC) addresses a fundamental challenge in collaborative data analysis: enabling joint calculations without compromising individual privacy. The core principle of MPC revolves around constructing protocols that allow several parties to compute a function of their combined inputs – such as determining an average salary or running a machine learning model – while ensuring that no party learns anything about the others’ private data beyond what can be inferred from the final result. This is achieved through cryptographic techniques, including secret sharing and homomorphic encryption, which allow computations to be performed on encrypted data. The power of MPC lies in its ability to unlock the value of distributed data without necessitating a trusted third party or exposing sensitive information, fostering collaboration in scenarios where data privacy is paramount, such as financial transactions, healthcare research, and secure voting systems.

Despite significant theoretical advancements in secure multi-party computation (SMPC), translating these protocols into real-world applications remains a considerable challenge. Many existing SMPC solutions suffer from performance bottlenecks, particularly when dealing with large datasets or complex computations; the cryptographic overhead often renders them impractical for time-sensitive scenarios. Furthermore, robustness against malicious adversaries – parties intentionally attempting to compromise the computation – frequently necessitates complex and computationally expensive techniques. Achieving a balance between efficiency, scalability, and strong security guarantees is a key hurdle; current implementations often excel in only one or two of these areas, limiting their utility in demanding fields like financial modeling, privacy-preserving machine learning, and secure data analytics. Consequently, a persistent need exists for SMPC protocols that are both theoretically sound and practically deployable, capable of handling the scale and complexity of modern computational tasks.

Deconstructing the Single Point: A Distributed Signature Approach

Threshold ML-DSA addresses the vulnerability of single-signer digital signature schemes by distributing the signing capability across multiple parties. Traditional ML-DSA relies on a single private key, creating a single point of failure; compromise of this key allows for forgery of signatures. By extending ML-DSA to a multi-party computation (MPC) setting, Threshold ML-DSA eliminates this risk. The private key is never fully reconstructed, and signature generation requires the combined effort of a defined threshold of participants. This distributed approach ensures that even if some parties are compromised, the overall security of the signature scheme remains intact, preventing unauthorized signature generation.

The Distributed Key Generation (DKG) protocol employed by Threshold ML-DSA facilitates the creation of a shared cryptographic key without any single party possessing the complete key. This process involves multiple parties each contributing to the key material through a series of communication rounds. Specifically, each party generates a private key share and a corresponding public key share. These shares are then exchanged and combined according to the DKG protocol’s rules, resulting in a collective public key and individual private key shares. The collective public key is used for signature verification, while signature generation requires a sufficient number of parties to combine their private key shares, ensuring no single party can compromise the system.

Requiring collaboration from a threshold number of parties for signature generation fundamentally alters the security profile of the ML-DSA scheme. Specifically, even if fewer than the defined threshold of parties are compromised or unavailable, a valid signature cannot be produced. This mitigates the risk of a single point of failure inherent in traditional signature schemes. The threshold, denoted as t, is a pre-defined parameter; a signature requires the participation of at least t out of n total parties. This approach distributes the responsibility and risk, increasing both the security and resilience of the signature process against both malicious attacks and accidental failures.

The Anatomy of Shared Secrets: DKG Foundations

Shamir Secret Sharing is a cryptographic algorithm central to the Distributed Key Generation (DKG) protocol, enabling the creation of a shared secret – the signing key – without any single participant possessing the complete key. This is achieved by dividing the secret into $n$ parts, or shares, such that any $t$ or more of these shares can be combined to reconstruct the original secret, while fewer than $t$ shares reveal no information about it. In the context of DKG, each participant receives one share, and the threshold $t$ is determined by the desired level of fault tolerance and security. This distribution mitigates the risk of a single point of failure or compromise, as collusion amongst fewer than $t$ participants is insufficient to reconstruct the signing key.

Feldman Commitments are utilized within the Distributed Key Generation (DKG) protocol to verifiably ensure the integrity and validity of each participant’s shared key fragment before reconstruction. Each participant commits to their secret share $s_i$ by computing a commitment $C_i = H(s_i, r_i)$ , where $H$ is a cryptographic hash function and $r_i$ is a randomly generated nonce. These commitments are then exchanged among all participants. Subsequently, each participant reveals their share $s_i$ , and others can verify that the revealed share matches the previously published commitment. This process guarantees that no participant can alter their share after the commitments are made public, preventing malicious attempts to compromise the final shared key.

Pairwise-Canceling Masks improve security during distributed key generation by adding random values, or ‘masks’, to each participant’s key fragment contribution. Each participant masks their share with a unique value, but crucially, masks are exchanged and canceled out in pairs during the reconstruction phase. This process ensures that no single participant’s original contribution is directly revealed, even to other participants or an external observer. The cancellation process effectively removes the influence of individual masks, leaving only the reconstructed secret key. This mitigates the risk of a malicious participant attempting to manipulate the key or revealing information about other participants’ shares, bolstering the overall robustness of the DKG protocol against collusion and compromise.

Under the Hood: Security and Practical Considerations

The robustness of Threshold ML-DSA stems from its grounding in well-established cryptographic assumptions: Module-SIS and PRF. Module-SIS, or Module Learning with Errors over Rings, posits the difficulty of solving certain lattice problems, a cornerstone of modern post-quantum cryptography. Similarly, the Pseudorandom Function (PRF) assumption asserts that a function appears indistinguishable from truly random output, even with computational access to its internal workings. These assumptions aren’t merely theoretical constructs; they’ve withstood decades of scrutiny from the cryptographic community and underpin the security of numerous deployed systems. By building Threshold ML-DSA upon these solid foundations, the scheme inherits a substantial level of confidence against potential attacks, ensuring the reliable and secure distribution of key management and multi-party signing operations even in a post-quantum computing landscape.

The signature generation process within this scheme can, in certain instances, employ Rejection Sampling to ensure the validity of generated signatures. However, the performance overhead associated with this technique is not insurmountable; through judicious selection of cryptographic parameters – specifically, the parameters governing the distribution used in the sampling process – the probability of rejection can be minimized. This careful parameterization effectively reduces the expected number of sampling iterations required, bringing the computational cost of signature generation to a practical level suitable for real-world applications. Consequently, the scheme maintains a balance between cryptographic rigor and efficient execution, avoiding significant performance penalties despite the potential use of Rejection Sampling.

This research details a threshold signature scheme specifically designed for ML-DSA, a leading post-quantum cryptographic standard recently selected by the National Institute of Standards and Technology (NIST). The scheme facilitates distributed key management, eliminating the single point of failure inherent in traditional cryptographic systems. By dividing the private key among multiple parties, signature generation requires the cooperation of a predefined threshold of these participants, bolstering security and resilience. This approach not only enhances protection against key compromise but also enables secure multi-party signing – a crucial capability for applications demanding collaborative authentication, such as secure voting systems or decentralized financial transactions. The practical implementation presented offers a robust and efficient solution for leveraging the security benefits of ML-DSA in distributed environments.

Beyond the Horizon: Universal Composability and Future Directions

This cryptographic scheme is built upon the principles of Universal Composability (UC) security, a robust standard that extends beyond traditional security definitions. Unlike systems evaluated in isolation, UC security rigorously analyzes a protocol’s behavior when seamlessly integrated with any other protocol. This means the scheme doesn’t just offer protection on its own; it maintains its security guarantees even within a larger, complex cryptographic ecosystem. The significance lies in the assurance that composing this scheme with other secure protocols won’t inadvertently introduce vulnerabilities or weaken the overall system’s integrity, fostering trust and reliability in increasingly interconnected applications like decentralized finance and secure multi-party computation. This composability is achieved through a careful design that simulates an ideal functionality, proving that any interaction with the scheme is indistinguishable from that ideal, regardless of the surrounding protocols.

While the current scheme demonstrates robust security and foundational functionality, practical implementation at scale necessitates a dedicated focus on efficiency. Future investigations will prioritize minimizing communication overhead – the amount of data exchanged between parties – and optimizing computational performance. This includes exploring techniques such as succinct non-interactive arguments of knowledge, improved data compression methods, and parallelization strategies to reduce latency and bandwidth requirements. Addressing these performance bottlenecks is crucial for enabling widespread adoption in resource-constrained environments and ensuring seamless integration with existing large-scale systems, ultimately unlocking the full potential of this cryptographic approach for applications like decentralized finance and secure voting platforms.

The advent of Threshold Multi-Party Digital Signature schemes, like ML-DSA, represents a significant leap towards constructing cryptographic systems capable of enduring compromise and scaling to meet modern demands. By distributing the signing key among multiple parties, the scheme eliminates single points of failure, ensuring continued operation even if a subset of key holders are corrupted or unavailable. This distributed trust model is particularly crucial for applications requiring high availability and robustness, such as decentralized finance platforms where uninterrupted transaction verification is paramount, and secure electronic voting systems demanding absolute integrity and resistance to manipulation. Beyond enhanced security, threshold signatures facilitate scalability by offloading computational burdens and reducing the risk of key exposure, thereby unlocking possibilities for broader adoption in resource-constrained environments and fostering a more secure and reliable digital future.

The presented work deliberately challenges established cryptographic boundaries. It isn’t enough to simply implement a post-quantum standard like ML-DSA; the research actively probes its limits within a distributed environment. This mirrors a core tenet of scientific inquiry – understanding isn’t passive acceptance, but rigorous testing. As Henri Poincaré observed, “Mathematics is the art of giving reasons.” This paper doesn’t merely accept the reasons for ML-DSA’s security, but meticulously constructs a system-threshold signatures via masked Lagrange reconstruction-to demonstrate that security, even when the underlying rules of key management are fractured and distributed. The masking techniques are not simply added for protection, but form the very basis of testing the robustness of the core concept – distributed key generation.

What Lies Ahead?

The construction detailed within reveals, predictably, more questions than resolutions. Achieving FIPS 204 compatibility is not an endpoint, but merely a successfully navigated regulatory hurdle. The inherent trade-offs between computational cost and security-particularly within the masking schemes-demand continued scrutiny. One suspects that optimizing for performance inevitably introduces subtle vulnerabilities, a dance with entropy that will require ongoing adversarial testing beyond current UC security models.

Further investigation should not shy away from exploring alternative masking strategies, or even architectures that deliberately introduce controlled noise-a form of cryptographic camouflage. The presented scheme, while functional, remains reliant on established assumptions regarding random number generation and the absence of side-channel attacks. To truly future-proof such a system, research must probe the limits of these assumptions, embracing a philosophy where breakage is not failure, but insight.

Ultimately, the true challenge lies not in building more complex cryptographic systems, but in understanding the fundamental limits of information itself. This work serves as a useful, if temporary, structure within that larger exploration-a scaffold upon which to test, and inevitably dismantle, the prevailing order.

Original article: https://arxiv.org/pdf/2601.20917.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/