Although they have been studied for a long time, distributed signature protocols have garnered renewed interest in recent years in view of novel applications to topics like blockchains. Most recent works have focused on distributed versions of ECDSA or variants of Schnorr signatures, however, and in particular, little attention has been given to constructions based on post-quantum secure assumptions like the hardness of lattice problems. A few lattice-based threshold signature and multi-signatureschemes have been proposed in the literature, but they either rely on hash-and-sign lattice signatures (which tend to be comparatively inefficient), use expensive generic transformations, or only come with incomplete security proofs.In this paper, we construct several lattice-based distributed signing protocols with low round complexity followingthe Fiat–Shamir with Aborts (FSwA) paradigm of Lyubashevsky (Asiacrypt 2009). Our protocols can be seen as distributed variants of the fast Dilithium-G signature scheme and the full security proof can be made assuming the hardness of module SIS and LWE problems. A key step to achieving security (unexplained in some earlier papers) is to prevent the leakage that can occur when parties abort after their first message—which can inevitably happen in the Fiat–Shamir with Aborts setting. We manage to do so using homomorphic commitments.Exploiting the similarities between FSwA and Schnorr-style signatures, our approach makes the most of observations from recent advancements in the discrete log setting, such as Drijvers et al.’s seminal work on two-round multi-signatures (S&P 2019). In particular, we observe that the use of commitment not only resolves the subtle issue with aborts, but also makes it possible to realize secure two-round n-out-of-n distributed signing and multi-signature in the plain public key model, by equipping the commitment with a trapdoor feature. The construction of suitable trapdoor commitment from lattices is a side contribution of this paper.
Although they have been studied for a long time, distributed signature protocols have garnered renewed interest in recent years in view of novel applications to topics like blockchains. Most recent works have focused on distributed versions of ECDSA or variants of Schnorr signatures, however, and in particular, little attention has been given to constructions based on post-quantum secure assumptions like the hardness of lattice problems. A few lattice-based threshold signature and multi-signatureschemes have been proposed in the literature, but they either rely on hash-and-sign lattice signatures (which tend to be comparatively inefficient), use expensive generic transformations, or only come with incomplete security proofs.In this paper, we construct several lattice-based distributed signing protocols with low round complexity followingthe Fiat–Shamir with Aborts (FSwA) paradigm of Lyubashevsky (Asiacrypt 2009). Our protocols can be seen as distributed variants of the fast Dilithium-G signature scheme and the full security proof can be made assuming the hardness of module SIS and LWE problems. A key step to achieving security (unexplained in some earlier papers) is to prevent the leakage that can occur when parties abort after their first message—which can inevitably happen in the Fiat–Shamir with Aborts setting. We manage to do so using homomorphic commitments.Exploiting the similarities between FSwA and Schnorr-style signatures, our approach makes the most of observations from recent advancements in the discrete log setting, such as Drijvers et al.’s seminal work on two-round multi-signatures (S&P 2019). In particular, we observe that the use of commitment not only resolves the subtle issue with aborts, but also makes it possible to realize secure two-round n-out-of-n distributed signing and multi-signature in the plain public key model, by equipping the commitment with a trapdoor feature. The construction of suitable trapdoor commitment from lattices is a side contribution of this paper.
Although they have been studied for a long time, distributed signature protocols have garnered renewed interest in recent years in view of novel applications to topics like blockchains. Most recent works have focused on distributed versions of ECDSA or variants of Schnorr signatures; however, and in particular, little attention has been given to constructions based on post-quantum secure assumptions like the hardness of lattice problems. A few lattice-based threshold signature and multi-signature schemes have been proposed in the literature, but they either rely on hash-and-sign lattice signatures (which tend to be comparatively inefficient), use expensive generic transformations, or only come with incomplete security proofs. In this paper, we construct several lattice-based distributed signing protocols with low round complexity following the Fiat–Shamir with Aborts (FSwA) paradigm of Lyubashevsky (Asiacrypt 2009). Our protocols can be seen as distributed variants of the fast Dilithium-G signature scheme and the full security proof can be made assuming the hardness of module SIS and LWE problems. A key step to achieving security (unexplained in some earlier papers) is to prevent the leakage that can occur when parties abort after their first message—which can inevitably happen in the Fiat–Shamir with Aborts setting. We manage to do so using homomorphic commitments. Exploiting the similarities between FSwA and Schnorr-style signatures, our approach makes the most of observations from recent advancements in the discrete log setting, such as Drijvers et al.’s seminal work on two-round multi-signatures (S&P 2019). In particular, we observe that the use of commitment not only resolves the subtle issue with aborts, but also makes it possible to realize secure two-round n -out-of- n distributed signing and multi-signature in the plain public key model , by equipping the commitment with a trapdoor feature. The construction of suitable trapdoor commitment from lattices is a side contribution of this paper.
The study of digital signatures with tight reductions has attracted considerable attention over the past two decades, as such schemes inherit essentially the same quantitative hardness as the underlying computational problem. In the context of lattice-based cryptography, the GPV approach (Gentry, Peikert, and Vaikuntanathan, STOC ’08) admits a simple tight reduction from the SIS problem, but relies on preimage sampling via trapdoors, which often leads to complex and brittle implementations. By contrast, tight proofs for trapdoor-free constructions, following the Fiat–Shamir paradigm, necessarily rely on decisional assumptions, most notably the decisional LWE assumption. From the perspective of concrete security, however, reliance on a search assumption is preferable, as known search-to-decision reductions for LWE provide only weak quantitative guarantees. \ind This paper presents the first efficient lattice-based signature scheme with a tight reduction to a search assumption (namely, the hardness of the search LWE problem) that does not require any trapdoor in the scheme itself (but merely uses, instead, a trapdoor in the proof). Our construction follows the Fiat-Shamir paradigm and can be viewed as a lattice analogue of Chevallier–Mames signatures (CRYPTO ’05). Establishing security in the lattice setting, however, requires overcoming significant technical obstacles: in particular, our proof develops several new techniques to cope with the inherently weak soundness guarantees of lattice-based interactive proofs.
Threshold signature schemes enable distribution of the signature issuing capability to multiple users, to mitigate the threat of signing key compromise. Though a classic primitive, these signatures have witnessed a surge of interest in recent times due to relevance to modern applications like blockchains and cryptocurrencies. In this work, we study round-optimal threshold signatures in the post-quantum regime and improve the only known lattice-based construction by Boneh et al. [CRYPTO'18] as follows: - Efficiency. We reduce the amount of noise flooding used in the construction from 2^Ω(λ) down to √Q, where Q is the bound on the number of generated signatures and λ is the security parameter. By using lattice hardness assumptions over polynomial rings, this allows to decrease the signature bit-lengths from Õ(λ³) to Õ(λ), bringing them significantly closer to practice. Our improvement relies on a careful analysis using Rényi divergence rather than statistical distance in the security proof. - Instantiation. The construction of Boneh et al. requires a standard signature scheme to be evaluated homomorphically. To instantiate this, we provide a homomorphism-friendly variant of Lyubashevsky’s signature [EUROCRYPT '12] which achieves low circuit depth by being "rejection-free" and uses an optimal, moderate noise flooding of √Q, matching the above. - Towards Adaptive Security. The construction of Boneh et al. satisfies only selective security, where all the corrupted parties must be announced before any signing query is made. We improve this in two ways: in the Random Oracle Model, we obtain partial adaptivity where signing queries can be made before the corrupted parties are announced but the set of corrupted parties must be announced all at once. In the standard model, we obtain full adaptivity, where parties can be corrupted at any time but this construction is in a weaker pre-processing model where signers must be provided correlated randomness of length proportional to the number of signatures, in an offline preprocessing phase.
With Dilithium and Falcon , NIST selected two lattice-based signature schemes during their post-quantum standardization project. Whereas Dilithium follows the Fiat-Shamir with Aborts (Lyubashevsky, Asiacrypt’09) blueprint, Falcon can be seen as an optimized version of the GPV-paradigm (Gentry et al., STOC’06). An important question now is whether those signatures allow additional features such as the aggregation of distinct signatures. One example are sequential aggregate signature ( SAS ) schemes (Boneh et al., Eurocrypt’04) which allow a group of signers to sequentially combine signatures on distinct messages in a compressed manner. The present work first reviews the state of the art of (sequentially) aggregating lattice-based signatures, points out the insecurity of one of the existing Falcon -based SAS (Wang and Wu, PROVSEC’19), and proposes a fix for it. We then construct the first Fiat-Shamir with Aborts based SAS by generalizing existing techniques from the discrete-log setting (Chen and Zhao, ESORICS’22) to the lattice framework. Going from the pre-quantum to the post-quantum world, however, does most often come with efficiency penalties. In our work, we also meet obstacles that seem inherent to lattice-based signatures, making the resulting scheme less efficient than what one would hope for. As a result, we only achieve quite small compression rates. We compare our construction with existing lattice-based SAS which all follow the GPV-paradigm. The bottom line is that none of the schemes achieves a good compression rate so far.
With the advent of quantum computing, research into post-quantum cryptography has gained significant attention. This is a novel branch of cryptography that utilizes algorithms and protocols designed to withstand attacks from quantum computers. Lattice theory represents a promising area within post-quantum cryptographic research. Two early examples of lattice-based cryptosystems are the GGH and NTRU schemes. These schemes are based on the challenge of finding the closest vector in a lattice and differ primarily in the type of lattice used. The NTRUSign protocol was developed by combining the strengths of both schemes. In 2008, another approach to lattice signatures was introduced by a group of authors. It is based on the hash-and-sign paradigm, in which a signature for a message is generated using a trapdoor. A year later, V. Lyubashevsky proposed another method for constructing lattice-based signatures that utilizes the Fiat — Shamir transform. However, due to the nature of the underlying lattice structure, the algorithm for signature generation produces a correct signature only with a certain probability. This is due to the use of a rejection sampling for security purposes. This paper presents an overview of existing lattice-based signature construction approaches and cryptographic schemes that are based on these approaches. A comparative analysis was conducted on these schemes, identifying the advantages and disadvantages of each method. Based on the results, optimal conditions for the application of each approach have been determined.
Digital signatures and key exchange protocols are the two most important public key cryptographic primitives used in the electronic transmission of data. The goal of key exchange is to preserve the secrecy of the communication, while the goal of digital signatures is to guarantee the authenticity of the exchanged messages. Constructions of digital signature schemes based on classical mathematical assumptions appeared shortly following the invention of public key cryptography in the late 1970s. And just like with key exchange, the most efficient variants are based on number-theoretic problems that are believed to be (sub)-exponentially hard for classical machines, but are solved in polynomial time by Shor’s algorithm on a powerful-enough quantum computer. Also, like for key exchange, the most efficient constructions that we believe to be quantum safe are based on the presumed hardness of lattice problems over polynomial rings. One interesting difference between key exchange and digital signatures is that key exchange appears to inherently require that some mathematical problem be computationally hard. Digital signatures, on the other hand, can be generically constructed from any one-way function [1,2]. So even though they certainly fall into the category of public key primitives based on their usage, their existence requires much weaker assumptions. Additionally, the transformation from a one-way function to a digital signature is not too inefficient. For example, the total parameter size (public key + signature) of the SPHINCS+ scheme [3] is around 40 kB. While these sizes are larger, and signing times considerably longer, than those of signatures based on factoring or discrete log, it is still a usable scheme for many applications. And being only based on symmetric assumptions (e.g. one wayness and collision resistance of cryptographic hash functions), its security is very attractive. In order to be considered an interesting alternative to the above-mentioned signature, a scheme based on a mathematical assumption would need to have significant performance advantages. Schemes based on factoring and the discrete logarithm problem were significantly shorter and faster, and so the generic approach lay dormant for over four decades. The new quantum-safe schemes will need to have similar performance advantages if they are to be used in lieu of this safe approach. Below, we describe two techniques for constructing lattice-based digital signatures with output sizes being just a few kilobytes. On a very high level, lattice-based signature constructions follow the two known approaches for constructing classical signatures. In the first approach, the signer outputs a function f and an image y = f(x) as his public key and keeps x as his secret key. To sign a message μ, he gives a non-interactive zero-knowledge proof that he knows an x satisfying y = f(x), using the message μ to create the ‘challenge’ H(μ) for the proof (where H is a public function that maps μ to something ‘random looking’). If the function f is one way then the verifier should be convinced that the proof could have only been created by the entity who knows x. A classic example of this type of scheme is the Schnorr signature scheme [4] based on the hardness of the discrete logarithm problem. The second approach is to create a function f together with a trapdoor f−1, output f as the public key and keep f−1 as the secret key. A message μ is signed by using the secret trapdoor to create a pre-image x such that f(x) = H(μ). Again, if the function f is one way then only someone in possession of a trapdoor should be able to invert it. An example of such a construction based on a ‘factoring-like’ assumption is the RSA signature scheme [5]. The high-level ideas for lattice-based signatures follow the above blueprints, but the technical details are significantly more involved. The main reason for the complications is the different algebraic structure of the hard one-way function underlying lattice cryptography. While the domain of the function in discrete log and RSA-based one-way functions are groups, the domains in lattice-based signatures are sets that are not closed under any operation—in particular, they are elements in a group that have small norms. This crucial small norm requirement precludes us from using uniformly random masking as in Schnorr signatures or having a trapdoor for a bijective one-way function as in RSA signatures. These barriers have, nevertheless, been overcome and the resulting digital signatures are quite practical. In 2017, the US National Institute of Standards and Technology (NIST) began a ‘competition’ for a quantum-safe key exchange and digital signatures standard. At the time of this writing, this process is in the third round and there are two lattice-based signatures remaining—each following one of the above high-level designs. The CRYSTALS-Dilithium [6] scheme follows the Schnorr framework, but adds a crucial rejection-sampling step to keep the size of the coefficients small. The FALCON scheme [7] utilizes a randomized trapdoor sampling technique that uses a secret trapdoor for f−1 to produce random pre-images from a particular distribution. Because there is no longer a bijection, it is crucial to also have the property that the distribution of the outputted pre-images does not leak information about the trapdoor. Both schemes are relatively fast and their parameters (public key + signature size) are the shortest of all quantum-safe signature schemes. While both schemes are based on lattices, they have rather different characteristics. FALCON has very short parameters (see Table 1), but entails a rather complicated procedure for signature generation. In particular, it uses (an optimized version of) the GPV sampler [8], which requires floating-point arithmetic with approximately 64 bits of precision. Requiring such high precision means that subtle implementation errors may not get detected even with rigorous testing. Dilithium, on the other hand, has larger parameters but a very simple implementation where all the sampling in the signing is done in a power-of-2 range; it is thus much less prone to implementation errors. It is quite possible that in the future both schemes will be used for different applications. Approximate parameter sizes (in bytes) for the CRYSTALS-Dilithium and FALCON digital signature schemes at approximately 128-bit security levels. Approximate parameter sizes (in bytes) for the CRYSTALS-Dilithium and FALCON digital signature schemes at approximately 128-bit security levels. Conflict of interest statement. None declared.