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Journal of Cryptology
Erschienen in: Journal of Cryptology 2/2022

01.04.2022 | Research Article

Two-Round n-out-of-n and Multi-Signatures and Trapdoor Commitment from Lattices

verfasst von: Ivan Damgård, Claudio Orlandi, Akira Takahashi, Mehdi Tibouchi

Erschienen in: Journal of Cryptology | Ausgabe 2/2022

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Abstract

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.
Vorheriger Artikel TinyKeys: A New Approach to Efficient Multi-Party Computation

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Fußnoten
1
It is still an open question whether the aborts issue can instead be resolved by careful parameter choice, allowing to simulate the rejected transcripts without any additional assumptions. But we are aware of on-going work in this direction. If the question is answered in the affirmative our three-round protocol could be proven secure even without a commitment. However, the use of homomorphic commitment is crucial for constructing our new two-round protocols, which is our main contribution.
 
2
We include this for completeness since, while the three-round protocol itself is not novel, to the best of our knowledge there has been no publicly available complete security proof solely relying on Module-LWE.
 
3
We remark that the “commitments” generated by \({\mathsf {H}}_1\) and \({\mathsf {H}}_2\) in Fig. 6 are not randomized, and therefore they are not hiding. In our protocol, however, all committed values have high min-entropy and this is indeed sufficient for the security proof to hold. Alternatively, one could cheaply turn them into full-fledged secure and extractable commitments by additionally hashing random strings that are to be sent out during the opening phase [80].
 
4
To be more precise, since the verification bound scales as \(n^{3/2}\), one should also increase q by the same bound to avoid arithmetic overflow. This makes the \(\mathsf {MSIS}_{}\) problem harder, but the \(\mathsf {MLWE}_{}\) easier if the dimension is kept unchanged. To keep the same security level, one should therefore also increase N by a factor of \(1+O(\frac{\log n}{\log q_0})\) where \(q_0\) is the value of q in the single-user setting. Therefore, one could in principle argue that signature size actually scales as \(O(\log ^2 n)\). However, one typically chooses \(q_0 > 2^{20}\), and therefore even in settings with billions of parties, \(\frac{\log n}{\log q_0} < 2\). Thus, one can effectively regard N as independent of n.
 
5
This condition could be actually relaxed somewhat by applying the result due to Nguyen [76]
 
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Metadaten
Titel
Two-Round n-out-of-n and Multi-Signatures and Trapdoor Commitment from Lattices
verfasst von
Ivan Damgård
Claudio Orlandi
Akira Takahashi
Mehdi Tibouchi
Publikationsdatum
01.04.2022
Verlag
Springer US
Erschienen in
Journal of Cryptology / Ausgabe 2/2022
Print ISSN: 0933-2790
Elektronische ISSN: 1432-1378
DOI
https://doi.org/10.1007/s00145-022-09425-3

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