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MuSig-L: Lattice-Based Multi-signature with Single-Round Online Phase

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

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Abstract

Multi-signatures are protocols that allow a group of signers to jointly produce a single signature on the same message. In recent years, a number of practical multi-signature schemes have been proposed in the discrete-log setting, such as MuSig2 (CRYPTO’21) and DWMS (CRYPTO’21). The main technical challenge in constructing a multi-signature scheme is to achieve a set of several desirable properties, such as (1) security in the plain public-key (PPK) model, (2) concurrent security, (3) low online round complexity, and (4) key aggregation. However, previous lattice-based, post-quantum counterparts to Schnorr multi-signatures fail to satisfy these properties.

In this paper, we introduce MuSig-L, a lattice-based multi-signature scheme simultaneously achieving these design goals for the first time. Unlike the recent, round-efficient proposal of Damgård et al. (PKC’21), which had to rely on lattice-based trapdoor commitments, we do not require any additional primitive in the protocol, while being able to prove security from the standard module-SIS and LWE assumptions. The resulting output signature of our scheme therefore looks closer to the usual Fiat–Shamir-with-abort signatures.

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Notes

  1. 1.

    Note in multi-signature every honest party behaves identically and thinks of themselves as “\(P_1\)” [7]. Other parties \(P_2, \ldots , P_n\) are called co-signers.

  2. 2.

    Observe that to avoid rejecting valid signatures due to arithmetic overflow q has to be larger than the size of the coefficients in the aggregated signature, i.e., the size of the ring has to grow linearly with \(\sqrt{n}\) too. This is inherent to additively aggregating signatures. As observed in [16], having a larger q makes \(\textsf{MSIS}_{}\) harder, but \(\textsf{MLWE}_{}\) easier. Compensating for it requires increasing N by a factor \(O\left( 1+\frac{\log n}{\log q_0}\right) \), where \(q_0\) is the modulus used in the single party case. However, one usually sets \(q>2^{20}\), which makes \(\frac{\log n}{\log q_0}\) less than 2 even for billions of users, and allows to neglect this factor in the signature size estimates.

  3. 3.

    This is not immediately evident from their analysis of the signature length. In fact, verifiability requires a signature to include the randomness used to generate the commitments. Such randomness is sampled from a discrete Gaussian of parameter s, which has to be large enough to be sampled using a trapdoor, i.e., linear in N (cf. [16, Theorem 2]) times square root of the number of parties (since the sum of n Gaussian randomness is output as a signature). This adds a factor \(O(\log (N\sqrt{n}))\) to their signature length, making it equivalent to ours.

  4. 4.

    Note that once \(b^{(j)}\)’s are simulated, finding corresponding uniform randomness \(r^{(j)}\)’s are easy assuming that the \(\textsf{Samp}\) algorithm is “sampleable” [14]. Such a property can be for example satisfied by simple CDT-based samplers.

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Acknowledgment

The authors are grateful to Claudio Orlandi for discussions in the earlier stages of this work. We thank Carsten Baum, Katharina Boudgoust, and Mark Simkin for helpful comments and discussions. Cecilia Boschini has been supported by the Università della Svizzera Italiana under the SNSF project number 182452, and by the Postdoc. Mobility grant No. P500PT_203075. Akira Takahashi has been supported by the Carlsberg Foundation under the Semper Ardens Research Project CF18-112 (BCM); the European Research Council (ERC) under the European Unions’s Horizon 2020 research and innovation programme under grant agreement No. 803096 (SPEC).

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Authors and Affiliations

  1. Technion, Haifa, Israel

    Cecilia Boschini

  2. Reichman University, Herzliya, Israel

    Cecilia Boschini

  3. Aarhus University, Aarhus, Denmark

    Akira Takahashi

  4. NTT Corporation, Tokyo, Japan

    Mehdi Tibouchi

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  1. Cecilia Boschini
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Correspondence to Akira Takahashi .

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  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Boschini, C., Takahashi, A., Tibouchi, M. (2022). MuSig-L: Lattice-Based Multi-signature with Single-Round Online Phase. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13508. Springer, Cham. https://doi.org/10.1007/978-3-031-15979-4_10

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