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Two-Round Trip Schnorr Multi-signatures via Delinearized Witnesses

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Advances in Cryptology – CRYPTO 2021 (CRYPTO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12825))

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Abstract

We construct a two-round Schnorr-based signature scheme (DWMS) by delinearizing two pre-commitments supplied by each signer. DWMS is a secure signature scheme in the algebraic group model (AGM) and the random oracle model (ROM) under the assumption of the hardness of the one-more discrete logarithm problem and the 2-entwined sum problem that we introduce in this paper. Our new m-entwined sum problem tweaks the k-sum problem in a scalar field using the associated group. We prove the hardness of our new problem in the AGM assuming the hardness of the discrete logarithm problem in the associated group. We believe that our new problem simplifies the security proofs of multi-signature schemes that use the delinearization of commitments.

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Notes

  1. 1.

    There can be a specified structure (e.g., tree structure [11]) between parties for more efficient communication.

  2. 2.

    Remark that \( \alpha _{11}^{(i)} = H_1(\mathsf {SID}^{(i)},1,1), \alpha _{12}^{(i)} = H_1(\mathsf {SID}^{(i)},1,2) \). So, the probability of having \(\alpha _{11}^{(i)} (\sum _{j =1}^{q_s} \eta ^{(i)}_{1j}y_j) =- \alpha _{12}^{(i)} (\sum _{j =1}^{q_s}\eta _{2j}^{(i)}y_j) \) given that \( T_{11}^{(i)} \ne 0, T_{11}^{(i)} \ne 0 \) is not a collision probability.

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Acknowledgement

We thank Raghav Bhaskar and Alistair Stewart for their extensive advise and extremely insightful conversations throughout the effort. We warmly thank Michele Orrù for his helpful conversations, especially around understanding the algebraic group models.

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Authors and Affiliations

  1. Web3 Foundation, Zug, Switzerland

    Handan Kılınç Alper & Jeffrey Burdges

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  1. Handan Kılınç Alper
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Correspondence to Handan Kılınç Alper .

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Editors and Affiliations

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

    Tal Malkin

  2. University of Michigan, Ann Arbor, MI, USA

    Chris Peikert

A Rank of a Random Matrix

A Rank of a Random Matrix

Assume that we have a random matrix \( \mathtt {{M}} \) of size \( (\ell \times \ell ') \). Given that \( \ell \le \ell ' \), the rank of \( \mathtt {{M}} \) can be at most \( \ell \).

Let’s define another event \( E_i \) where the first i row vectors of \( \mathtt {{M}} \) are linearly independent. In this case, which is the probability that a random vector equals to \( {\pmb {0}} \) (vector consisting of 0). In this case,

So, the probability of \( \mathtt {{M}} \)’s rank is less than \( \ell \) is at most \( \frac{\ell }{p}\).

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Kılınç Alper, H., Burdges, J. (2021). Two-Round Trip Schnorr Multi-signatures via Delinearized Witnesses. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12825. Springer, Cham. https://doi.org/10.1007/978-3-030-84242-0_7

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