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Shorter Ring Signatures from Standard Assumptions

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Public-Key Cryptography – PKC 2019 (PKC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11442))

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Abstract

Ring signatures, introduced by Rivest, Shamir and Tauman (ASIACRYPT 2001), allow to sign a message on behalf of a set of users while guaranteeing authenticity and anonymity. Groth and Kohlweiss (EUROCRYPT 2015) and Libert et al. (EUROCRYPT 2016) constructed schemes with signatures of size logarithmic in the number of users. An even shorter ring signature, of size independent from the number of users, was recently proposed by Malavolta and Schröder (ASIACRYPT 2017). However, all these short signatures are obtained relying on strong and controversial assumptions. Namely, the former schemes are both proven secure in the random oracle model while the later requires non-falsifiable assumptions.

The most efficient construction under mild assumptions remains the construction of Chandran et al. (ICALP 2007) with a signature of size \(\varTheta (\sqrt{n})\), where n is the number of users, and security is based on the Diffie-Hellman assumption in bilinear groups (the SXDH assumption in asymmetric bilinear groups).

In this work we construct an asymptotically shorter ring signature from the hardness of the Diffie-Hellman assumption in bilinear groups. Each signature comprises \(\varTheta (\root 3 \of {n})\) group elements, signing a message requires computing \(\varTheta (\root 3 \of {n})\) exponentiations, and verifying a signature requires \(\varTheta (n^{2/3})\) pairing operations. To the best of our knowledge, this is the first ring signature based on bilinear groups with \(o(\sqrt{n})\) signatures and sublinear verification complexity.

This work was funded in part by the French ANR ALAMBIC project (ANR-16-CE39-0006).

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Notes

  1. 1.

    Although any signature scheme compatible with Groth-Sahai proofs suffices (e.g. structure preserving signatures), we would rather keep it simple and stick to Boneh-Boyen signature which, since the verification key is just one group element, simplifies the notation and reduces the size of the final signature.

  2. 2.

    Even when the adversary only knows a commitment to \({\varvec{\beta }}\), as it will be in our case, g is not collision resistant. For small rings, the adversary may guess \({\varvec{\beta }}\) with non-negligible probability and solve \(\sum _i\beta _i(vk_i-vk'_i)=0\) for some non trivial \({\varvec{vk}}'\). However, this adversary is not even not aware that it has found a collision.

  3. 3.

    We elaborate more on the erasures assumption for ring signatures in the full version of this work [15].

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Acknowledgments

We thank to the anonymous reviewers for the constructive feedback. It was very useful for simplifying the SXDH-based construction. We also thanks Carla Ràfols and Mojtaba Khalili for their comments on earlier versions of this work. This work was funded in part by the French ANR ALAMBIC project (ANR-16-CE39-0006).

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González, A. (2019). Shorter Ring Signatures from Standard Assumptions. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11442. Springer, Cham. https://doi.org/10.1007/978-3-030-17253-4_4

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