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Group Signatures Without NIZK: From Lattices in the Standard Model

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Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11478))

Abstract

In a group signature scheme, users can anonymously sign messages on behalf of the group they belong to, yet it is possible to trace the signer when needed. Since the first proposal of lattice-based group signatures in the random oracle model by Gordon, Katz, and Vaikuntanathan (ASIACRYPT 2010), the realization of them in the standard model from lattices has attracted much research interest, however, it has remained unsolved. In this paper, we make progress on this problem by giving the first such construction. Our schemes satisfy CCA-selfless anonymity and full traceability, which are the standard security requirements for group signatures proposed by Bellare, Micciancio, and Warinschi (EUROCRYPT 2003) with a slight relaxation in the anonymity requirement suggested by Camenisch and Groth (SCN 2004). We emphasize that even with this relaxed anonymity requirement, all previous group signature constructions rely on random oracles or NIZKs, where currently NIZKs are not known to be implied from lattice-based assumptions. We propose two constructions that provide tradeoffs regarding the security assumption and efficiency:

  • Our first construction is proven secure assuming the standard LWE and the SIS assumption. The sizes of the public parameters and the signatures grow linearly in the number of users in the system.

  • Our second construction is proven secure assuming the standard LWE and the subexponential hardness of the SIS problem. The sizes of the public parameters and the signatures are independent of the number of users in the system.

Technically, we obtain the above schemes by combining a secret key encryption scheme with additional properties and a special type of attribute-based signature (ABS) scheme, thus bypassing the utilization of NIZKs. More specifically, we introduce the notion of indexed ABS, which is a relaxation of standard ABS. The above two schemes are obtained by instantiating the indexed ABS with different constructions. One is a direct construction we propose and the other is based on previous work.

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Notes

  1. 1.

    By LWE and SIS problems with polynomial approximation factors, we mean they are problems which are as hard as certain worst case lattice problems with polynomial approximation factor.

  2. 2.

    As mentioned in Sect. 4 of [KW18], their scheme is only publicly verifiable when considering a slightly weaker notion of zero-knowledge than the standard notion of zero-knowledge for preprocessing NIZKs. In our work, the weaker notion suffices.

  3. 3.

    Our core idea of fixing the witness can also be realized by instead embedding \(i \in I\) and a (weak) PRF seed into the ABS signing key, and using a public key encryption scheme. We provide detailed discussions on our choice of using SKEs in the full version.

  4. 4.

    Actually, the paper proposes constructions of constrained signature (CS), which is a slightly different primitive from ABS. However, this primitive readily implies ABS.

  5. 5.

    More specifically, the first scheme only achieves a so-called weakly-hiding property, where the key attribute is not leaked from a signature, but two signatures that are signed by the same user can be linked. Translated into the setting of group signature, this allows an adversary to link two different signatures by the same user, which trivially breaks anonymity.

  6. 6.

    During construction, we fix n and consider this very weak bound for one-dimensional discrete Gaussian samples for simplicity of analysis.

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Acknowledgement

The authors would like to thank Yusuke Sakai and Ai Ishida for helpful discussions and anonymous reviewers of Eurocrypt 2019 for their valuable comments. The first author was partially supported by JST CREST Grant NumberJPMJCR1302 and JSPS KAKENHI Grant Number 17J05603. The second author was supportedby JST CREST Grant No. JPMJCR1688 and JSPS KAKENHI Grant Number 16K16068.

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  1. National Institute of Advanced Industrial Science and Technology (AIST), The University of Tokyo, Tokyo, Japan

    Shuichi Katsumata

  2. National Institute of Advanced Industrial Science and Technology (AIST), Tokyo, Japan

    Shota Yamada

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  1. Shuichi Katsumata
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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Katsumata, S., Yamada, S. (2019). Group Signatures Without NIZK: From Lattices in the Standard Model. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11478. Springer, Cham. https://doi.org/10.1007/978-3-030-17659-4_11

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