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2019 | OriginalPaper | Chapter

Group Signatures Without NIZK: From Lattices in the Standard Model

Authors : Shuichi Katsumata, Shota Yamada

Published in: Advances in Cryptology – EUROCRYPT 2019

Publisher: Springer International Publishing

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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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previous chapter Ring Signatures: Logarithmic-Size, No Setup—from Standard Assumptions

next chapter A Modular Treatment of Blind Signatures from Identification Schemes

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.

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.

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.

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

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.

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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Title: Group Signatures Without NIZK: From Lattices in the Standard Model
Authors: Shuichi Katsumata
Shota Yamada
Publisher: Springer International Publishing
Book: Advances in Cryptology – EUROCRYPT 2019
Print ISBN: 978-3-030-17658-7

Electronic ISBN: 978-3-030-17659-4

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-17659-4_11

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