Abstract
Ring signatures allow a signer to anonymously sign on behalf of a group of users, the so-called ring; the only condition is that the signer is a member of the ring. At PKC 2007, Shacham and Waters left an open problem, “obtain a ring signature secure without random oracles and its signature size is independent of the number of signers implicated in the ring”, which has not been solved yet. In this paper, by using a powerful tool, indistinguishability obfuscator (\(\mathsf i \mathcal {O}\)), we construct a constant size ring signature scheme without random oracles and thus answer Shacham et al.’s open problem. Furthermore, we construct an identity-based ring signature scheme which also has constant signature size in the standard model. However, we stress that due to the low efficiency of the existing \(\mathsf i \mathcal {O}\) candidates, we mainly focus on the existence of the constant size ring signature schemes without random oracles, but do not care about their practicability. A shortcoming of our approach is that the ring unforgeability merely is selective but not adaptive.
Keywords
This research is supported by the National Natural Science Foundation of China (Grant No. 60970139) and the Strategic Priority Program of Chinese Academy of Sciences (Grant No. XDA06010702).
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Notes
- 1.
For ease of notation on the reader, we suppress repeated \(\mathbb {PP}\) arguments that are provided to all of the following algorithms. For example, we will write \((SK, VK)\leftarrow \mathsf{KeyGen }()\) instead of \((SK, VK) \leftarrow \mathsf{KeyGen }(\mathbb {PP})\).
- 2.
- 3.
In the beginning, \(\mathcal {A}\) does not given the keys \(S=\{VK_i\}_{i=1}^{n(\lambda )}\). In order to obtain the forgery ring \(R^{*}\), we require that \(\mathcal {A}\) submits a set of index \(I_{R^{*}}=\{i_1, \ldots , i_{|R^{*}|}\}\subseteq [n(\lambda )]\). Then after the keys \(S=\{VK_i\}_{i=1}^{n(\lambda )}\) are generated, the forgery ring \(R^{*}=\{VK_{i_1}, \ldots , VK_{i_{|R^{*}|}}\}\subseteq S\) is also obtained.
- 4.
This idea is from [9] where Boneh and Zhandry constructed a non-interactive key exchange protocol.
- 5.
The idea of our identity-based ring signature scheme is from Boneh and Zhandry’s [9] identity-based non-interactive key exchange scheme.
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The authors would like to thank anonymous reviewers for their helpful comments and suggestions.
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Tang, F., Li, H. (2015). Ring Signatures of Constant Size Without Random Oracles. In: Lin, D., Yung, M., Zhou, J. (eds) Information Security and Cryptology. Inscrypt 2014. Lecture Notes in Computer Science(), vol 8957. Springer, Cham. https://doi.org/10.1007/978-3-319-16745-9_6
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