Abstract
A cryptographic watermarking scheme embeds a message into a program while preserving its functionality. Recently, a number of watermarking schemes have been proposed, which are proven secure in the sense that given one marked program, any attempt to remove the embedded message will substantially change its functionality.
In this paper, we formally initiate the study of collusion attacks for watermarking schemes, where the attacker’s goal is to remove the embedded messages given multiple copies of the same program, each with a different embedded message. This is motivated by practical scenarios, where a program may be marked multiple times with different messages.
The results of this work are twofold. First, we examine existing cryptographic watermarking schemes and observe that all of them are vulnerable to collusion attacks. Second, we construct collusion resistant watermarking schemes for various cryptographic functionalities (e.g., pseudorandom function evaluation, decryption, etc.). To achieve our second result, we present a new primitive called puncturable functional encryption scheme, which may be of independent interest.
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Notes
- 1.
In this paper, we focus on watermarking schemes for programs and only consider those with provable security against arbitrary removal strategies. We refer readers to Sect. 1.2 for an extended introduction to the area.
- 2.
In a concurrent work [GKM+19], collusion resistant watermarking schemes for public-key cryptographic primitives are presented. However, their constructions are under a relaxed notion of functionality-preserving. In this work, we achieve collusion resistance while preserving the original “statistical functionality-preserving” proposed in [CHN+16].
- 3.
One could find some punctured points via generating them from public information, but cannot distinguish a random punctured point from a random point in the input space.
- 4.
We remark that this will not affect the claimed security of \({\mathsf {WM}}_0\). The attacks only show that \({\mathsf {WM}}_0\) is not applicable in scenarios where collusion attacks are a legit threat.
- 5.
The watermarking scheme proposed in [BKS17] is constructed in a different approach, however, it cannot embed different messages into the same program.
- 6.
This seems unavoidable since circuits embedded with different messages should be run differently on some points to enable message extraction.
- 7.
Here, we assume that N is polynomial in the security parameter and will show how to remove this restriction later.
- 8.
In this work, we concentrate on schemes supporting function family \(\mathcal {F}\) of polynomial-size circuit with output space \(\{0,1\}^m\).
- 9.
The circuit \({\mathtt {Ext}}\), as well as all circuits \({\mathtt {Ext^{(\cdot )}}}\) appeared in the security proofs for \({\mathsf {WM}}\) will be padded to the same size.
- 10.
The circuit \({\mathtt {M}}\), as well as all circuits \({\mathtt {M^{(\cdot )}}}\) appeared in the security proof for \({\mathsf {WM}}\) will be padded to the same size.
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Acknowledgement
We appreciate the anonymous reviewers for their valuable suggestions. Part of this work was supported by the National Natural Science Foundation of China (Grant No. 61572294, 61602396, 61632020, U1636205), Early Career Scheme research grant (ECS Grant No. 25206317) from the Research Grant Council of Hong Kong, the Innovation and Technology Support Programme of Innovation and Technology Fund of Hong Kong (Grant No. ITS/356/17), and the MonashU-PolyU-Collinstar Capital Joint Lab on Blockchain. Junzuo Lai was supported by National Natural Science Foundation of China (Grant No. 61922036, 61572235), and Guangdong Natural Science Funds for Distinguished Young Scholar (No. 2015A030306045).
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Yang, R., Au, M.H., Lai, J., Xu, Q., Yu, Z. (2019). Collusion Resistant Watermarking Schemes for Cryptographic Functionalities. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11921. Springer, Cham. https://doi.org/10.1007/978-3-030-34578-5_14
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