Abstract
Zero-knowledge proofs (ZKP) are a fundamental notion in modern cryptography and an essential building block for countless privacy-preserving constructions. Recent years have witnessed a rapid development in the designs of ZKP for general statements, in which, for a publicly given Boolean function \(f: \{0,1\}^n \rightarrow \{0,1\}\), one’s goal is to prove knowledge of a secret input \(\mathbf {x} \in \{0,1\}^n\) satisfying \(f(\mathbf {x}) = b\), for a given bit b. Nevertheless, in many interesting application scenarios, not only the input \(\mathbf {x}\) but also the underlying function f should be kept private. The problem of designing ZKP for the setting where both \(\mathbf {x}\) and f are hidden, however, has not received much attention.
This work addresses the above-mentioned problem for the class of symmetric Boolean functions, namely, Boolean functions f whose output value is determined solely by the Hamming weight of the n-bit input \(\mathbf {x}\). Although this class might sound restrictive, it has exponential cardinality \(2^{n+1}\) and captures a number of well-known Boolean functions, such as threshold, sorting and counting functions. Specifically, with respect to a commitment scheme secure under the Learning-Parity-with-Noise (LPN) assumption, we show how to prove in zero-knowledge that \(f(\mathbf {x}) = b\), for a committed symmetric function f, a committed input \(\mathbf {x}\) and a bit b. The security of our protocol relies on that of an auxiliary commitment scheme which can be instantiated under quantum-resistant assumptions (including LPN). The protocol also achieves reasonable communication cost: the variant with soundness error \(2^{-\lambda }\) has proof size \(c\cdot \lambda \cdot n\), where c is a relatively small constant. The protocol can potentially find appealing privacy-preserving applications in the area of post-quantum cryptography, and particularly in code-based cryptography.
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Notes
- 1.
The size of \(\mathsf {UC}\) is \(t \cdot \log |C|\) times larger than the size C of the original circuit for computing f, where t could be a non-small constant.
- 2.
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Acknowledgements
We thank the anonymous reviewers of PQCrypto 2021 for helpful comments. The research is supported by Singapore Ministry of Education under Research Grant MOE2019-T2-2-083 and by A*STAR, Singapore under research grant SERC A19E3b0099. Duong Hieu Phan was supported in part by the French ANR ALAMBIC (ANR16-CE39-0006).
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Ling, S., Nguyen, K., Phan, D.H., Tang, H., Wang, H. (2021). Zero-Knowledge Proofs for Committed Symmetric Boolean Functions. In: Cheon, J.H., Tillich, JP. (eds) Post-Quantum Cryptography. PQCrypto 2021. Lecture Notes in Computer Science(), vol 12841. Springer, Cham. https://doi.org/10.1007/978-3-030-81293-5_18
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