Welcome to the resource topic for
**2024/256**

**Title:**

Fiat-Shamir for Bounded-Depth Adversaries

**Authors:**
Liyan Chen, Yilei Chen, Zikuan Huang, Nuozhou Sun, Tianqi Yang, Yiding Zhang

**Abstract:**

We study how to construct hash functions that can securely instantiate the Fiat-Shamir transformation against bounded-depth adversaries. The motivation is twofold. First, given the recent fruitful line of research of constructing cryptographic primitives against bounded-depth adversaries under worst-case complexity assumptions, and the rich applications of Fiat-Shamir, instantiating Fiat-Shamir hash functions against bounded-depth adversaries under worst-case complexity assumptions might lead to further applications (such as SNARG for P, showing the cryptographic hardness of PPAD, etc.) against bounded-depth adversaries. Second, we wonder whether it is possible to overcome the impossibility results of constructing Fiat-Shamir for arguments [Goldwasser, Kalai, FOCS ’03] in the setting where the depth of the adversary is bounded, given that the known impossibility results (against p.p.t. adversaries) are contrived.

Our main results give new insights for Fiat-Shamir against bounded-depth adversaries in both the positive and negative directions. On the positive side, for Fiat-Shamir for proofs with certain properties, we show that weak worst-case assumptions are enough for constructing explicit hash functions that give \mathsf{AC}^0[2]-soundness. In particular, we construct an \mathsf{AC}^0[2]-computable correlation-intractable hash family for constant-degree polynomials against \mathsf{AC}^0[2] adversaries, assuming \oplus \mathsf{L}/\mathsf{poly} \not\subseteq \widetilde{\mathsf{Sum}}_{n^{-c}} \circ\mathsf{AC}^0[2] for some c > 0. This is incomparable to all currently-known constructions, which are typically useful for larger classes and against stronger adversaries, but based on arguably stronger assumptions. Our construction is inspired by the Fiat-Shamir hash function by Peikert and Shiehian [CRYPTO ’19] and the fully-homomorphic encryption scheme against bounded-depth adversaries by Wang and Pan [EUROCRYPT ’22].

On the negative side, we show Fiat-Shamir for arguments is still impossible to achieve against bounded-depth adversaries. In particular,

• Assuming the existence of \mathsf{AC}^0[2]-computable CRHF against p.p.t. adversaries, for every poly-size hash function, there is a (p.p.t.-sound) interactive argument that is not \mathsf{AC}^0[2]-sound after applying Fiat-Shamir with this hash function.

• Assuming the existence of \mathsf{AC}^0[2]-computable CRHF against \mathsf{AC}^0[2] adversaries, there is an \mathsf{AC}^0[2]-sound interactive argument such that for every hash function computable by \mathsf{AC}^0[2] circuits the argument does not preserve \mathsf{AC}^0[2]-soundness when applying Fiat-Shamir with this hash function. This is a low-depth variant of the result of Goldwasser and Kalai.

**ePrint:**
https://eprint.iacr.org/2024/256

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