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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 AC⁰[2]-soundness. In particular, we construct an AC⁰[2]-computable correlation-intractable hash family for constant-degree polynomials against AC⁰[2] adversaries, assuming ⊕L/poly ⊈ Sum̃_{n^{-c}}∘AC⁰[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 AC⁰[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 AC⁰[2]-sound after applying Fiat-Shamir with this hash function. - Assuming the existence of AC⁰[2]-computable CRHF against AC⁰[2] adversaries, there is an AC⁰[2]-sound interactive argument such that for every hash function computable by AC⁰[2] circuits, the argument does not preserve AC⁰[2]-soundness when applying Fiat-Shamir with this hash function. This is a low-depth variant of Goldwasser and Kalai.

The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [X. Chen et al., 2025; Chen et al., 2025; Chen et al., 2025]. In this paper we consider the problem of relative-error testing an unknown and arbitrary f: {0,1}ⁿ → {0,1} for the property of being a unate function, i.e. a function that is either monotone non-increasing or monotone non-decreasing in each of the n input variables. Our first result is a one-sided non-adaptive algorithm for this problem that makes Õ(log(N)/ε) samples and queries, where N = |f^{-1}(1)| is the number of satisfying assignments of the function that is being tested and the value of N is given as an input parameter to the algorithm. Building on this algorithm, we next give a one-sided adaptive algorithm for this problem that does not need to be given the value of N and with high probability makes Õ(log(N)/ε) samples and queries. We also give lower bounds for both adaptive and non-adaptive two-sided algorithms that are given the value of N up to a constant multiplicative factor. In the non-adaptive case, our lower bounds essentially match the complexity of the algorithm that we provide.