International Association
for Cryptologic Research

We study the black-box function inversion problem, which is the problem of finding x \in [N] such that f(x) = y, given as input some challenge point y in the image of a function f : [N] \to [N], using T oracle queries to f \emph{and} preprocessed advice \sigma \in \{0,1\}^S depending on f. We prove a number of new results about this problem, as follows. 1. We show an algorithm that works for any T and S satisfying T S^2 \cdot \max\{S,T\} = \widetilde{\Theta}(N^3) In the important setting when S < T, this improves on the celebrated algorithm of Fiat and Naor [STOC, 1991], which requires T S^3 \gtrsim N^3. E.g., Fiat and Naor's algorithm is only non-trivial for S \gg N^{2/3}, while our algorithm gives a non-trivial tradeoff for any S \gg N^{1/2}. (Our algorithm and analysis are quite simple. As a consequence of this, we also give a self-contained and simple proof of Fiat and Naor's original result, with certain optimizations left out for simplicity.) 2. We observe that there is a very simple \emph{non-adaptive} algorithm (i.e., an algorithm whose i-th query x_i is chosen based entirely on \sigma and y, and not on the f(x_1),\ldots, f(x_{i-1})) that improves slightly on the trivial algorithm. It works for any T and S satisfying S = \Theta(N \log(N/T)), for example, T = N /\polylog(N), S = \Theta(N/\log \log N). This answers a question due to Corrigan-Gibbs and Kogan [TCC, 2019], who asked whether non-trivial non-adaptive algorithms exist; namely, algorithms that work with parameters T and S satisfying T + S/\log N < o(N). We also observe that our non-adaptive algorithm is what we call a \emph{guess-and-check} algorithm, that is, it is non-adaptive \emph{and} its final output is always one of the oracle queries x_1,\ldots, x_T. For guess-and-check algorithms, we prove a matching lower bound, therefore completely characterizing the achievable parameters (S,T) for this natural class of algorithms. (Corrigan-Gibbs and Kogan showed that any such lower bound for \emph{arbitrary} non-adaptive algorithms would imply new circuit lower bounds.) 3. We show equivalence between function inversion and a natural decision version of the problem in both the worst case and the average case, and similarly for functions f : [N] \to [M] with different ranges. Some of these equivalence results are deferred to the full version [ECCC, 2022]. All of the above results are most naturally described in a model with \emph{shared randomness} (i.e., random coins shared between the preprocessing algorithm and the online algorithm). However, as an additional contribution, we show (using a technique from communication complexity due to Newman [IPL, 1991]) how to generically convert any algorithm that uses shared randomness into one that does not.

This paper introduces a SNARK called Brakedown. Brakedown targets R1CS, a popular NP-complete problem that generalizes circuit-satisfiability. It is the first built system that provides a linear-time prover, meaning the prover incurs O(N) finite field operations to prove the satisfiability of an N-sized R1CS instance. Brakedown’s prover is faster, both concretely and asymptotically, than prior SNARK implementations. It does not require a trusted setup and may be post-quantum secure. Furthermore, it is compatible with arbitrary finite fields of sufficient size; this property is new among built proof systems with sublinear proof sizes. To design Brakedown, we observe that recent work of Bootle, Chiesa, and Groth (BCG, TCC 2020) provides a polynomial commitment scheme that, when combined with the linear-time interactive proof system of Spartan (CRYPTO 2020), yields linear-time IOPs and SNARKs for R1CS (a similar theoretical result was previously established by BCG, but our approach is conceptually simpler, and crucial for achieving high-speed SNARKs). A core ingredient in the polynomial commitment scheme that we distill from BCG is a linear-time encodable code. Existing constructions of such codes are believed to be impractical. Nonetheless, we design and engineer a new one that is practical in our context. We also implement a variant of Brakedown that uses Reed-Solomon codes instead of our linear-time encodable codes; we refer to this variant as Shockwave. Shockwave is not a linear-time SNARK, but it provides shorter proofs and lower verification times than Brakedown, and also provides a faster prover than prior plausibly post-quantum SNARKs.