Abstract
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 \(f : [N] \rightarrow [N]\), using T oracle queries to f 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
$$ T S^2 \cdot \max \{S,T\} = \widetilde{\varTheta }(N^3) \; . $$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.)
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2.
We observe that there is a very simple non-adaptive algorithm (i.e., an algorithm whose ith 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 \( S = \varTheta (N \log (N/T))\), for example, \(T = N /\mathrm {poly\,log}(N)\), \(S = \varTheta (N/\log \log N)\). 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 guess-and-check algorithm, that is, it is non-adaptive 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 arbitrary non-adaptive algorithms would imply new circuit lower bounds.)
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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 \(f : [N] \rightarrow [M]\) 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 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.
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Notes
- 1.
However, a big part of the reason that advice is considered to be expensive is because memory is often considered to be more expensive than computing time. Unfortunately, though our algorithm can use much less than T bits of advice, our online algorithm still must use roughly T bits of space. So, though we do show an algorithm that uses less advice, we do not show an algorithm that uses less space.
- 2.
Admittedly, this simplicity is partially (though not entirely) due to the fact that we chose not to optimize for parameters other than S and T, while Fiat and Naor were quite careful to optimize, e.g., the actual running time and space of both the query algorithm and the preprocessing algorithm. See Sect. 1.4 for more discussion.
- 3.
In fact, we also missed this algorithm. An earlier version of this paper described a much more complicated algorithm that achieves the same parameters. We are very grateful to the anonymous CRYPTO reviewer who reviewed that version and discovered the simple algorithm.
- 4.
At first, this statement might sound trivial, since we started with an algorithm that works with shared randomness r, and we seem to have converted into an algorithm with more shared randomness. The difference, however, is in the order of quantifiers. In the shared randomness model, we ask that for any function f with high probability over the randomness r, the algorithm inverts f. Here, we show that with high probability over the random strings \(r_1,\ldots , r_k\), for every function f there exists i such that the algorithm inverts f with randomness \(r_i\).
- 5.
We are oversimplifying quite a bit here and leaving out many important details. Perhaps most importantly, we are assuming here for simplicity that the DFI oracle always outputs the correct answer, while Corrigan-Gibbs and Kogan worked with a much weaker DFI oracle. They were also careful to keep the domain of the functions \(f_i\) the same as the domain of the function f, while we are not concerned with this.
- 6.
Indeed, this is the whole purpose of this rather subtle construction of g (which is only a slight variant of the construction in Fiat and Naor [12])—to provide \(\mathcal {P}'\) and \(\mathcal {A}'\) with access to a shared random function from [N] to D without requiring \(\mathcal {A}'\) to make too many queries. Notice that this is non-trivial because the set D is not known to \(\mathcal {A}'\) and might not have a succinct description. (\(\mathcal {A}'\) instead only knows the image \(\widehat{L}\) of \([N] - D\) under f.).
- 7.
The requirement of uniqueness substantially simplifies the analysis. However, it is possible to use a weaker condition.
- 8.
We remark that the result for injective functions is very similar to [8, Theorem 8]. We simply include it for completeness.
- 9.
One could reduce the latter probability of failure to 0 with an adaptive reduction, but we prefer to keep the reduction non-adaptive with a small probability of error.
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Acknowledgements
Siyao Guo was supported by National Natural Science Foundation of China Grant No. 62102260, Shanghai Municipal Education Commission (SMEC) Grant No. 0920000169, NYTP Grant No. 20121201 and NYU Shanghai Boost Fund. Spencer Peters and Noah Stephens-Davidowitz were supported in part by the NSF under Grant No. CCF-2122230. We are indebted to all reviewers of this paper, but we would like to acknowledge specifically the anonymous CRYPTO reviewer who pointed out the existence of the very simple non-adaptive algorithm.
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Golovnev, A., Guo, S., Peters, S., Stephens-Davidowitz, N. (2023). Revisiting Time-Space Tradeoffs for Function Inversion. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14082. Springer, Cham. https://doi.org/10.1007/978-3-031-38545-2_15
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