Abstract
Addition of n inputs is often the easiest nontrivial function to compute securely. Motivated by several open questions, we ask what can be computed securely given only an oracle that computes the sum. Namely, what functions can be computed in a model where parties can only encode their input locally, then sum up the encodings over some Abelian group \({\mathbb G}\), and decode the result to get the function output.
An additive randomized encoding (ARE) of a function \(f(x_1,\ldots ,x_n)\) maps every input \(x_i\) independently into a randomized encoding \(\hat{x}_i\), such that \(\sum _{i=1}^n\) \(\hat{x}_i\) reveals \(f(x_1,\ldots ,x_n)\) and nothing else about the inputs. In a robust ARE, the sum of any subset of the \(\hat{x}_i\) only reveals the residual function obtained by restricting the corresponding inputs.
We obtain positive and negative results on ARE. In particular:
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Information-theoretic ARE. We fully characterize the 2-party functions \(f:X_1\times X_2\rightarrow \{0,1\}\) admitting a perfectly secure ARE. For \(n\ge 3\) parties, we show a useful “capped sum” function that separates statistical security from perfect security.
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Computational ARE. We present a general feasibility result, showing that all functions can be computed in this model, under a standard hardness assumption in bilinear groups. We also describe a heuristic lattice-based construction.
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Robust ARE. We present a similar feasibility result for robust computational ARE based on ideal obfuscation along with standard cryptographic assumptions.
We then describe several applications of ARE and the above results.
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Under a standard cryptographic assumption, our computational ARE schemes imply the feasibility of general non-interactive secure computation in the shuffle model, where messages from different parties are shuffled. This implies a general utility-preserving compiler from differential privacy in the central model to computational differential privacy in the (non-robust) shuffle model.
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The existence of information-theoretic robust ARE implies “best-possible” information-theoretic MPC protocols (Halevi et al., TCC 2018) and degree-2 multiparty randomized encodings (Applebaum et al., TCC 2018). This yields new positive results for specific functions in the former model, as well as a simple unifying barrier for obtaining negative results in both models.
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Notes
- 1.
- 2.
In the following we will not refer to robustness, though all of our ARE constructions in the information-theoretic setting are in fact robust.
- 3.
In the context of information-theoretic AREs it is often convenient to replace the notation \(\text{ Enc }(\textsf{pp}, i, x_i)\) by \(\text{ Enc}_i(x_i)\).
- 4.
The notation \({\widehat{f}}\) is a standard notation for the Fourier representation of f and is used only in Sect. 4.2 of this paper. It is unrelated to the notation of encoding (e.g., \(\hat{x}_i\) denotes the encoding of \(x_i\)) that we use in other parts of the paper, and is standard in the randomized-encoding literature.
- 5.
Actually, convolution can be defined not just for functions that correspond to distributions and also the theorem applies to the more general case, but in this paper we will only be interested in the restricted case of distributions.
- 6.
Since this definition is used for proving negative results, weakening the definition only makes the results stronger.
- 7.
In fact, the proof rules out even the case with \(D_1=\{0,1\},D_2=\{0,1,2\}\).
- 8.
For standard internal-output MPRE, this can be improved to \(t\le 2n/3\).
- 9.
The lemma from [3] applies to degree-2 polynomials. Here we replace each monomial by a 2-local function.
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Acknowledgements
We thank Jonathan Ullman for helpful discussions on differential privacy in the shuffle model and the anonymous reviewers for their comments. Y. Ishai and E. Kushilevitz were supported by ISF grant 2774/20 and BSF grant 2018393. Y. Ishai was additionally supported by ERC Project NTSC (742754).
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Halevi, S., Ishai, Y., Kushilevitz, E., Rabin, T. (2023). Additive Randomized Encodings and Their Applications. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14081. Springer, Cham. https://doi.org/10.1007/978-3-031-38557-5_7
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