CryptoDB
Additive Randomized Encodings and Their Applications
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Conference:  CRYPTO 2023 
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 $\G$, and decode the result to get the function output. An {\em 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 {\em robust} ARE, the sum of {\em any subset} of the $\hat x_i$ only reveals the {\em residual function} obtained by restricting the corresponding inputs. We obtain positive and negative results on ARE. In particular: \begin{itemize} \item {\em Informationtheoretic ARE.} We fully characterize the 2party functions $f:X_1\times X_2\to\{0,1\}$ admitting a perfectly secure ARE. For $n\ge 3$ parties, we show a useful ``capped sum'' function that separates statistical security from perfect security. \item {\em Computational ARE.} We present a general feasibility result, showing that \emph{all functions} can be computed in this model, under a standard hardness assumption in bilinear groups. We also describe a heuristic latticebased construction. \item {\em Robust ARE.} We present a similar feasibility result for {\em robust} computational ARE based on ideal obfuscation along with standard cryptographic assumptions. \end{itemize} We then describe several applications of ARE and the above results. \begin{itemize} \item Under a standard cryptographic assumption, our computational ARE schemes imply the feasibility of general noninteractive secure computation in the {\em shuffle model}, where messages from different parties are shuffled. This implies a general utilitypreserving compiler from differential privacy in the central model to computational differential privacy in the (nonrobust) shuffle model. \item The existence of informationtheoretic {\em robust} ARE implies ``bestpossible'' informationtheoretic MPC protocols (Halevi et al., TCC 2018) and degree2 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. \end{itemize} 
BibTeX
@inproceedings{crypto202333190, title={Additive Randomized Encodings and Their Applications}, publisher={SpringerVerlag}, doi={10.1007/9783031385575_7}, author={Shai Halevi and Yuval Ishai and Eyal Kushilevitz and Tal Rabin}, year=2023 }