Talks
Summer 2015

# Spooky Encryption and its Applications

Friday, August 12th, 2016 11:40 am12:00 pm

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Consider encrypting $n$ inputs under $n$ independent public keys. Given the ciphertexts $\{c_i=\Enc_{\pk_i}(x_i)\}_i$, Alice outputs ciphertexts $c'_1,\ldots,c'_n$ that decrypt to $y_1,\ldots,y_n$ respectively. What relationships between the $x_i$'s and $y_i$'s can Alice induce?

Motivated by applications to delegating computations, Dwork, Langberg, Naor, Nissim and Reingold \cite{DLNNR04} showed that a semantically secure scheme disallows \emph{signaling} in this setting, meaning that $y_i$ cannot depend on $x_j$ for $j \neq i$ . On the other hand if the scheme is homomorphic then any \emph{local} (component-wise) relationship is achievable, meaning that each $y_i$ can be an arbitrary function of $x_i$. However, there are also relationships which are neither signaling nor local. Dwork et al. asked if it is possible to have encryption schemes that support such spooky'' relationships. Answering this question is the focus of our work.

Our first result shows that, under the $\LWE$ assumption, there exist encryption schemes supporting a large class of spooky'' relationships, which we call \emph{additive function sharing} (AFS) spooky. In particular, for any polynomial-time function $f$, Alice can ensure that $y_1,\ldots,y_n$ are random subject to $\sum_{i=1}^n y_i = f(x_1,\ldots,x_n)$. For this result, the public keys all depend on common public randomness.  Our second result shows that, assuming sub-exponentially hard indistinguishability obfuscation (iO) (and additional more standard assumptions), we can remove the common randomness and choose the public keys completely independently. Furthermore, in the case of $n=2$ inputs, we get a scheme that supports an even larger class of spooky relationships.

We discuss several implications of AFS-spooky encryption. Firstly, it gives a strong counter-example to a method proposed by Aiello et~al. \cite{ABOR00} for building arguments for $\NP$ from homomorphic encryption. Secondly, it gives a simple 2-round multi-party computation protocol where, at the end of the first round, the parties can locally compute an additive secret sharing of the output. Lastly, it immediately yields a function secret sharing (FSS) scheme for all functions.

We also define a notion of \emph{spooky-free} encryption, which ensures that no spooky relationship is achievable. We show that any non-malleable encryption scheme is spooky-free. Furthermore, we can construct spooky-free \emph{homomorphic} encryption schemes from SNARKs, and it remains an open problem whether it is possible to do so from falsifiable assumptions.