# Trapdoor Hash Functions and Their Applications

Döttling, Nico and Garg, Sanjam and Ishai, Yuval and Malavolta, Giulio and Mour, Tamer and Ostrovsky, Rafail
(2019) Trapdoor Hash Functions and Their Applications.
In: CRYPTO 2019.
We introduce a new primitive, called {\em trapdoor hash functions} (TDH), which are hash functions $\mathsf{H}: \{0,1\}^n \rightarrow \{0,1\}^\sec$ with additional trapdoor function-like properties. Specifically, given an index $i\in[n]$, TDHs allow for sampling an encoding key $\ek$ (that hides $i$) along with a corresponding trapdoor. Furthermore, given $\mathsf{H}(x)$, a hint value $\mathsf{E}(\ek,x)$, and the trapdoor corresponding to $\ek$, the $i^{th}$ bit of $x$ can be efficiently recovered. In this setting, one of our main questions is: How small can the hint value $\mathsf{E}(\ek,x)$ be? We obtain constructions where the hint is only \emph{one bit} long based on DDH, QR, DCR, or LWE. This primitive opens a floodgate of applications for low-communication secure computation. We mainly focus on two-message protocols between a receiver and a sender, with private inputs $x$ and $y$, resp., where the receiver should learn $f(x,y)$. We wish to optimize the \emph{(download) rate} of such protocols, namely the asymptotic ratio between the size of the output and the sender's message. Using TDHs, we obtain: \begin{enumerate} \item The first protocols for (two-message) \emph{rate-1 string OT} based on DDH, QR, or LWE. This has several useful consequences, such as: \begin{enumerate} \item The first constructions of PIR with communication cost poly-logarithmic in the database size based on DDH or QR. These protocols are in fact rate-1 when considering block PIR. \item The first constructions of a \emph{semi-compact} homomorphic encryption scheme for branching programs, where the encrypted output grows only with the program {\em length}, based on DDH or QR. \item The first constructions of lossy trapdoor functions with input to output ratio approaching 1 based on DDH, QR or LWE. \item The first {\em constant-rate} LWE-based construction of a 2-message statistically sender-private'' OT protocol in the plain model. \end{enumerate} \item The first {\em rate-1} protocols (under any assumption) for $n$ parallel OTs and matrix-vector products from DDH, QR or LWE. \end{enumerate} We further consider the setting where $f$ evaluates a RAM program $y$ with running time $T\ll |x|$ on $x$. We obtain the first protocols with communication sublinear in the size of $x$, namely $T\cdot\sqrt{|x|}$ or $T\cdot\sqrt[3]{|x|}$, based on DDH or, resp., pairings (and correlated-input secure hash functions).