A function $f$ is extractable if it is possible to algorithmically ``extract,'' from any program that outputs a value $y$ in the image of $f,$ a preimage of $y$. % under $f$.
When combined with hardness properties such as one-wayness or collision-resistance, extractability has proven to be a powerful tool. However, so far, extractability has not been explicitly shown. Instead, it has only been considered as a non-standard {\em knowledge assumption} on certain functions.
We give the first construction of extractable one-way functions assuming only standard hardness assumptions (e.g. the subexponential Learning with Errors Assumption).
Our functions are extractable against adversaries with bounded polynomial advice and unbounded polynomial running time. We then use these functions to construct the first 2-message zero-knowledge arguments and 3-message zero-knowledge arguments of knowledge, against the same class of adversarial verifiers, from essentially the same assumptions.
The construction uses ideas from [Barak, FOCS01] and [Barak, Lindell, and Vadhan, FOCS03], and rely on the recent breakthrough construction of privately verifiable $\P$-delegation schemes [Kalai, Raz, and Rothblum]. The extraction procedure uses the program evaluating $f$ in a non-black-box way, which we show to be necessary.
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