**Must you know the code of
f to securely compute f?**

Mike Rosulek (

**Abstract:**

When Alice and Bob want to securely evaluate a function of their shared
inputs, they typically first express the function as a (boolean
or arithmetic) circuit, and then they securely evaluate that circuit
gate-by-gate. In other words, a secure protocol for evaluating $f$ is
typically obtained in a *non-black-box-way*
from $f$ itself. As a consequence, secure computation protocols have high
overhead (in communication and computation) that is directly linked to the
circuit-description complexity of $f$.

In other settings throughout cryptography, black-box constructions invariably
lead to better practical efficiency than comparable non-black-box
constructions. Could secure computation protocols similarly be made more
practical by eliminating their dependence on a circuit representation of the
target function? Or, in other words, *must
one know the code of $f$ to securely evaluate $f$?*

In this work we initiate the theoretical study of this question. We show the
following:

1. A complete characterization of the 2-party tasks which admit such security
against semi-honest adversaries. The characterization is inspired by notions
of *autoreducibility*
from computational complexity theory. From this characterization, we show a
class of pseudorandom functions that *cannot*
be securely evaluated (when one party holds the seed and the other holds the
input) without “knowing” the code of the function in question. On
the positive side, we show a class of functions (related to blind signatures)
that can indeed be securely computed without “knowing” the code
of the function.

2. Sufficient conditions for such security against malicious adversaries,
also based on autoreducibility. We show that it is
not possible to prove membership in the image of a one-way function in
zero-knowledge, without “knowing” the code of the one-way
function. We also describe a variant of the GMW compiler for transforming
semi-honest to malicious security while preserving the specific black-box
property considered here.