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TR22-113 | 11th August 2022 23:53

Leakage-Resilient Hardness v.s. Randomness


Authors: Yanyi Liu, Rafael Pass
Publication: 14th August 2022 11:05
Downloads: 162


A central open problem in complexity theory concerns the question of
whether all efficient randomized algorithms can be simulated by
efficient deterministic algorithms. The celebrated ``hardness
v.s. randomness” paradigm pioneered by Blum-Micali (SIAM JoC’84),
Yao (FOCS’84) and Nisan-Wigderson (JCSS’94) presents hardness
assumptions under which $\prBPP = \prP$, but these hardness assumptions
are not known to also be necessary for such derandomization.

In this work, following the recent work by Chen and Tell (FOCS’21) that
considers “almost-all-input” hardness of a function $f$ (i.e., hardness
of computing $f$ on more than a finite number of inputs), we consider
“almost-all-input” \emph{leakage-resilient hardness} of a function $f$
—that is, hardness of computing $f(x)$ even given, say, $\sqrt{|x|}$
bits of leakage of $f(x)$. We show that leakage-resilient hardness
characterizes derandomization of $\prBPP$ (i.e., gives a both \emph{
necessary} and \emph{sufficient} condition for derandomization). In more
detail, we show that there exists a constant $c$ such that the following
are equivalent:

- $\prBPP = \prP$;
- Existence of a poly-time computable function $f :\{0, 1\}^n \rightarrow \{0,1\}^n$
that is almost-all-input leakage-resilient hard with respect to
$n^c$-time probabilistic algorithms.

Our characterization naturally extends also to the low-end
derandomization regime, to derandomization of $\prMA$, and also to
average-case derandomization, by appropriately weakening the
requirements on the function $f$.

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