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Revision #3 to TR19-099 | 7th September 2020 04:23

#### Nearly Optimal Pseudorandomness From Hardness

Revision #3
Authors: Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckerman
Accepted on: 7th September 2020 04:23
Keywords:

Abstract:

Existing proofs that deduce $\mathbf{BPP}=\mathbf{P}$ from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against nondeterministic circuits, we convert any randomized algorithm over inputs of length $n$ running in time $t \ge n$ to a deterministic one running in time $t^{2+\alpha}$ for an arbitrarily small constant $\alpha > 0$. Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing).

Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size $s$ with seed length $(1+\alpha)\log s$, under the assumption that there exists a function $f \in \mathbf{E}$ that requires nondeterministic circuits of size at least $2^{(1-\alpha')n}$, where $\alpha = O(\alpha')$. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

Changes to previous version:

Mostly in the introduction.

Revision #2 to TR19-099 | 16th April 2020 05:25

#### Nearly Optimal Pseudorandomness From Hardness

Revision #2
Authors: Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckerman
Accepted on: 16th April 2020 05:25
Keywords:

Abstract:

Existing proofs that deduce $\mathbf{BPP}=\mathbf{P}$ from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against randomized single-valued nondeterministic (SVN) circuits, we convert any randomized algorithm over inputs of length $n$ running in time $t \ge n$ to a deterministic one running in time $t^{2+\alpha}$ for an arbitrarily small constant $\alpha > 0$. Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits.

Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size $s$ with seed length $(1+\alpha)\log s$, under the assumption that there exists a function $f \in \mathbf{E}$ that requires randomized SVN circuits of size at least $2^{(1-\alpha')n}$, where $\alpha = O(\alpha')$. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

Changes to previous version:

Rephrased the hardness assumption and added support for lower errors.

Revision #1 to TR19-099 | 2nd November 2019 05:09

#### Nearly Optimal Pseudorandomness From Hardness

Revision #1
Authors: Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckerman
Accepted on: 2nd November 2019 05:09
Keywords:

Abstract:

Existing proofs that deduce $\mathbf{BPP}=\mathbf{P}$ from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against nondeterministic circuits, we convert any randomized algorithm over inputs of length $n$ running in time $t \ge n$ to a deterministic one running in time $t^{2+\alpha}$ for an arbitrarily small constant $\alpha > 0$. Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing).

Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size $s$ with seed length $(1+\alpha)\log s$, under the assumption that there exists a function $f \in \mathbf{E}$ that requires nondeterministic circuits of size at least $2^{(1-\alpha')n}$, where $\alpha = O(\alpha')$. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.

Changes to previous version:

Minor revisions

### Paper:

TR19-099 | 29th July 2019 20:12

#### Nearly Optimal Pseudorandomness From Hardness

TR19-099
Authors: Dean Doron, Dana Moshkovitz, Justin Oh, David Zuckerman
Publication: 29th July 2019 22:55
Existing proofs that deduce $\mathbf{BPP}=\mathbf{P}$ from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against nondeterministic circuits, we convert any randomized algorithm over inputs of length $n$ running in time $t \ge n$ to a deterministic one running in time $t^{2+\alpha}$ for an arbitrarily small constant $\alpha > 0$. Such a slowdown is nearly optimal, as, under complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing).
Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size $s$ with seed length $(1+\alpha)\log s$, under the assumption that there exists a function $f \in \mathbf{E}$ that requires nondeterministic circuits of size at least $2^{(1-\alpha')n}$, where $\alpha = O(\alpha')$. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.