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Blanco avatar Crypto

The existence of OWFs vs $\mathbf{EXP} \neq \mathbf{BPP}$

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Blanco

In CRYPTO 2021, Liu and Pass published a paper with title "On the Possibility of Basing Cryptography on $\mathbf{EXP} \neq \mathbf{BPP}$.

One of the main results of this work can be interpreted as an indication that the existence of OWFs is equivalent to $\mathbf{EXP} \neq \mathbf{BPP}$. $\mathbf{EXP} \neq \mathbf{BPP}$ is a weak assumpation, what is the relation between this assumpation and the average-case hardness? Any introduction and comment about this work is welcome.

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Mark avatar Crypto
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Mark

$\mathsf{EXP}\neq \mathsf{BPP}$ is the typical assumption used in the derandomization literature. Words to search are "Nisan-Widgerson" and "Impagliazzo-Widgerson". For example, the abstract of Impagliazzo-Widgerson reads:

We prove that if $\mathsf{BPP}\neq \mathsf{EXP}$, then every problem in $\mathsf{BPP}$ can be solved deterministically in subexponential time on almost every input (on every samplable ensemble for infinitely many input sizes). This is the first derandomization result for $\mathsf{BPP}$ based on uniform, non-cryptographic hardness assumptions. It implies the following gap in the average-instance complexities of problems in $\mathsf{BPP}$ : either these complexities are always sub-exponential or they contain arbitrarily large exponential functions.

In general, when looking into a new area PhD thesis can be mildly easier to read than papers, so if you're interested more Marco Carmosino's thesis may be useful.

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