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TR23-046 | 13th April 2023 03:34

NP-Hardness of Approximating Meta-Complexity: A Cryptographic Approach



It is a long-standing open problem whether the Minimum Circuit Size Problem ($\mathrm{MCSP}$) and related meta-complexity problems are NP-complete. Even for the rare cases where the NP-hardness of meta-complexity problems are known, we only know very weak hardness of approximation.

In this work, we prove NP-hardness of approximating meta-complexity with nearly-optimal approximation gaps. Our key idea is to use *cryptographic constructions* in our reductions, where the security of the cryptographic construction implies the correctness of the reduction. We present both conditional and unconditional hardness of approximation results as follows.

$\bullet$ Assuming subexponentially-secure witness encryption exists, we prove essentially optimal NP-hardness of approximating conditional time-bounded Kolmogorov complexity ($\mathrm{K}^t(x \mid y)$) in the regime where $t \gg |y|$. Previously, the best hardness of approximation known was a $|x|^{1/ \mathrm{poly}(\log \log |x|)}$ factor and only in the sublinear regime ($t \ll |y|$).
$\bullet$ Unconditionally, we show near-optimal NP-hardness of approximation for the Minimum Oracle Circuit Size Problem (MOCSP), where Yes instances have circuit complexity at most $2^{\varepsilon n}$, and No instances are essentially as hard as random truth tables. Our reduction builds on a witness encryption construction proposed by Garg, Gentry, Sahai, and Waters (STOC'13). Previously, it was unknown whether it is NP-hard to distinguish between oracle circuit complexity $s$ versus $10s\log N$.
$\bullet$ Finally, we define a "multi-valued" version of $\mathrm{MCSP}$, called $\mathrm{mvMCSP}$, and show that w.p. $1$ over a random oracle $O$, $\mathrm{mvMCSP}^O$ is NP-hard to approximate under quasi-polynomial-time reductions with $O$ oracle access. Intriguingly, this result follows almost directly from the security of Micali's CS proofs (Micali, SICOMP'00).

In conclusion, we give three results convincingly demonstrating the power of cryptographic techniques in proving NP-hardness of approximating meta-complexity.

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