On the Average-Case Complexity of MCSP and Its Variants

Shuichi Hirahara, R. Santhanam
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引用次数: 52

Abstract

We prove various results on the complexity of MCSP (Minimum Circuit Size Problem) and the related MKTP (Minimum Kolmogorov Time-Bounded Complexity Problem): * We observe that under standard cryptographic assumptions, MCSP has a pseudorandom self-reduction. This is a new notion we define by relaxing the notion of a random self-reduction to allow queries to be pseudorandom rather than uniformly random. As a consequence we derive a weak form of a worst-case to average-case reduction for (a promise version of) MCSP. Our result also distinguishes MCSP from natural NP-complete problems, which are not known to have worst-case to average-case reductions. Indeed, it is known that strong forms of worst-case to average-case reductions for NP-complete problems collapse the Polynomial Hierarchy. * We prove the first non-trivial formula size lower bounds for MCSP by showing that MCSP requires nearly quadratic-size De Morgan formulas. * We show average-case superpolynomial size lower bounds for MKTP against AC0[p] for any prime p. * We show the hardness of MKTP on average under assumptions that have been used in much recent work, such as Feige's assumptions, Alekhnovich's assumption and the Planted Clique conjecture. In addition, MCSP is hard under Alekhnovich's assumption. Using a version of Feige's assumption against co-nondeterministic algorithms that has been conjectured recently, we provide evidence for the first time that MKTP is not in coNP. Our results suggest that it might worthwhile to focus on the average-case hardness of MKTP and MCSP when approaching the question of whether these problems are NP-hard.
MCSP及其变体的平均情况复杂度
我们证明了MCSP(最小电路尺寸问题)和相关的MKTP(最小Kolmogorov时限复杂性问题)的复杂性的各种结果:*我们观察到在标准密码假设下,MCSP具有伪随机自约简。这是我们通过放松随机自约简的概念来定义的一个新概念,允许查询是伪随机的,而不是均匀随机的。因此,我们得到了MCSP(承诺版本)的最坏情况到平均情况约简的弱形式。我们的结果还将MCSP与自然np完全问题区分开来,后者不知道有最坏情况到平均情况的约简。事实上,已知np完全问题的最坏情况到平均情况约简的强形式会使多项式层次崩溃。*通过证明MCSP需要近二次大小的De Morgan公式,我们证明了MCSP的第一个非平凡公式大小下界。*我们展示了对于任意素数p, MKTP对AC0[p]的平均情况下的超多项式大小下界。*我们展示了MKTP在最近许多工作中使用的假设下的平均硬度,例如Feige的假设,Alekhnovich的假设和plantclique猜想。此外,在Alekhnovich的假设下,MCSP是困难的。使用Feige对最近推测的共不确定性算法的假设的一个版本,我们首次提供了MKTP不在coNP中的证据。我们的研究结果表明,在处理MKTP和MCSP的平均硬度问题时,可能值得关注这些问题是否是np困难的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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