有时Kolmogorov复杂度投影下的密码硬度

E. Allender, John Gouwar, Shuichi Hirahara, Caleb Robelle
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引用次数: 7

摘要

有界Kolmogorov复杂度的一个版本,记作KT,在过去几年中受到了关注,因为它与电路复杂度和最小电路尺寸问题MCSP密切相关。基本上,所有关于MCSP复杂度的结果也适用于MKTP(计算字符串的KT复杂度的问题)。在BPP -Turing约简下,MKTP和MCSP对SZK(统计零知识)都是困难的;两者都不是NP完全的。最近,MKTP的一些硬度结果得到了证明,而MCSP的硬度结果并不成立。特别是,在非均匀≤NC 0 m约简下,DET (P的一个子类)很难实现MKTP。在本文中,我们改进了这一点,证明了MKTP对(明显更大的)NISZK L类不仅在≤NC 0 m约简下而且在投影下都是困难的。MKTP对NISZK在≤P /聚m的还原下也有一定的困难。其中,NISZK是一类具有非交互零知识证明的问题,NISZK L是Dvir等人研究过的一类SZK L的非交互版本。作为一个应用,我们提供了几个改进的NP问题的最坏情况到平均情况的约简,并且我们获得了MKTP的一个新的下界(目前还不知道它是否适用于MCSP)。
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Cryptographic Hardness under Projections for Time-Bounded Kolmogorov Complexity
A version of time-bounded Kolmogorov complexity, denoted KT , has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP . Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP -Turing reductions; neither is known to be NP -complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP . In particular, MKTP is hard for DET (a subclass of P ) under nonuniform ≤ NC 0 m reductions. In this paper, we improve this, to show that MKTP is hard for the (apparently larger) class NISZK L under not only ≤ NC 0 m reductions but even under projections. Also MKTP is hard for NISZK under ≤ P / poly m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK L is the non-interactive version of the class SZK L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP , and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP ).
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