[论文解读] Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness

Iddo Tzameret, Luming Zhang|arXiv (Cornell University)|Apr 28, 2023

Benford’s Law and Fraud Detection被引用 1

一句话总结

本文通过证明每个半比特生成器均可被拉伸为次线性数量的半比特，推进了非确定性安全伪随机性的理论，解决了自Rudich（1997年）以来的一个开放问题。此外，本文通过非确定性不可预测性提出了对超比特的新表征，并引入了超核心谓词，将其与非确定性环境下的核心谓词联系起来，其应用涵盖平均情况复杂度与证明复杂度。

ABSTRACT

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich’s original work [S. Rudich, 1997], and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: Demi-bit stretch: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests [S. Rudich, 1997]. They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier of Razborov and Rudich [A. A. Razborov and S. Rudich, 1997]. Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit b:{0,1}ⁿ → {0,1}^{n+1} can be stretched into sublinear many demi-bits b':{0,1}ⁿ → {0,1}^{n+n^{c}}, for every constant 0 < c < 1. Average-case hardness: Using work by Santhanam [Rahul Santhanam, 2020], we apply our results to obtain new average-case Kolmogorov complexity results: we show that K^{poly}[n-O(1)] is zero-error average-case hard against NP/poly machines iff K^{poly}[n-o(n)] is, where for a function s(n):ℕ → ℕ, K^{poly}[s(n)] denotes the languages of all strings x ∈ {0,1}ⁿ for which there are (fixed) polytime Turing machines of description-length at most s(n) that output x. Characterising super-bits by nondeterministic unpredictability: In the deterministic setting, Yao [Yao, 1982] proved that super-polynomial hardness of pseudorandom generators is equivalent to ("next-bit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability: NP/poly-unpredictability, coNP/poly-unpredictability, ∩-unpredictability and ∪-unpredictability, and show that super-polynomial nondeterministic hardness of generators lies between ∩-unpredictability and ∪-unpredictability. Characterising super-bits by nondeterministic hard-core predicates: We introduce a nondeterministic variant of hard-core predicates, called super-core predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function [Goldreich and Levin, 1989; Håstad et al., 1999], and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core.

研究动机与目标

• 解决半比特是否能超越其原始输出大小被拉伸的开放问题。
• 通过更强的不可预测性概念表征伪随机生成器的非确定性困难性。
• 引入并研究超核心谓词，作为核心谓词在非确定性环境下的类比。
• 将结果应用于推导新的平均情况Kolmogorov复杂度结果，并在证明复杂度中产生影响。
• 通过与PAC学习和电路学习的联系，在非存在性假设下证明半比特存在的合理性。

提出的方法

• 引入四种新的不可预测性概念：NP/poly-、coNP/poly-、∩-和∪-不可预测性，以表征非确定性困难性。
• 开发了一种拉伸构造，将任意半比特 b: {0,1}^n → {0,1}^{n+1} 转换为 b': {0,1}^n → {0,1}^{n+nc}，其中 0 < c < 1。
• 提出超核心谓词作为核心谓词的非确定性变体，证明了超比特的存在性与超核心谓词的存在性等价。
• 应用Santhanam的平均情况复杂度结果，表明在给定假设下，Kpoly[n−O(1)] 对 NP/poly 的零误差平均情况困难性当且仅当 Kpoly[n−o(n)] 也是如此。
• 利用半比特非存在性，通过大小为 2^{no(1)} 的随机电路在高置信度下推导出 Circuit[nc] 的学习算法。
• 利用 i.o. 半比特非存在性与PAC学习之间的联系，表明除非半比特存在，否则学习小电路是可行的。

实验结果

研究问题

• RQ1 每个半比特是否都能被拉伸为次线性数量的半比特，从而解决伪随机性中的一个开放问题？
• RQ2 在超越确定性下一比特模型的背景下，如何通过非确定性不可预测性来表征超比特？
• RQ3 在非确定性环境下，核心谓词的作用是什么，它们与超比特之间有何关系？
• RQ4 超比特与半比特拉伸性在平均情况复杂度方面有何后果？
• RQ5 半比特非存在性对学习小电路和PAC学习有何影响？

主要发现

• 每个半比特 b: {0,1}^n → {0,1}^{n+1} 均可被拉伸为 b': {0,1}^n → {0,1}^{n+nc}，其中 0 < c < 1，从而解决了拉伸性问题。
• 存在一个超比特当且仅当对某个非压缩函数存在一个超核心谓词。
• 生成器的非确定性困难性严格介于 ∩-不可预测性与 ∪-不可预测性之间，本文引入了四种新的不可预测性概念。
• 在给定假设下，Kpoly[n−O(1)] 对 NP/poly 的零误差平均情况困难性当且仅当 Kpoly[n−o(n)] 也是如此。
• 若 i.o. 半比特非存在，则 Circuit[nc] 可被大小为 2^{no(1)} 的随机电路以置信度 1/2^{no(1)} 和误差 1/2−1/2^{no(1)} 学习。
• 若 i.o. 半比特存在，则在 ÑP/qpoly-自然证明不存在的假设下，i.o. 超比特也必须存在。