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Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness

Iddo Tzameret, Lu-Ming Zhang

TL;DR

This paper develops a foundational theory of nondeterministic-secure pseudorandomness, extending Rudich’s barrier framework to nondeterministic adversaries and introducing demi-bits as a weaker barrier primitive than super-bits. It proves a key stretchability result: every demi-bit $b:\{0,1 extasciicircum n ightarrow\\{0,1 extasciicircum n+1 ight ight}$ can be efficiently expanded to $g:\{0,1 extasciicircum n ightarrow\\{0,1 extasciicircum n+n^c ight ight}$ for any fixed $0<c<1$, using a novel nondeterministic-hybrid argument. The work then connects this construction to broader implications in average-case and proof complexity, via equivalences with hitting-set generators and zero-error average-case hardness for time-bounded Kolmogorov complexity, and shows that stretched demi-bits yield barrier-like results for proof complexity generators. It also develops a refined landscape of nondeterministic unpredictability (including ${ m NP}/{ m poly}$-, ${ m coNP}/{ m poly}$-, union, and intersection forms) and defines super-core predicates to articulate nondeterministic hard-core phenomena, establishing a deep link between existence of demi-bits, super-bits, and potential nondeterministic one-wayness. Overall, the results illuminate how weaker, nondeterministic barriers may constrain strong lower-bound approaches and open pathways to new foundational understanding in cryptography and complexity theory.

Abstract

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work (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 (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 (Rudich and Razborov 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 $b:\{0,1\}^n\to \{0,1\}^{n+1}$ can be stretched into sublinear many demi-bits $b':\{0,1\}^{n}\to \{0,1\}^{n+n^{c}}$, for every constant $0<c<1$. >>> see rest of abstract in paper.

Stretching Demi-Bits and Nondeterministic-Secure Pseudorandomness

TL;DR

This paper develops a foundational theory of nondeterministic-secure pseudorandomness, extending Rudich’s barrier framework to nondeterministic adversaries and introducing demi-bits as a weaker barrier primitive than super-bits. It proves a key stretchability result: every demi-bit can be efficiently expanded to for any fixed , using a novel nondeterministic-hybrid argument. The work then connects this construction to broader implications in average-case and proof complexity, via equivalences with hitting-set generators and zero-error average-case hardness for time-bounded Kolmogorov complexity, and shows that stretched demi-bits yield barrier-like results for proof complexity generators. It also develops a refined landscape of nondeterministic unpredictability (including -, -, union, and intersection forms) and defines super-core predicates to articulate nondeterministic hard-core phenomena, establishing a deep link between existence of demi-bits, super-bits, and potential nondeterministic one-wayness. Overall, the results illuminate how weaker, nondeterministic barriers may constrain strong lower-bound approaches and open pathways to new foundational understanding in cryptography and complexity theory.

Abstract

We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work (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 (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 (Rudich and Razborov 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 can be stretched into sublinear many demi-bits , for every constant . >>> see rest of abstract in paper.
Paper Structure (34 sections, 35 theorems, 64 equations, 1 figure)

This paper contains 34 sections, 35 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The theory of nondeterministic-secure pseudorandomness
  3. Relations to barrier results
  4. Contributions, significance and context
  5. Stretching demi-bits
  6. Discussion and significance of stretching demi-bits to barrier results
  7. Applications in average-case complexity
  8. Applications in proof complexity
  9. Technique overview
  10. Fine-grained characterisation of nondeterministic security based on unpredictability
  11. Super-cores: hard-core predicates in the nondeterministic setting
  12. Justification for the existence of demi-bits and PAC-learning
  13. Preliminaries and basic concepts
  14. Notations and conventions
  15. Computational models
  16. ...and 19 more sections

Key Result

Theorem 1.5

If super-bits exist, then there are no $N\tilde{P}/qpoly$-natural properties useful against ${\mathsf{P}} /\sf{poly}$, where $N\tilde{P}/qpoly$ is the class of languages recognised by non-uniform, quasi-polynomial-size circuit families.

Figures (1)

  • Figure 1: Super-polynomial nondeterministic hardness here refers to \ref{['def:nondeterministic-hardness']}. Note that $\cup$-unpredictability is at least as strong as ${\mathsf{NP}} /{\mathsf{poly}}\textup{-unpredictability}$, because it rules out predictors in both${\mathsf{NP}} /{\mathsf{poly}}$ and ${\mathsf{coNP}} /{\mathsf{poly}}$. And similarly, $\cup$-unpredictability is at least as strong as ${\mathsf{coNP}} /{\mathsf{poly}}$-unpredictability.

Theorems & Definitions (81)

  • Definition 1.1: Nondeterministic hardness superbit
  • Definition 1.2: Super-bits superbit
  • Definition 1.3: Demi-hardness superbit
  • Definition 1.4: Demi-bits superbit
  • Remark : Cryptographic vs. complexity-theoretic regime
  • Theorem 1.5: superbit
  • Theorem : Informal; \ref{['thm:alg']}
  • Proposition : \ref{['prop:santh-claim']}; San20
  • Definition 2.1: the language $\mathrm{K}^t[s]$ and $\mathrm{K}^{{\mathsf{poly}}}[s(n)]$
  • Theorem : equivalence for average-case time-bounded Kolmogorov Complexity; \ref{['thm:ac']}
  • ...and 71 more