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Resilient functions: Optimized, simplified, and generalized

Peter Ivanov, Emanuele Viola

TL;DR

This work advances the theory of resilient Boolean functions by providing explicit depth-3 circuits that are resilient to coalitions of size $c n/\log^{2} n$ with bias $n^{-c}$ and extending the construction to biased product distributions $B_{\sigma}$. The core method combines expander-walk samplers with a Reed-Solomon-based design and a refined Janson inequality to bound bias, yielding a simpler, explicit construction that nearly matches nonexplicit results. It also achieves exact balance (up to constants) and matches the KKL-type influence bounds, addressing longstanding questions about tradeoffs between resilience, balance, and bias. The results have implications for randomness extractors and related combinatorial constructions by providing robust, efficient circuit realizations under nonuniform distributions.

Abstract

An $n$-bit boolean function is resilient to coalitions of size $q$ if any fixed set of $q$ bits is unlikely to influence the function when the other $n-q$ bits are chosen uniformly. We give explicit constructions of depth-$3$ circuits that are resilient to coalitions of size $cn/\log^{2}n$ with bias $n^{-c}$. Previous explicit constructions with the same resilience had constant bias. Our construction is simpler and we generalize it to biased product distributions. Our proof builds on previous work; the main differences are the use of a tail bound for expander walks in combination with a refined analysis based on Janson's inequality.

Resilient functions: Optimized, simplified, and generalized

TL;DR

This work advances the theory of resilient Boolean functions by providing explicit depth-3 circuits that are resilient to coalitions of size with bias and extending the construction to biased product distributions . The core method combines expander-walk samplers with a Reed-Solomon-based design and a refined Janson inequality to bound bias, yielding a simpler, explicit construction that nearly matches nonexplicit results. It also achieves exact balance (up to constants) and matches the KKL-type influence bounds, addressing longstanding questions about tradeoffs between resilience, balance, and bias. The results have implications for randomness extractors and related combinatorial constructions by providing robust, efficient circuit realizations under nonuniform distributions.

Abstract

An -bit boolean function is resilient to coalitions of size if any fixed set of bits is unlikely to influence the function when the other bits are chosen uniformly. We give explicit constructions of depth- circuits that are resilient to coalitions of size with bias . Previous explicit constructions with the same resilience had constant bias. Our construction is simpler and we generalize it to biased product distributions. Our proof builds on previous work; the main differences are the use of a tail bound for expander walks in combination with a refined analysis based on Janson's inequality.
Paper Structure (22 sections, 22 theorems, 60 equations, 4 figures)

This paper contains 22 sections, 22 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:resilient_explicit']}
  3. The design property
  4. Future Directions
  5. Organization
  6. Nonexplicit tradeoff resilient circuits
  7. Proof of Lemma \ref{['lem:exists-shifted-sampler-1']}
  8. Explicit tradeoff resilient circuits
  9. Proof of Lemma \ref{['thm:explicit-sampler-design']}
  10. Remaining proof
  11. Sampler to resilience
  12. Sampler and design to small bias
  13. Proof of Lemmas \ref{['lem:post-janson-1']}, \ref{['cor:max-power']}
  14. Proof of Proposition \ref{['prop:powers-to-sampler']}
  15. Balanced circuits matching KKL
  16. ...and 7 more sections

Key Result

Theorem 2

For infinitely many $n$ there exist monotone depth 3, size $n^{c}$ circuits $C:\mathrm{\{0,1\}}^{n}\to\mathrm{\{0,1\}}$ which are $\left(\frac{c\log^{2}n}{n}\right)$-tradeoff resilient with bias $n^{-1+o(1)}.$

Figures (4)

  • Figure : $G(i)$ defines a read-once OR$_v$-AND$_w$ subcircuit:
  • Figure : $S(G(i),0)$,$S(G(i),1)$, $S(G(i),v-1)$
  • Figure : $w$
  • Figure : $C_{G(i)}$

Theorems & Definitions (40)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Definition 9
  • Claim 10
  • Theorem 13: RR17
  • ...and 30 more