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Pseudorandomness, symmetry, smoothing: I

Harm Derksen, Peter Ivanov, Chin Ho Lee, Emanuele Viola

TL;DR

This work analyzes how small-bias distributions interact with smoothing and symmetry in the context of pseudorandomness. By developing tight bounds via Krawtchouk polynomials and exploring smoothed tests, the authors show that small-bias plus noise can be as ineffective as bounded uniformity against several natural tests, including threshold, small-space, and constant-depth circuits. They provide both negative results (limitations of space-bounded PRGs and AC0 fooling with noise) and positive results (limits of XOR paradigms and robustness of sums of small-bias distributions). The findings advance understanding of when simple, natural pseudorandom constructions rival or fail against classical PRG paradigms, with implications for seed-length trade-offs and the design of space-bounded pseudorandom generators.

Abstract

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that 1) achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. 2) have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 3) rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.

Pseudorandomness, symmetry, smoothing: I

TL;DR

This work analyzes how small-bias distributions interact with smoothing and symmetry in the context of pseudorandomness. By developing tight bounds via Krawtchouk polynomials and exploring smoothed tests, the authors show that small-bias plus noise can be as ineffective as bounded uniformity against several natural tests, including threshold, small-space, and constant-depth circuits. They provide both negative results (limitations of space-bounded PRGs and AC0 fooling with noise) and positive results (limits of XOR paradigms and robustness of sums of small-bias distributions). The findings advance understanding of when simple, natural pseudorandom constructions rival or fail against classical PRG paradigms, with implications for seed-length trade-offs and the design of space-bounded pseudorandom generators.

Abstract

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that 1) achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. 2) have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 3) rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.
Paper Structure (24 sections, 20 theorems, 91 equations)

This paper contains 24 sections, 20 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Smoothed tests and distributions.
  3. Small-bias plus (pseudorandom) noise.
  4. Xor-ing small-bias distributions.
  5. Our results
  6. One-way small space.
  7. Constant-depth circuits.
  8. Sum of small-bias distributions.
  9. Typical shifts.
  10. Krawtchouk polynomials
  11. Small-bias plus noise is far from bounded uniformity
  12. Distinguishing $D_\alpha$ from uniform with a threshold
  13. Distinguishing $D_\alpha$ from bounded uniformity with a symmetric test
  14. Bounded uniformity plus noise fools symmetric tests
  15. Shifted symmetric small-bias fools symmetric tests
  16. ...and 9 more sections

Key Result

Lemma 2

Any $(\varepsilon,k)$-biased distribution is $((\frac{e^3n}{k})^{k/2} \varepsilon)$-close to a $k$-wise uniform distribution in total variation distance.

Theorems & Definitions (59)

  • Definition 1
  • Lemma 2: Theorem 1.1 OZ18
  • Definition 5
  • Theorem 7
  • Lemma 8
  • Theorem 9
  • Theorem 10
  • Corollary 11
  • proof : Proof of \ref{['cor:thr-thr-const-advant']} from \ref{['thm:sbpn-vs-uniform']}
  • Corollary 12
  • ...and 49 more