ECCC-Report TR21-110https://eccc.weizmann.ac.il/report/2021/110Comments and Revisions published for TR21-110en-usSun, 25 Jul 2021 14:28:58 +0300
Paper TR21-110
| Fourier growth of structured $\mathbb{F}_2$-polynomials and applications |
Jaroslaw Blasiok,
Peter Ivanov,
Yaonan Jin,
Chin Ho Lee,
Rocco Servedio,
Emanuele Viola
https://eccc.weizmann.ac.il/report/2021/110 We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of "structured" $\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at level $k=2$) give unconditional pseudorandom generators.
Our main structural results on Fourier growth are as follows:
- We show that any symmetric degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot O(d)^k$, and this is tight for any constant $k$. This quadratically strengthens an earlier bound that was implicit in [RSV13].
- We show that any read-$\Delta$ degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot (k \Delta d)^{O(k)}$.
- We establish a composition theorem which gives $L_{1,k}$ bounds on disjoint compositions of functions that are closed under restrictions and admit $L_{1,k}$ bounds.
Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of $\mathbb{F}_2$-polynomials.Sun, 25 Jul 2021 14:28:58 +0300https://eccc.weizmann.ac.il/report/2021/110