A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912336996990976 |
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| author | Yao, Penghui Zhao, Mingnan |
| author_facet | Yao, Penghui Zhao, Mingnan |
| contents | Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-$d$ polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length $\mathrm{poly}(k,d,1/ε)\cdot\log n$ that can fool any function of $k$ degree-$d$ PTFs with probability at least $1-\varepsilon$. More specifically, we show that the summation of $L$ independent $R$-moment-matching Gaussian vectors $ε$-fools functions of $k$ degree-$d$ PTFs, where $L=\mathrm{poly}( k, d, \frac{1}ε)$ and $R = O({\log \frac{kd}ε})$. The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10904 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions Yao, Penghui Zhao, Mingnan Computational Complexity Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-$d$ polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length $\mathrm{poly}(k,d,1/ε)\cdot\log n$ that can fool any function of $k$ degree-$d$ PTFs with probability at least $1-\varepsilon$. More specifically, we show that the summation of $L$ independent $R$-moment-matching Gaussian vectors $ε$-fools functions of $k$ degree-$d$ PTFs, where $L=\mathrm{poly}( k, d, \frac{1}ε)$ and $R = O({\log \frac{kd}ε})$. The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence. |
| title | A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2504.10904 |