Pseudorandom generators for polynomials

In theoretical computer science, a pseudorandom generator for low-degree polynomials is an efficient procedure that maps a short truly random seed to a longer pseudorandom string in such a way that low-degree polynomials cannot distinguish the output distribution of the generator from the truly random distribution. That is, evaluating any low-degree polynomial at a point determined by the pseudorandom string is statistically close to evaluating the same polynomial at a point that is chosen uniformly at random. Pseudorandom generators for low-degree polynomials are a particular instance of pseudorandom generators for statistical tests, where the statistical tests considered are evaluations of low-degree polynomials.

A pseudorandom generator ${\displaystyle G:{\mathbb{𝔽}}^{\ell }\to {\mathbb{𝔽}}^n}$ for polynomials of degree ${\displaystyle d}$ over a finite field ${\displaystyle \mathbb{𝔽}}$ is an efficient procedure that maps a sequence of ${\displaystyle \ell }$ field elements to a sequence of ${\displaystyle n}$ field elements such that any ${\displaystyle n}$-variate polynomial over ${\displaystyle \mathbb{𝔽}}$ of degree ${\displaystyle d}$ is fooled by the output distribution of ${\displaystyle G}$. In other words, for every such polynomial ${\displaystyle p(x_1,\dots ,x_n)}$, the statistical distance between the distributions ${\displaystyle p(U_n)}$ and ${\displaystyle p(G(U_{\ell }))}$ is at most a small ${\displaystyle ϵ}$, where ${\displaystyle U_k}$ is the uniform distribution over ${\displaystyle {\mathbb{𝔽}}^k}$.

The case ${\displaystyle d=1}$ corresponds to pseudorandom generators for linear functions and is solved by small-bias generators. For example, the construction of (Naor Naor) achieves a seed length of ${\displaystyle \ell =\log n+O(\log ({ϵ}^{-1}))}$, which is optimal up to constant factors.

(Bogdanov Viola) conjectured that the sum of small-bias generators fools low-degree polynomials and were able to prove this under the Gowers inverse conjecture. (Lovett 2009) proved unconditionally that the sum of ${\displaystyle 2^d}$ small-bias spaces fools polynomials of degree ${\displaystyle d}$. (Viola 2008) proves that, in fact, taking the sum of only ${\displaystyle d}$ small-bias generators is sufficient to fool polynomials of degree ${\displaystyle d}$. The analysis of (Viola 2008) gives a seed length of ${\displaystyle \ell =d\cdot \log n+O(2^d\cdot \log ({ϵ}^{-1}))}$.

• Bogdanov, Andrej; Viola, Emanuele (2007). "Spectral Graph Theory and its Applications". 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07). pp. 41–51. doi:10.1109/FOCS.2007.56. ISBN 978-0-7695-3010-9. http://eccc.hpi-web.de/report/2007/081/download. 
• Lovett, Shachar (2009), "Unconditional Pseudorandom Generators for Low Degree Polynomials", Theory of Computing 5 (3): 69–82, doi:10.4086/toc.2009.v005a003 
• Naor, Joseph; Naor, Moni (1990), "Small-bias probability spaces: Efficient constructions and applications", Proceedings of the twenty-second annual ACM symposium on Theory of computing - STOC '90, pp. 213–223, doi:10.1145/100216.100244, ISBN 978-0897913614, http://www.wisdom.weizmann.ac.il/~naor/PAPERS/bias_abs.html 
• Viola, Emanuele (2008). "The Sum of d Small-Bias Generators Fools Polynomials of Degree D". 2008 23rd Annual IEEE Conference on Computational Complexity. pp. 124–127. doi:10.1109/CCC.2008.16. ISBN 978-0-7695-3169-4. http://www.ccs.neu.edu/home/viola/papers/d.pdf. 

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