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.

Definition

A pseudorandom generator $G:\mathbb {F} ^{\ell }\rightarrow \mathbb {F} ^{n}$ for polynomials of degree $d$ over a finite field $\mathbb {F}$ is an efficient procedure that maps a sequence of $\ell$ field elements to a sequence of $n$ field elements such that any $n$-variate polynomial over $\mathbb {F}$ of degree $d$ is fooled by the output distribution of $G$. In other words, for every such polynomial $p(x_{1},\dots ,x_{n})$, the statistical distance between the distributions $p(U_{n})$ and $p(G(U_{\ell }))$ is at most a small $\epsilon$, where $U_{k}$ is the uniform distribution over $\mathbb {F} ^{k}$.

Construction

Lovett (3rd from left) in 2009

The case $d=1$ corresponds to pseudorandom generators for linear functions and is solved by small-bias generators. For example, the construction of Naor & Naor (1990) achieves a seed length of $\ell =\log n+O(\log(\epsilon ^{-1}))$, which is optimal up to constant factors.

Bogdanov & Viola (2007) 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 $2^{d}$ small-bias spaces fools polynomials of degree $d$. Viola (2008) proves that, in fact, taking the sum of only $d$ small-bias generators is sufficient to fool polynomials of degree $d$. The analysis of Viola (2008) gives a seed length of $\ell =d\cdot \log n+O(2^{d}\cdot \log(\epsilon ^{-1}))$.

References

• 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. S2CID 1142549.
• 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, CiteSeerX 10.1.1.421.2784, doi:10.1145/100216.100244, ISBN 978-0897913614, S2CID 14031194{{citation}}: CS1 maint: date and year (link)
• Viola, Emanuele (2008). "The Sum of d Small-Bias Generators Fools Polynomials of Degree D". 2008 23rd Annual IEEE Conference on Computational Complexity (PDF). pp. 124–127. CiteSeerX 10.1.1.220.1554. doi:10.1109/CCC.2008.16. ISBN 978-0-7695-3169-4.