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 ${\displaystyle G:\mathbb {F} ^{\ell }\rightarrow \mathbb {F} ^{n}}$ 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 ${\displaystyle \epsilon }$, where ${\displaystyle U_{k}}$ is the uniform distribution over ${\displaystyle \mathbb {F} ^{k}}$.

Construction

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 (1990) achieves a seed length of ${\displaystyle \ell =\log n+O(\log(\epsilon ^{-1}))}$, 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 ${\displaystyle 2^{d}}$ 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 ${\displaystyle \ell =d\cdot \log n+O(2^{d}\cdot \log(\epsilon ^{-1}))}$.

References

• Bogdanov, Andrej; Viola, Emanuele (2007), "Pseudorandom Bits for Polynomials", Proceedings of the 48th Annual Symposium on Foundations of Computer Science (FOCS 2007): 41–51, doi:10.1109/FOCS.2007.56
• Lovett, Shachar (2009), "Unconditional Pseudorandom Generators for Low Degree Polynomials" (PDF), 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 22nd annual ACM symposium on Theory of computing, STOC 1990: 213–223, doi:10.1145/100216.100244, ISBN 0897913612
• Viola, Emanuele (2008), "The sum of d small-bias generators fools polynomials of degree d" (PDF), Proceedings of the 23rd Annual Conference on Computational Complexity (CCC 2008): 124–127, doi:10.1109/CCC.2008.16