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:𝔽ℓ->𝔽n for polynomials of degree d over a finite field 𝔽 is an efficient procedure that maps a sequence of ℓ field elements to a sequence of n field elements such that any n-variate polynomial over 𝔽 of degree d is fooled by the output distribution of G. In other words, for every such polynomial p(x1,…,xn), the statistical distance between the distributions p(Un) and p(G(Uℓ)) is at most a small ϵ, where Uk is the uniform distribution over 𝔽k.

## 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) achieves a seed length of ℓ=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 2d 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 ℓ=d⋅log⁡n+O(2d⋅log⁡(ϵ−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. 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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