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

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A pseudorandom generator   for polynomials of degree   over a finite field   is an efficient procedure that maps a sequence of   field elements to a sequence of   field elements such that any  -variate polynomial over   of degree   is fooled by the output distribution of  . In other words, for every such polynomial  , the statistical distance between the distributions   and   is at most a small  , where   is the uniform distribution over  .

Construction

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Lovett (3rd from left) in 2009

The case   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  , 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   small-bias spaces fools polynomials of degree  . Viola (2008) proves that, in fact, taking the sum of only   small-bias generators is sufficient to fool polynomials of degree  . The analysis of Viola (2008) gives a seed length of  .

References

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  • 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.