Pseudorandom generators for polynomials

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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[edit]

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[edit]

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[edit]

  • 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