Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
Journal of Computer and System Sciences
Finite fields
MODp-tests, almost independence and small probability spaces
Random Structures & Algorithms
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Pseudorandom generators for low degree polynomials
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Gowers uniformity, influence of variables, and PCPs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Low-degree tests at large distances
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates
SIAM Journal on Computing
Pseudorandom Bits for Polynomials
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Inverse conjecture for the gowers norm is false
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The Sum of d Small-Bias Generators Fools Polynomials of Degree d
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Small Sample Spaces Cannot Fool Low Degree Polynomials
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Concentration of Measure for the Analysis of Randomized Algorithms
Concentration of Measure for the Analysis of Randomized Algorithms
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We present a new approach to constructing pseudorandom generators that fool low-degree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators $G:\mathbb{F}^s\to\mathbb{F}^n$ that fool polynomials over a finite field $\mathbb{F}$: We stress that the results in (1) and (2) are unconditional, i.e., do not rely on any unproven assumption. Moreover, the results in (3) rely on a special case of the conjecture which may be easier to prove. Our generator for degree-$d$ polynomials is the componentwise sum of $d$ generators for degree-1 polynomials (on independent seeds). Prior to our work, generators with logarithmic seed length were only known for degree-1 (i.e., linear) polynomials [J. Naor and M. Naor, SIAM J. Comput., 22 (1993), pp. 838-856]. In fact, over small fields such as $\mathbb{F}_2=\{0,1\}$, our results constitute the first progress on these problems since the long-standing generator by Luby, Veličković, and Wigderson [Deterministic approximate counting of depth-2 circuits, in Proceedings of the 2nd Israeli Symposium on Theoretical Computer Science (ISTCS), 1993, pp. 18-24], whose seed length is much bigger: $s=\exp\left(\Omega\left(\sqrt{\log n}\right)\right)$, even for the case of degree-2 polynomials over $\mathbb{F}_2$.