Pseudorandomness via the Discrete Fourier Transform

Abstract

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests, and combinatorial shapes. For all these classes, our generator is the first to achieve near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize $\mathsf{RL}$---a central question in complexity theory. Our construction combines ideas from a large body of prior work, ranging from the classical construction of [J. Naor and M. Naor, SIAM J. Comput., 22 (1993), pp. 838--856] to the recent gradually increasing independence paradigm of [D. M. Kane, R. Meka, and J. Nelson, Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Comput. Sci. 6845, Springer, Heidelberg, 2011, pp. 628--639; L. E. Celis et al., SIAM J. Comput., 42 (2013), pp. 1030--1050; P. Gopalan et al., Proceedings of the $53$rd Annual IEEE Symposium on Foundations of Computer Science, 2012, pp. 120--129], while also introducing some novel analytic machinery which might find other applications.