One central theme of complexity theory is the rich interplay between hardness (functions that are hard to compute) and pseudorandomness (procedures that convert randomized algorithms into equivalent deterministic algorithms). In one direction, from the classic works of Nisan-Wigderson and Impagliazzo-Wigderson, we know certain hardness hypothesis (circuit lower bounds) implies that all randomized algorithms can be "derandomized" with a polynomial overhead. In another direction, a decade ago, Williams proved that certain circuit lower bounds follow from non-trivial derandomization.
In this thesis we establish many new connections between hardness and pseudorandomness, strengthening and refining the classic works mentioned above.
(New circuit lower bounds from non-trivial derandomization.) Following Williams' algorithmic method, we prove several new circuit lower bounds using various non-trivial derandomization algorithms, including almost everywhere and strongly average-case lower bounds against ACC^0 circuits, and a new construction of rigid matrices.
(Super-fast and non-black-box derandomization from plausible hardness assumptions.) Under plausible hardness hypotheses, we obtain almost optimal worst-case derandomization of both randomized algorithms and constant-round Arthur-Merlin protocols. We also propose a new framework for non-black-box derandomization and demonstrate its usefulness by showing (1) it connects derandomization to a new type of hardness assumption against uniform algorithms and (2) (from plausible assumptions) it gives derandomization of both randomized algorithms and constant-round Arthur-Merlin protocols with almost no overhead such that no polynomial-time adversary can find a mistake.