Given an efficient algorithm that correctly computes a tiny fraction of the entries of the matrix multiplication of a small fraction of two matrices, can one design an efficient algorithm that computes matrix multiplication exactly for all the matrices? In this paper, we present such ``worst-case exact to average-case approximate'' ... more >>>

In this work we observe a tight connection between three topics: $NC^0$ cryptography, $NC^0$ range avoidance, and static data structure lower bounds. Using this connection, we leverage techniques from the cryptanalysis of $NC^0$ PRGs to prove state-of-the-art results in the latter two subjects. Our main result is a quadratic improvement ... more >>>

We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved in quasipolynomial time). The starting point is a simple NP-hardness result without a gap, and is thus "PCP-free." Our approach ... more >>>