On Matrix Multiplication and Polynomial Identity Testing
Robert Andrews

TL;DR
This paper links lower bounds on matrix multiplication border rank to derandomizing polynomial identity testing, providing new hitting set generators with seed lengths dependent on matrix multiplication complexity.
Contribution
It introduces a novel approach connecting matrix multiplication lower bounds to polynomial identity testing derandomization, yielding new explicit hitting set generators.
Findings
Hitting set generator with seed length $O(\sqrt{n} imes ext{border rank}^{-1}(s))$
Generator hits circuits of size $O(n^{1+ ext{small }\delta})$
Border rank lower bounds imply non-trivial hitting sets for small circuits.
Abstract
We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting denote the border rank of matrix multiplication, we construct a hitting set generator with seed length that hits -variate circuits of multiplicative complexity . If the matrix multiplication exponent is not 2, our generator has seed length and hits circuits of size for sufficiently small . Surprisingly, the fact that already yields new, non-trivial hitting set generators for circuits of sublinear multiplicative complexity.
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Taxonomy
TopicsCryptography and Residue Arithmetic · Polynomial and algebraic computation · Numerical Methods and Algorithms
