We study the possibility of basing fine-grained cryptography on the hardness of fundamental problems in linear algebra. Our main technical contribution is a hardness self-amplification method based on random low-rank updates: we show how to turn an algorithm that solves an inverse-polynomial fraction of random linear systems into a solver which succeeds for every right-hand side of almost every matrix. Our reduction increases the runtime only linearly in the inverse of the original success probability. In particular, this yields fine-grained one-way functions from the almost worst-case hardness of linear systems over large enough fields. Using the same approach, we also obtain tighter worst-case to average-case reductions for matrix multiplication and online matrix-vector multiplication, as well as variants of our reductions that recover exact solutions from approximate algorithms, whose expected coordinate agreement is only $1/q+\varepsilon$, with inverse-quadratic dependence on $\varepsilon$ up to logarithmic factors, over every finite field. Finally, we also use it to connect the existence of fine-grained key agreement to the problem of computing the rank of a matrix: over a suitable sequence of growing characteristic-two fields, the almost-everywhere average-case hardness of rank computation (for any of a large class of distributions) implies infinitely-often fine-grained key agreement.

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