Optimal Random Self-Reductions for All Linear Problems

The linear problem specified by an n × n matrix M over a finite field is the problem of computing the product of M and a given vector x. We present optimal error-tolerant random self-reductions (also known as worst-case to average-case reductions) for all linear problems: Given a linear-size circuit that computes M x on an ε-fraction of inputs x for a positive constant ε, we construct a randomized linear-size circuit that computes M x for all inputs x with high probability. This resolves the open problem posed by Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA'24), who presented quantum n 1.5 -time random self-reductions for all linear problems. Somewhat surprisingly, we also demonstrate the quantum advantage of their quantum reduction over classical uniform algorithms, by proving that any classical subquadratic-time random self-reduction requires the advice complexity of Ω(log(1/ε) • log n), as long as the field size is at most 1/ε. We complement this advice complexity lower bound by presenting (1) a random self-reduction with the optimal advice complexity of O(log(1/ε)•log n) and (2) a uniform random self-reduction over a large finite field.