Near-Optimal Time-Sparsity Trade-Offs for Solving Noisy Linear Equations

We present a polynomial-time reduction from solving noisy linear equations over $\mathbb{Z}/q\mathbb{Z}$ in dimension $\Theta(k\log n/\mathsf{poly}(\log k,\log q,\log\log n))$ with a uniformly random coefficient matrix to noisy linear equations over $\mathbb{Z}/q\mathbb{Z}$ in dimension $n$ where each row of the coefficient matrix has uniformly random support of size $k$. This allows us to deduce the hardness of sparse problems from their dense counterparts. In particular, we derive hardness results in the following canonical settings. 1) Assuming the $\ell$-dimensional (dense) LWE over a polynomial-size field takes time $2^{\Omega(\ell)}$, $k$-sparse LWE in dimension $n$ takes time $n^{\Omega({k}/{(\log k \cdot (\log k + \log \log n))})}.$ 2) Assuming the $\ell$-dimensional (dense) LPN over $\mathbb{F}_2$ takes time $2^{\Omega(\ell/\log \ell)}$, $k$-sparse LPN in dimension $n$ takes time $n^{\Omega(k/(\log k \cdot (\log k + \log \log n)^2))}~.$ These running time lower bounds are nearly tight as both sparse problems can be solved in time $n^{O(k)},$ given sufficiently many samples. We further give a reduction from $k$-sparse LWE to noisy tensor completion. Concretely, composing the two reductions implies that order-$k$ rank-$2^{k-1}$ noisy tensor completion in $\mathbb{R}^{n^{\otimes k}}$ takes time $n^{\Omega(k/ \log k \cdot (\log k + \log \log n))}$, assuming the exponential hardness of standard worst-case lattice problems.