Just How Hard Are Rotations of $\mathbb {Z}^n$? Algorithms and Cryptography with the Simplest Lattice.

We study the computational problem of finding a shortest non-zero vector in a rotation of $$\mathbb {Z}^n$$ , which we call $$\mathbb {Z}$$ SVP. It has been a long-standing open problem to determine if a polynomial-time algorithm for $$\mathbb {Z}$$ SVP exists, and there is by now a beautiful line of work showing how to solve it efficiently in certain very special cases. However, despite all of this work, the fastest known algorithm that is proven to solve $$\mathbb {Z}$$ SVP is still simply the fastest known algorithm for solving SVP (i.e., the problem of finding shortest non-zero vectors in arbitrary lattices), which runs in $$2^{n + o(n)}$$ time. We therefore set aside the (perhaps impossible) goal of finding an efficient algorithm for $$\mathbb {Z}$$ SVP and instead ask what else we can say about the problem. E.g., can we find any non-trivial speedup over the best known SVP algorithm? And, if $$\mathbb {Z}$$ SVP actually is hard, then what consequences would follow? Our results are as follows.