Provable Lattice Reduction of $mathbb Z^n$ with Blocksize $n/2$

The Lattice Isomorphism Problem (LIP) is the computational task of recovering, assuming it exists, a orthogonal linear transformation sending one lattice to another. For cryptographic purposes, the case of the trivial lattice $mathbb Z^n$ is of particular interest ($mathbb Z$LIP). Heuristic analysis suggests that the BKZ algorithm with blocksize $beta = n/2 + o(n)$ solves such instances (Ducas, Postlethwaite, Pulles, van Woerden, ASIACRYPT 2022).

In this work, I propose a provable version of this statement, namely, that $mathbb Z$LIP can indeed be solved by making polynomially many calls to a Shortest Vector Problem (SVP) oracle in dimension at most $n/2 + 1$.