Recent results showed that point cloud registration with given correspondences can be made robust to outlier rates of up to 95\% using the truncated least squares (TLS) formulation. However, solving this combinatorial optimization problem to global optimality is challenging. Provably globally optimal approaches using semidefinite programming (SDP) relaxations take hundreds of seconds for 100 points. In this paper, we propose a novel linear time convex relaxation as well as a contractor method to speed up Branch and Bound (BnB). Our solver can register two 3D point clouds with 100 points to provable global optimality in less than half a second when the axis of rotation is provided. Although it currently cannot solve the full 6DoF problem, it is two orders of magnitude faster than the state-of-the-art SDP solver STRIDE when solving the rotation-only TLS problem. In addition to providing a formal proof for global optimality, we present empirical evidence of global optimality using adversarial instances with local minimas close to the global minimum.